This application calculates the average transaction price and the final price (marginal price after the transaction) for a constant-product AMM defined by x × y = k. The following convention is used: buy means that the user buys asset x and pays with asset y, while sell means that the user sells asset x and receives asset y.
In this article, Saral BINDAL (Indian Institute of Technology Kharagpur, Metallurgical and Materials Engineering, 2024-2028 & Research assistant at ESSEC Business School) examines the statistical properties of historical VIX data and applies statistical methods to model its dynamics.
The Chicago Board Options Exchange (CBOE) Volatility Index, or VIX, is a real-time market index designed to measure the market’s expectation of 30-day forward-looking annualized volatility in the US equity market. It is option-based, calculated using the market prices of S&P 500 index options to gauge expected volatility. The VIX construction methodology can be found in the article CBOE Volatility Index
In this article, we analyze the VIX index using historical market data to study its statistical properties, and discuss the methods used to model its dynamics in quantitative finance.
The VIX Index Historical data is publicly available from the Chicago Board Options Exchange (CBOE) and is updated on a daily basis. The analysis in this article is based on daily closing values of the VIX obtained from this data.
Figure 1 below illustrates the time series plot of VIX daily closing values from January 1990 to August 2026. From the figure, we can observe that during periods of high uncertainty, the VIX tends to rise significantly, as seen during the Global Financial Crisis (2007–2009) and the COVID-19 pandemic (2019–2020). We can also see the VIX rising as of now amid the ongoing Middle East crisis, although not to the same extent as observed during previous periods of heightened uncertainty.
Figure 1. VIX Index Historical Data (1990-2026)
Source: computation by the author.
Using historical VIX data from 1990 to 2026, Figure 2 illustrates the empirical distribution of daily VIX closing values. The histogram, overlaid with a kernel density estimate (KDE), provides a visual representation of the distribution’s underlying probability density. The corresponding first four moments of the distribution are reported below:
Table 1. VIX Distribution Statistics
Source: computation by the author.
From the above table, we can observe that the data is right-skewed, which indicates that volatility tends to remain relatively low under normal market conditions but can rise sharply during periods of financial stress. The long right tail therefore reflects the occurrence of volatility shocks and tail-risk events, such as the Global Financial Crisis (2007–2009), the COVID-19 market shock (2020), and other episodes of significant financial or geopolitical uncertainty. Moreover, a high excess kurtosis indicates fatter tails compared to a normal distribution, implying a higher probability of observing extreme VIX levels. From the histogram, we observe that the VIX is concentrated around a level of 15 for a large proportion of the sample. This suggests that, under normal market conditions, the VIX tends to fluctuate around this level, which can be interpreted as its long-term level of the market uncertainty
Figure 2. Historical Distribution and Kernel Density Estimation of the Daily VIX levels
Source: computation by the author.
The VIX exhibits several well-documented time-series properties that distinguish it from traditional financial asset prices. Unlike equity prices, whose levels are generally characterized by non-stationary dynamics, the VIX exhibits pronounced mean reversion and persistence, together with sharp spikes during episodes of market stress. The mean-reverting behaviour of the VIX is particularly noteworthy because it resembles a fundamental feature of interest-rate dynamics, which has long been incorporated into models such as the Vasicek (1977) and Cox, Ingersoll, and Ross (1985). The mean-reverting specifications have similarly been used to capture the tendency of volatility to return toward a long-run level.
However, the two processes differ substantially in their temporal dynamics. Interest-rate persistence typically reflects gradual adjustments in response to macroeconomic conditions and monetary policy, with mean reversion occurring over relatively longer horizons. In contrast, the VIX responds rapidly to changes in market expectations: episodes of financial stress can trigger abrupt upward spikes, followed by relatively rapid mean reversion toward lower levels. Thus, while both exhibit persistent and mean-reverting dynamics, the VIX is distinguished by its faster adjustment, pronounced asymmetry, and sharp responses to market stress.
VIX dynamics exhibit several distinctive characteristics, particularly mean reversion and persistence, which require different modelling approaches. This section examines how various econometric models capture these features, ranging from mean-reverting and Log-VIX models to HAR, ARCH/GARCH, and stochastic volatility models.
Early approaches to modelling volatility indices treated volatility as a mean-reverting stochastic process. A commonly used specification is the Cox-Ingersoll-Ross (CIR) process, originally developed for interest-rate modelling by Cox, Ingersoll and Ross (1985) and subsequently applied to volatility derivatives by Grünbichler and Longstaff (1996). Grünbichler and Longstaff modelled the volatility index using a mean-reverting square-root process and derived pricing formulas for volatility futures and options.
The dynamics are given by
The term κ(θ − Vt)dt represents the mean-reverting component, which pulls the process towards its long-run level θ, while σ√VtdWt represents the diffusion component, capturing random fluctuations in the volatility index.
The CIR framework was subsequently examined empirically in the context of VIX futures by Zhang and Zhu (2006). They estimated a stochastic variance model using historical VIX data and used it to derive and evaluate VIX futures prices.
An alternative mean-reverting modelling approach is the Ornstein-Uhlenbeck (OU) process (Uhlenbeck and Ornstein, 1930), which assumes a constant diffusion coefficient:
Unlike the CIR process, the OU process is Gaussian and can theoretically take negative values. This makes the OU process unsuitable for modelling the VIX level directly, since the VIX is strictly positive. A common way to retain the mean-reverting OU structure while ensuring a positive VIX is therefore to model log(VIX) rather than the VIX level itself.
A natural way to address the possibility of negative values is to model the logarithm of the VIX. Let
A mean-reverting logarithmic process can then be written as
or equivalently,
Since
the resulting modelled VIX is strictly positive. The logarithmic transformation therefore preserves the mean-reverting structure while avoiding the negative-value problem associated with modelling the VIX level using a Gaussian process.
Mean-reverting in the log models were studied by Detemple and Osakwe (2000) in the context of volatility option valuation. Their model provides an early theoretical foundation for modelling volatility through a log-normal mean-reverting process. The approach was subsequently applied directly to the VIX by Bao (2013), who developed a mean-reverting logarithmic model for the spot VIX and extended it to incorporate jumps and stochastic volatility.
Although continuous-time mean-reverting models capture the tendency of volatility to move towards a long-run level, empirical evidence indicates that the VIX also exhibits substantial persistence across different time horizons. Fernandes, Medeiros and Scharth (2014) conducted a detailed analysis of the time-series properties of the VIX and found evidence of long-range dependence. They therefore employed Heterogeneous Autoregressive (HAR) models to model and forecast the VIX.
The HAR framework captures persistence by allowing past VIX observations over different horizons to affect the current value. A simplified HAR model can be expressed as
where
and
The three components represent information from daily, weekly, and monthly horizons, respectively. This allows the HAR-VIX model to capture the persistence of the VIX across different time horizons while remaining relatively simple and parsimonious. The HAR framework was originally introduced by Corsi (2009) to model the heterogeneous dynamics of realized volatility. Fernandes, Medeiros and Scharth (2014) adapted this framework to the VIX and showed that its multi-horizon structure provides a useful representation of the strong persistence and long-range dependence in the VIX.
The volatility of the VIX itself can vary over time. Rather than assuming that the diffusion coefficient is constant, stochastic-volatility models allow the volatility governing VIX fluctuations to evolve as a separate stochastic process.
Kaeck and Alexander (2013) investigate continuous-time models of VIX dynamics that explicitly incorporate stochastic volatility of volatility. Their analysis considers several one- and two-factor continuous-time models, including affine and non-affine specifications and models with jumps, using VIX data over an extended period.
Conceptually, the model can be represented as
where
The key distinction is therefore that the variance of VIX fluctuations is no longer constant. It becomes a state variable that evolves over time.
This additional source of randomness allows the model to capture changes in the intensity of VIX fluctuations and provides a more flexible representation of the sharp and persistent movements observed during periods of financial stress. Kaeck and Alexander’s analysis specifically examines whether stochastic volatility of volatility improves the ability of continuous-time models to describe VIX dynamics.
The VIX also exhibits volatility clustering, where periods of high volatility tend to be followed by high volatility, while periods of low volatility tend to be followed by low volatility. This suggests that the variance of log change in VIX is not constant over time. ARCH and GARCH models capture this feature by allowing the conditional variance of log change in VIX to evolve over time.
Let the log change in the VIX be defined as
An ARCH or GARCH model can then be used to model the conditional variance of these changes. The standard GARCH(1,1) specification is
where
The αεt−12 term captures the immediate impact of a new shock, while the βσt−12 term captures the persistence of previously elevated variance. A high value of β therefore indicates that periods of high variation in the VIX tend to persist.
The ARCH model was introduced by Engle (1982), while Bollerslev (1986) extended it to the more general GARCH framework. For the VIX, these models are useful for studying time-varying conditional variance and volatility clustering in VIX changes, rather than the mean-reverting behaviour of the VIX level itself.
We use the complete historical daily VIX closing values (1990 – 2026) to estimate the parameters of the CIR model described above. Since the CIR model is specified in continuous time while the data are observed at daily intervals, the process is first discretized using the Euler-Maruyama approximation:
where Δt; = 1/252 for daily observations and εt+1 ∼ N(0,1). This discretization implies that, conditional on the previous day’s VIX value, the expected value and variance of the next observation are:
Assuming the discretized process is conditionally normally distributed, these expressions allow us to construct a likelihood for the observed VIX data. The log-likelihood is:
The CIR parameters are then estimated by choosing the values that maximize this log-likelihood:
This gives an estimated mean-reversion speed of κ = 4.8618, a long-run VIX level of θ = 19.4237, and a diffusion parameter of σ = 5.1952.
These estimated parameters are then used to simulate a possible future path of the VIX. The simulation starts from the observed VIX value of 14.63 on 13th August 2026 and projects the VIX forward over 252 trading days using the CIR dynamics. The mean-reverting component pulls the process towards the estimated long-run level, while the diffusion component introduces random fluctuations around this tendency.
Figure 3 compares the historical VIX with one simulated path generated by the estimated CIR model. The historical series shows the large fluctuations and sharp spikes observed in the VIX over the sample period, while the simulated path represents one possible future realization starting from the current VIX level of 14.63. The dotted line at 19.42 represents the estimated long-run level of the VIX. The simulated path fluctuates around this level and exhibits a tendency to move towards it, illustrating the mean-reverting behavior implied by the CIR model.
You can download the Excel file with complete historical data used for the above calculations below.
Figure 3. Historical Path and Multiple CIR Simulated Paths for the VIX
Source: computation by the author.
You can download the Python code below to reproduce the CIR parameter estimation and simulated VIX paths presented above.
Alternatively, you can download the R code below with the same functionality as in the Python file.
For anyone interested in finance or a career in trading, understanding the statistical properties of VIX and how it is modelled is very important. As one of the most widely used measures of market uncertainty and expected volatility, it serves as an important tool for market analysis, risk assessment and numerous volatility-based trading strategies.
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Corsi F. (2009) A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics, 7(2), 174-196.
Cox J. C., J. E. Ingersoll and S. A. Ross (1985) A theory of the term structure of interest rates. Econometrica, 53(2), 385-407.
Detemple J. and C. Osakwe (2000) The valuation of volatility options. European Finance Review, 4(1), 21-50.
Engle R. F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4), 987-1007.
Fernandes M., M. C. Medeiros and M. Scharth (2014) Modelling and predicting the CBOE market volatility index. Journal of Banking & Finance, 40, 1-10.
Grünbichler A. and F. A. Longstaff (1996) Valuing futures and options on volatility. Journal of Banking & Finance, 20(6), 985-1001.
Jiang G. J. and Y. S. Tian (2005) The model-free implied volatility and its information content. The Review of Financial Studies, 18(4), 1305-1342.
Kaeck A. and C. Alexander (2013) Continuous-time VIX dynamics: On the role of stochastic volatility of volatility. International Review of Financial Analysis, 28, 46-56.
Uhlenbeck G. E. and L. S. Ornstein (1930) On the theory of the Brownian motion. Physical Review, 36(5), 823-841.
Whaley R. E. (2009) Understanding the VIX. Journal of Portfolio Management, 35(3), 98-105.
Zhang J. E. and Y. Zhu Y. (2006) VIX futures. Journal of Futures Markets, 26(6), 521-531.
CBOE Global Markets (2026) VIX Historical Data.
The article was written in September 2026 by Saral BINDAL (Indian Institute of Technology Kharagpur, Metallurgical and Materials Engineering, 2024-2028 & Research assistant at ESSEC Business School). His interests include tracking geopolitical developments and analyzing their direct impact on macroeconomic factors such as inflation, trade balances, and currency volatility, with a focus on using data to quantify these global economic ripple effects.
Discover all posts written by Saral BINDAL.
In this article, Bochen LIU (Queen’s Smith School of Business, BCom 2023–2027; ESSEC BBA Exchange Program, Fall 2025) explains how Volkswagen’s localization strategy has become a key marketing approach for maintaining competitiveness in the Chinese automobile market. By adapting products, branding, digital communication, and consumer engagement to local preferences, Volkswagen demonstrates how international companies can strengthen their market position through localization rather than standardization.
Volkswagen has been one of the most successful foreign automobile brands in China since entering the market in the 1980s. Through joint ventures with SAIC and FAW, Volkswagen developed a strong business presence by combining its global automotive expertise with local market knowledge. Rather than simply exporting global models, Volkswagen adopted a localization strategy by adapting its products, production processes, and marketing activities to better meet the needs and preferences of Chinese consumers. This approach allowed Volkswagen to establish a strong market position and become one of the leading international automobile brands in China.
The success of Volkswagen in China was highly dependent on its ability to make products that could suit the taste of the Chinese market. For instance, one of the first successful products made by Volkswagen in China was the Santana, which had high reliability and suitability for use by the Chinese consumers. Later, another model known as Lavida was designed specifically for the Chinese market. This model was made taking into consideration the needs of the Chinese consumers for spaciousness and comfort. This shows that the company managed to localize its global brand in the local market by making products that could suit the needs of the Chinese customers.
The Chinese automobile market has been transformed in recent years. In the past, VW could take advantage of its global reputation, engineering skills, and reliability to gain success in the Chinese market. However, today, the customers give more attention to electric cars, intelligent systems, and digital products. The development of Chinese companies producing electric cars, such as BYD and NIO, is posing new challenges for the company. In conclusion, while Volkswagen’s localization strategy has been successful in helping the company enter the Chinese market, it needs to adapt constantly.
Volkswagen’s localization strategy is important because Chinese consumers increasingly evaluate vehicles based on factors beyond traditional brand reputation. While Volkswagen previously benefited from its global reputation for reliability and engineering quality, the Chinese automotive market has shifted toward digital technology, intelligent driving features, and user experience. Therefore, adapting products and marketing communication to local preferences has become essential for maintaining customer relevance.
For example, Volkswagen has invested in localized technology and product development to better respond to Chinese consumers’ expectations for intelligent vehicles. Features such as intelligent cockpit systems, localized infotainment functions, and digital services allow Volkswagen to compete with domestic brands that have strong technological advantages. Without these adaptations, Volkswagen risks losing younger Chinese consumers who increasingly prioritize innovation and connectivity over traditional brand value.
Market localization requires market analysis procedures. As part of its business strategy, the company constantly gathers consumer data, analyzes competitors, and tracks shifts in consumer demands. Tools like the STP model (segmentation, targeting, and positioning), SWOT analysis, and consumer journey mapping make this easier.
As a result, Volkswagen uses market analysis tools to identify potential customer groups and create more relevant marketing campaigns.
The effectiveness of localization can be evaluated through measurable business indicators, including market share, customer acquisition, sales growth, and brand preference. These metrics allow managers to determine whether localized marketing investments generate sustainable returns.
Improved localization often increases customer satisfaction and strengthens long-term brand loyalty, ultimately reducing marketing costs while improving overall business performance.
The Chinese automotive market has experienced significant uncertainty due to the rapid development of electric vehicles, government support for new energy vehicles, and changing consumer expectations. Unlike traditional market risks that can be estimated through historical data, these changes have created new competitive conditions that Volkswagen could not fully predict based on its previous success.
These changes require Volkswagen to continuously adapt its marketing strategy rather than relying only on its previous competitive advantages. For decades, Volkswagen benefited from its strong reputation for quality and reliability in China. However, the rise of domestic EV manufacturers has shifted competition toward technology, connectivity, and user experience. Therefore, Volkswagen’s future competitiveness depends on its ability to localize not only its products but also its digital and innovation strategies.
Volkswagen builds long-term resilience by making localization an ongoing strategic capability rather than a temporary marketing campaign. Continuous investment in local research and development, partnerships with Chinese technology companies, and localized communication channels enable the company to respond more quickly to changing market conditions.
This adaptive approach allows Volkswagen to strengthen customer relationships while maintaining competitiveness in an increasingly dynamic automotive industry.
Volkswagen’s experience demonstrates that successful international marketing requires more than global brand recognition. Companies must understand local consumers, adapt their value proposition, and continuously refine their marketing strategy to remain competitive.
For business and marketing students, this case illustrates how localization, consumer insight, and strategic positioning can transform global brands into locally relevant market leaders, providing valuable lessons for international marketing and brand management.
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American Marketing Association (AMA)
The article was written in August 2026 by Bochen LIU (Queen’s Smith School of Business, BCom 2023–2027; ESSEC BBA Exchange Program, Fall 2025).
▶ Discover all posts by Bochen LIU
In this article, Feitong GUO (The Chinese University of Hong Kong, Shenzhen, Accounting and Data Analytics, 2023–2027; ESSEC Business School, BBA Exchange Program, Spring 2026) explains how SimTrade changed her understanding of patience, risk and decision-making under uncertainty.
Confucius warned that “欲速则不达，见小利则大事不成”: if we pursue speed or become distracted by small gains, we may fail to achieve the larger objective. Before taking SimTrade, I was often impatient in simulations. A short-term profit could make me close a position too early, while an unexpected loss could make me abandon my reasoning and rush into the next round. The course taught me that good trading is not constant action. It is the discipline to observe, form a view, define acceptable risk and wait until the evidence justifies a decision.
Across market-making exercises in an order-driven market with a limit book, I experienced both premature exits and positions that moved sharply against me. I also learned that quoting a wider spread does not automatically create profit: other participants may offer better prices, execution is uncertain and inventory can become risky when the market changes direction. I therefore stopped treating every price movement as a command to act. Instead, I began to ask what information had changed, whether the movement was ordinary noise or a regime shift, and whether my original assumptions still held.
This distinction also clarified the difference between patience and procrastination. Procrastination means neither acting nor thinking. Patience is active: observing the market, updating expectations, defining price and risk limits, and being ready to act. It is not stubbornly holding a losing position. When evidence invalidates the original thesis, discipline means accepting the loss rather than defending a sunk cost.
The following FRED charts connect my simulation experience with real-market evidence. They do not provide a trading rule or predict the next crash. Instead, they show why patience must be combined with preparation for rare but consequential changes.
Figure 1. CBOE Volatility Index (VIX) and the S&P 500.
Source: Federal Reserve Bank of St. Louis FRED, using CBOE and S&P Dow Jones Indices data.
Figure 1 shows that sharp increases in expected volatility can coincide with steep equity-market declines, most visibly during the 2020 shock. The relationship is not a mechanical buy-or-sell signal, but it illustrates how quickly the market environment can change. My lesson is that patience cannot mean ignoring downside risk: a trader should establish risk limits before uncertainty rises, because calm observation becomes harder once prices move abruptly.
Figure 2. Long-run history of the CBOE Volatility Index, with a high-volatility threshold.
Source: Federal Reserve Bank of St. Louis FRED, using CBOE data.
Figure 2 places volatility spikes in a longer historical context. High-volatility episodes are intermittent, while calmer conditions occupy much of the sample. This helps explain why overreacting to every small movement can be costly, but also why a routine trading rhythm must include contingency plans. We cannot know the exact timing of the next extreme episode; we can only avoid confusing the absence of a shock with the absence of risk.
Figure 3. Absolute daily changes in the S&P 500 and the VIX.
Source: Federal Reserve Bank of St. Louis FRED, using CBOE and S&P Dow Jones Indices data.
Figure 3 compares realized daily market movements with the VIX, a forward-looking measure based on options prices. The two tend to rise together in stressed periods, but they do not match exactly because they describe different horizons and information sets. This distinction mirrors SimTrade: observing what has already happened is not the same as forecasting what may happen next. Better judgment requires both evidence from realized outcomes and an explicit view of future uncertainty.
Selling a winning position too quickly while hesitating over a losing one resembles the disposition effect documented by Shefrin and Statman. Short-term gains feel concrete and easy to secure, whereas realizing a loss feels like admitting that the original judgment was wrong. Recognizing this bias helped me separate emotional comfort from decision quality.
Time, money and effort inevitably influence emotions, even though sunk costs should not determine the next decision. Perseverance is rational only when the expected future benefit still justifies the remaining risk. This principle is relevant beyond trading: in analytics projects, I should not defend a model merely because I spent time building it. I should keep, revise or abandon it according to evidence.
SimTrade strengthened three qualities that I want to bring to business and data analytics: patient reasoning, reflection and a long-term orientation. Data rarely explains itself. Analysts must distinguish signal from noise, test assumptions, investigate unexpected results and wait for sufficient evidence without becoming passive. The course showed me that a good decision is not defined only by whether one trade makes money. It is defined by whether the process was consistent, explainable and responsive to new information.
For students interested in finance, consulting or analytics, simulations offer a safe environment in which to discover how emotion enters supposedly rational decisions. The most transferable lesson is not a particular order or strategy. It is the habit of slowing down before acting: specify the objective, interpret the evidence, control the downside and review the result. “Haste does not bring success” is therefore not an argument for inactivity; it is a framework for disciplined action.
▶ Bochen LIU Know yourself and know your opponent, and you will never be defeated – Sun Tzu
▶ What I learned during my time in the stock market
▶ Posts about behavioral finance
Arkes, H. R., & Blumer, C. (1985) The Psychology of Sunk Cost, Organizational Behavior and Human Decision Processes, 35(1), 124–140.
Kahneman, D., & Tversky, A. (1979) Prospect Theory: An Analysis of Decision under Risk, Econometrica, 47(2), 263–291.
Shefrin, H., & Statman, M. (1985) The Disposition to Sell Winners Too Early and Ride Losers Too Long: Theory and Evidence, The Journal of Finance, 40(3), 777–790.
Confucius, The Analects, Book XIII, “Zi Lu”.
Federal Reserve Bank of St. Louis FRED: Measuring uncertainty and volatility with FRED data
The article was written in August 2026 by Feitong GUO (The Chinese University of Hong Kong, Shenzhen, Accounting and Data Analytics, 2023–2027; ESSEC Business School, BBA Exchange Program, Spring 2026).
▶ Discover all articles by Feitong GUO .
In this article, Feitong GUO (The Chinese University of Hong Kong, Shenzhen, Accounting and Data Analytics, 2023-2027; ESSEC Business School, BBA Exchange Program, Spring 2026) reflects on how her experience as an Audit Assistant at Ernst & Young (EY) and her work as a Project Consultant at A Better Community (ABC) taught her two complementary ways of supporting reliable decisions: verifying evidence against established standards and designing an evaluation framework when the standards themselves must first be defined.
EY and ABC operate in very different environments, but both transform complex information into structured evidence for decision-makers. EY does so through established assurance and review procedures, while ABC provides professional services that help social organizations and funders make more informed decisions.
Ernst & Young (EY) is a global professional-services organization providing assurance, consulting, strategy and transactions, and tax services. I worked as an Audit Assistant in July and August 2025. To respect confidentiality, I do not identify the financial institutions or funds involved.
Logo of EY.
Source: the company.
A Better Community (ABC), founded in 2008, mobilizes professional volunteers to provide consulting, research, digital, impact-investment and philanthropic-advisory services to social organizations. Since August 2025, I have served as a Volunteer Management Team Member and Project Consultant. I have participated in three volunteer-recruitment seasons and two project seasons, supporting a nonprofit educational research organization and a foundation investment project.
Logo of A Better Community.
Source: the organization.
At EY, my tasks included assisting with audit working papers, reviewing credit files and checking financial information. In credit-file reviews, I examined borrower information such as loan amounts, interest rates, collateral or guarantees, repayment capacity and the bank’s internal loan-risk classification. Completeness meant more than confirming that every required field had been filled in. I also checked whether the information was supported by relevant documents and remained consistent across client reports, audit materials and credible public sources. When the sources did not agree, I recorded the discrepancy and sought clarification rather than selecting the figure that appeared most convenient.
I also supported valuation-related checks for selected investment projects held by funds. The purpose was to verify parts of the existing estimates, not to perform the valuation independently. I organized project information and valuation schedules, traced operational inputs such as annual production volumes, prices and forecasts to annual reports and other supporting materials, and checked formulas and cross-references between different tables. When figures did not match, I followed their sources and calculation paths to locate the problem. If the discrepancy could not be resolved, I documented it and raised a specific question with the client team or my mentor.
These tasks could be repetitive, but they were not trivial. A small inconsistency in a date, amount, formula or supporting document could affect the reliability of the output. Audit taught me to preserve an evidence trail, distinguish a fact from an inference and know when escalation was necessary. I contributed to the process rather than owning the final audit conclusion, yet that role showed me how disciplined verification supports trust.
In ABC’s foundation investment project, the question moved upstream. The client wanted a reusable methodology to screen potential partner organizations and inform future funding decisions. Our team was not merely checking whether information met an existing standard; we first had to decide what a reasonable and usable standard should contain. I was responsible for researching nonprofit-screening methodologies and contributing to the indicator framework and its validation.
Our emerging framework has three sequential layers: compliance, communication and value. Compliance is an entry gate. It may include checks of registration status, annual reports, financial disclosures, penalties and abnormal operating information through credible public sources. If an organization does not pass the required compliance checks, it does not proceed to the communication assessment. This prevents strong publicity from compensating for a fundamental compliance concern.
For organizations that pass the first gate, the communication layer distinguishes current capacity from future potential. Current capacity may be reflected in the volume, frequency, clarity and audience response of existing content. Potential depends less on polished videos or follower counts and more on whether the organization operates a credible project, can provide verifiable stories and evidence, is willing to cooperate in content development, and has a mission that can be meaningfully communicated to the foundation’s target audience. The value layer then asks whether the project addresses a genuine need, has a reasonable intervention logic and is aligned with the funder’s objectives.
At first sight, my work at EY and ABC followed a similar process. In both cases, information had to be collected, entered into a structured table, assessed and transformed into a decision-oriented output. In EY credit-file reviews, the output contributed to an internal loan-risk classification. At ABC, the framework is intended to support the screening of nonprofit organizations and future funding decisions.
The fundamental difference lies in who defines the methodology. At EY, the required fields, supporting documents, assessment standards and classification rules had already been established. My responsibility was to research and verify the information, complete the required fields, check consistency and identify issues that needed clarification. At ABC, our team is helping to design the table itself: its questions, evidence requirements, gates, scoring logic and final decision rules.
The two settings also use different forms of internal consistency. In valuation checks at EY, I examined numerical relationships: for example, how operational assumptions entered one schedule and affected another calculation. In the ABC framework, the relationship is procedural. An organization must pass the compliance gate before its communication potential and project value are assessed. In both cases, a result is credible only when the path from evidence to conclusion can be traced.
Figure 1. Evaluation criteria developed by the Organisation for Economic Co-operation and Development’s Development Assistance Committee (OECD-DAC).
Source: KfW Development Bank, based on the OECD-DAC evaluation criteria.
The OECD-DAC framework evaluates development interventions through six complementary criteria: relevance, coherence, effectiveness, efficiency, impact and sustainability. It appears here because our team first examined established evaluation methodologies before designing a framework adapted to nonprofit screening and funding decisions. The framework shows that evaluation requires several distinct questions and that each criterion should clarify what evidence an evaluator needs.
However, we did not copy the OECD-DAC model directly. It is primarily designed to evaluate development interventions, whereas our client needs to screen organizations before selecting partners and allocating future funding. We therefore used it as a methodological reference and adapted its multi-dimensional logic to our sequential compliance, communication and value framework. It informed our thinking, but it is not our final framework.
Consider a nonprofit organization with a legally registered status, an active social-media account and an education project that appears attractive to potential donors. These characteristics would not immediately produce a positive recommendation. The compliance layer would first verify its registration, annual reports, financial disclosures and relevant risk information. If the organization passed this gate, the communication layer would distinguish current performance, such as publication frequency and audience engagement, from future potential, including the availability of credible project stories, supporting evidence and willingness to cooperate. The value layer would then examine whether the project addresses a genuine need and is aligned with the foundation’s objectives.
This staged process is intended to avoid two common errors: treating polished communication as proof of project value, and treating a lack of current communication resources as proof that an organization has no communication potential. The detailed indicators are still being developed and will need practical testing before the methodology is delivered.
A framework may look logical in a presentation and still fail in practice. Another project group will apply the draft indicators to real cases. Their work may reveal unavailable data, overlapping criteria, vague scoring anchors or outputs that do not match the client’s decision needs. We will then revise the definitions, evidence requirements, gates and scoring rules. This process resembles audit in an important way: a conclusion must be traceable to evidence. It also resembles analytics because the model must be tested against observed results rather than defended simply because effort has already been invested in it.
Both experiences required attention to detail, information organization and professional judgment. Audit placed greater emphasis on accuracy, documentation, reconciliation and escalation. Nonprofit consulting required methodology research, framework design, stakeholder alignment and the ability to translate broad ideas into observable indicators. Across both settings, the most important skill was asking what decision the information was meant to support before deciding how to collect or process it.
EY taught me how to use a methodology responsibly, while ABC taught me how difficult it is to design one responsibly. Applying an existing framework requires accuracy, traceability and consistency. Designing a new framework requires the same discipline, but also demands decisions about definitions, evidence, thresholds and trade-offs. Together, the two experiences changed my understanding of evaluation: a reliable result depends not only on the quality of the information entered into a table, but also on the quality of the methodology behind the table.
Due diligence reduces information asymmetry by checking the reliability, completeness and relevance of evidence before a decision. Credit-file review, valuation checks and nonprofit compliance screening all apply this logic, even though their procedures and stakeholders differ. In each setting, important claims should be traceable to identifiable sources.
Funding decisions rarely depend on a single metric. Multi-criteria decision analysis makes trade-offs explicit by defining dimensions, evidence and scoring rules. Some conditions may function as gates rather than compensable scores. In our emerging framework, compliance serves this role: communication strength cannot compensate for failure to meet a mandatory compliance requirement.
An evaluation framework is a model of reality. Validation asks whether different reviewers can apply it consistently, whether the necessary evidence is available and whether its outputs are useful for the intended decision. Testing and iteration are therefore part of responsible design, not signs that the first draft failed.
For students interested in audit, consulting, impact investing or business analytics, these experiences show a progression from checking information to structuring judgment. Entry-level work may begin with documents, confirmations and meeting records, but those tasks build the discipline needed for higher-level analysis. The challenge is to connect every task to its decision purpose: what is being verified, what uncertainty remains and what evidence would justify the next step?
▶ Iris ORHAND My apprenticeship experience as a Junior Financial Auditor at EY
▶ Iris ORHAND Risk-based Audit: From Risks to Assertions to Audit Procedures
▶ Anant JAIN Impact Investing
Gamvros, I., Nidel, M., & Raghavan, N. R. S. (2006) Investment Analysis and Budget Allocation at Catholic Relief Services, Interfaces, 36(5), 400-406.
Salimi, N., & ten Have, W. (2024) Aiding Philanthropic Venture Capitalists in Selecting Nonprofit Organisations, Journal of Philanthropy and Marketing, 29(1), e1834.
Bayarri, M. J., Berger, J. O., Higdon, D., Kennedy, M. C., Kottas, A., Paulo, R., Sacks, J., Cafeo, J. A., Cavendish, J., Lin, C. H., & Tu, J. (2002) A Framework for Validation of Computer Models, NISS Technical Report 128.
A Better Community: Consulting services
KfW Development Bank: Evaluation criteria and rating scale
The article was written in August 2026 by Feitong GUO (The Chinese University of Hong Kong, Shenzhen, Accounting and Data Analytics, 2023-2027; ESSEC Business School, BBA Exchange Program, Spring 2026).
▶ Discover all articles by Feitong GUO .
In this article, Bochen LIU (Queen’s Smith School of Business, BCom 2023–2027; ESSEC BBA Exchange Program, Fall 2025) analyzes how the famous Chinese strategic quote “知己知彼，百战不殆” (know yourself and know your opponent, and you will never be defeated), derived from Sun Tzu’s The Art of War, functions as a foundational principle in modern Business strategy, particularly in positioning, segmentation, and competitive analysis.
The Chinese strategic philosophy “知己知彼，百战不殆” (know yourself and know your opponent, and you will never be defeated), originating from Sun Tzu’s The Art of War, continues to provide valuable insights for modern business strategy and competitive analysis. Although this principle was originally developed in the context of military strategy, its underlying logic remains highly relevant in today’s corporate environment. Companies must understand their own capabilities while continuously analyzing competitors, customers, and external market conditions. Knowledge of both internal resources and external competition enables organizations to develop stronger positioning strategies and maintain competitive advantages.
In the current business environment, companies face uncertainty caused by changing consumer behaviour, technological development, and increasing competition. Having a high-quality product or service alone does not guarantee success. Organizations must understand their own strengths, recognize their weaknesses, and evaluate how competitors influence customer expectations and market conditions. Therefore, the principle of “knowing yourself and knowing your opponent” can be transformed into a practical business framework through Strengths, Weaknesses, Opportunities, and Threats (SWOT) analysis, which allows companies to evaluate internal capabilities and external challenges.
“Know yourself” refers to the process through which companies understand their internal resources, capabilities, and limitations. From a business perspective, this involves evaluating internal factors such as brand reputation, product quality, pricing strategies, operational efficiency, financial resources, and customer relationships.
Understanding internal strengths allows companies to identify areas where they possess competitive advantages. For example, companies with strong brand recognition may not compete with competitors through lower prices. Instead, they can create value through customer loyalty, premium positioning, and differentiated customer experiences. Similarly, organizations with unique technologies or specialized expertise can use these capabilities to distinguish themselves from competitors.
However, knowing oneself also requires recognizing internal weaknesses. Companies that fail to understand their limitations may develop unrealistic strategies that cannot be effectively implemented. For example, a business may provide high-quality products but lack sufficient distribution channels, marketing resources, or operational capacity to support expansion.
Therefore, “know yourself” does not simply mean identifying positive characteristics. It requires an objective evaluation of both strengths and weaknesses so that companies can develop realistic and effective strategies.
“Know your opponent” refers to understanding competitors, customers, and external market conditions. Businesses operate in environments where competitors continuously adjust their strategies, introduce new products, and influence consumer expectations. Without sufficient knowledge of the external environment, companies may lose their competitive position.
Competitor analysis allows businesses to understand how other organizations attract customers, differentiate their products, and respond to market changes. For example, companies operating in the education industry need to analyze competitors’ pricing strategies, teaching methods, customer acquisition approaches, and brand positioning before developing their own marketing strategies.
Knowing your opponent does not mean copying competitors’ strategies. Instead, it enables businesses to identify opportunities for differentiation and develop unique value propositions. By understanding competitors’ strengths and weaknesses, companies can discover underserved customer segments and create stronger market positions.
Therefore, “know your opponent” represents a continuous process of monitoring external changes and adapting business strategies according to evolving market conditions.
SWOT analysis is one of the most effective frameworks for applying Sun Tzu’s principle to modern business strategy. The framework evaluates four key areas: Strengths, Weaknesses, Opportunities, and Threats. Strengths and weaknesses represent internal factors related to “knowing yourself,” while opportunities and threats represent external factors related to “knowing your opponent.”
Source: the author.
Strengths refer to internal advantages that allow companies to compete effectively. These advantages may include strong brand recognition, innovative products, loyal customers, advanced technology, or efficient operations. By identifying strengths, companies can develop strategies that maximize their existing competitive advantages.
Weaknesses refer to internal limitations that may reduce competitiveness. These may include high operating costs, limited resources, weak brand awareness, or inefficient processes. Recognizing weaknesses allows organizations to improve internal capabilities and prevent potential problems from affecting future growth.
Opportunities refer to external conditions that create potential growth possibilities. These may include emerging customer demands, technological development, new market segments, or changing consumer preferences. Companies that successfully identify opportunities can adapt their strategies and expand their market presence.
Threats refer to external factors that may negatively influence business performance. These include new competitors, changing consumer behaviour, economic uncertainty, and increasing price competition. Understanding threats allows companies to prepare appropriate responses before competitors gain an advantage.
Through SWOT analysis, businesses can transform the abstract principle of “know yourself and know your opponent” into a structured decision-making process. This framework allows organizations to connect internal capabilities with external market conditions and develop more effective strategies.
Starbucks Corporation provides a clear example of how “know yourself and know your opponent” can be applied through SWOT analysis. The company demonstrates how understanding internal capabilities and external competition can support long-term competitive success.
From an internal perspective, Starbucks’ major strengths are its global brand recognition and unique customer experience. Starbucks is not simply a company that sells coffee; it creates value through store atmosphere, personalized beverages, customer service, and strong brand identity. These advantages allow Starbucks to build customer loyalty and maintain premium pricing compared with many traditional coffee providers.
However, Starbucks also faces internal weaknesses. Operating thousands of locations worldwide requires significant investment in employees, store management, supply chains, and quality control. These high operating costs create challenges when Starbucks competes with lower-cost coffee providers. Recognizing these weaknesses allows the company to improve efficiency and adapt its operations.
From an external perspective, Starbucks identifies opportunities by analyzing changes in consumer behaviour. The increasing popularity of digital ordering, mobile payment systems, and personalized products provides opportunities for Starbucks to strengthen customer relationships. By understanding customer expectations, Starbucks can develop services that better match market demands.
At the same time, Starbucks must understand competitive threats. In markets such as China, Starbucks faces increasing competition from local coffee brands that compete through lower prices, rapid expansion, and digital platforms. Instead of competing only through price, Starbucks relies on its strengths, including global branding, customer experience, and product innovation, to maintain differentiation.
This example demonstrates that successful businesses need both internal and external analysis. Starbucks maintains competitiveness because it understands its own capabilities while continuously monitoring competitor strategies and market changes.
The principle of “know yourself and know your opponent” remains an important concept in contemporary business strategy. Organizations cannot achieve sustainable success by focusing only on their internal capabilities. They must also understand competitors, customers, and external environmental factors that influence market performance.
SWOT analysis provides a practical method for implementing this principle because it combines internal evaluation with external market analysis. Through SWOT analysis, businesses can create more effective strategies, improve market positioning, and develop sustainable competitive advantages.
Sun Tzu’s strategic thinking therefore continues to provide valuable lessons for modern organizations. Companies that accurately understand themselves and their opponents are better prepared to compete in dynamic markets.
▶ Mathis HOUROU Client Segmentation and Private Banking: Marketing Strategy or Risk Shield?
▶ Camille KELLER My Apprenticeship Experience as Digital Strategy Officer at Gan Assurances
▶ Liner SHI My Intern Experience in Tencent Strategy Department
The Art of War (Chinese Text Project)
Encyclopaedia Britannica – The Art of War
American Marketing Association (AMA)
The article was written in August 2026 by Bochen LIU (Queen’s Smith School of Business, BCom 2023–2027; ESSEC BBA Exchange Program, Fall 2025).
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In this article, Anirudh KUMAR (B.S. Economics, IIT Kanpur, with a minor in Artificial Intelligence and Machine Learning) shares his experience as a machine learning research intern. From February to August 2024, he worked under the supervision of Professor Swagato Chatterjee on a study of how people respond to Facebook communication about hydrogen fuel cell vehicles. The project later became a paper in IEEE Transactions on Engineering Management.
The project examined a practical communication problem. When an organization posts about a sustainable technology, which parts of the message are associated with likes, comments, shares, and other interactions? We chose hydrogen fuel cell vehicles because the technology is promising but still unfamiliar to many people. Organizations therefore have to explain both its immediate uses and its longer-term potential.
A single subject can be framed in very different ways. A post about a hydrogen-powered bus might discuss lower tailpipe emissions, give technical details about fuel-cell efficiency, announce a government pilot project, or warn about the cost of delaying cleaner transport. The technology is the same, but the topic, readability, and emotional tone are different. Our job was to turn those differences into variables that we could test.
My mission was to help build a reproducible pipeline from raw Facebook posts to research findings. I worked on text cleaning, topic modelling, sentiment and emotion measures, model comparison, charts, and short research briefs. Each stage had to answer a specific question: what was measured, why was it measured that way, and could another researcher reproduce the result?
Figure 1. From Facebook posts to research findings
Source: Author’s own work.
The study used Facebook posts collected through CrowdTangle from February 2022 to February 2024. The search combined “hydrogen fuel” with terms such as vehicle, car, transport, train, airplane, traffic, and truck. We kept English-language posts and used the country of the page administrators to divide the sample.
The final dataset contained 9,672 posts: 4,106 observations from developed countries and 5,566 from developing countries. The developed-country sample was led by the United States, the United Kingdom, and Australia. The developing-country sample included India, China, the Philippines, South Africa, Pakistan, Myanmar, and observations from 72 other countries.
Our outcome variable was the total number of interactions with a post. We also recorded factors that could affect engagement even before considering the words, including the page’s followers and likes at the time of posting and whether the post was a photo, video, status update, or another format.
NLTK and spaCy are Python libraries for natural language processing. I used them to split text into tokens, remove material that did not help the analysis, and reduce related word forms to a common lemma. For example, lemmatization can map vehicles to vehicle and emissions to emission. This prevents the model from treating simple grammatical variations as unrelated ideas.
Illustrative post before cleaning: “Hydrogen-powered buses are not yet cost competitive, but they can reduce urban emissions.”
Illustrative tokens after cleaning: “hydrogen”, “power”, “bus”, “not”, “yet”, “cost”, “competitive”, “reduce”, “urban”, “emission”.
The word not is important. If a standard stop-word list removed it, the sentence could appear more positive than the writer intended. I therefore did not treat preprocessing as a one-click operation. I inspected examples after URL and punctuation removal, after tokenization, after stop-word filtering, after lemmatization, and after the final features had been created. This is what I meant by checking the data after every major transformation.
Consistency mattered too. The forms EV, electric vehicle, and electric vehicles can refer to the same idea. If they are left as separate terms without review, topic frequencies and model inputs can become harder to interpret.
Latent Dirichlet Allocation, or LDA, is a topic model. It searches for groups of words that often occur together and represents each post as a mixture of those groups. I compared specifications with different numbers of topics and used coherence scores as one diagnostic. The final choice also had to produce topics that a reader could distinguish and name.
For developed countries, the model identified 12 topics. Examples included “Battery Technology and Toxicity” with words such as battery, lithium, toxic, electricity, EV, and metal; “Electric Aircraft and Aviation” with aircraft, plane, aviation, and flight; and “Electric Cars and Sustainable Transportation” with vehicle, electric, car, and powered. These three short-term topics had the strongest positive relationships with engagement relative to the reference topic in the final Poisson model.
For developing countries, the model identified 11 topics. Examples included “Government Green Energy Projects”, “Electric Vehicles and Battery Technology”, and “Future Energy Technologies”. The mix was different from the developed-country sample: immediate transport projects mattered, but longer-term energy planning also attracted attention.
This step taught me the difference between an output and a finding. LDA will always return word groups. The researcher still has to check whether those groups are stable, distinct, and useful for answering the research question.
I used TextBlob and NLTK WordNet for sentiment measures and NRCLex for emotion categories. Sentiment summarizes whether the language is more positive or negative. NRCLex adds categories such as anticipation, trust, joy, surprise, fear, anger, disgust, and sadness.
The distinction matters because two negative messages can invite different reactions. A post expressing sadness about slow adoption may encourage passive reading, while a post warning about an urgent climate or energy risk may evoke fear and prompt comments or shares. In the final models, fear had the strongest positive relationship among the negative emotions in both country groups. Among positive emotions, anticipation was important in developed countries and trust was important in developing countries.
Dictionary-based scores are constructed variables, not direct readings of a person’s feelings. Negation, technical language, and context can change a sentence’s meaning. I therefore checked sample classifications instead of assuming that every automated label was correct.
The study measured text complexity with the Gunning Fog Index. The index combines average sentence length with the share of complex words, usually words with three or more syllables:
Fog Index = 0.4 x [(words / sentences) + 100 x (complex words / words)]
Consider three ways to introduce the same subject. “Hydrogen cars are good” is easy to read but says little. “Hydrogen fuel cells produce electricity without tailpipe carbon emissions, although storage and refuelling remain costly” gives the reader both an accessible explanation and useful detail. A paragraph filled with unexplained terms such as proton-exchange membranes, electrochemical conversion, and volumetric energy density may be accurate but difficult for a general audience.
The results showed an inverted U-shaped relationship between the Fog Index and interactions. Engagement rose as messages became more informative, then fell when the text became too complex. The precise turning point differed by context. In the developing-country analysis, the partial-dependence curve reached its maximum near a Fog score of 20.
We used Poisson regression to explain how readability, topics, and emotions were associated with the count of interactions. We then compared Random Forest, Support Vector Regression, XGBoost, and Poisson-based predictive models. The predictive exercise used both in-sample and out-of-sample root mean squared error, or RMSE. A lower out-of-sample RMSE means the model predicted unseen posts more accurately.
The comparison also showed why the training score cannot be the only criterion. In the developed-country sample, XGBoost achieved a low in-sample error but showed signs of overfitting. Random Forest provided a better balance. Its out-of-sample RMSE fell from 4,898 in the baseline model to 3,056 after topic information was added. Adding emotion variables changed the error to 3,171, so the larger feature set was not automatically better.
Posts that were very simple could lack useful information, while highly technical posts could demand too much effort from a general reader. The highest engagement appeared between those extremes. For an organization, the practical lesson is to explain the technology clearly without removing the detail that makes the message informative.
In developed countries, short-term topics such as electric vehicles, battery technology, and aviation were more engaging than distant or futuristic themes. In developing countries, the pattern was mixed. Government green-energy projects and battery technology mattered, but future energy technologies also resonated. A single global content plan would miss these differences.
Trust can make a new technology feel credible, while fear can draw attention to the consequences of inaction. Neither result means that organizations should exaggerate. It means the emotional tone of a message is measurable and should be considered alongside the topic and readability.
I converted the analysis into topic tables, model-comparison charts, feature-importance plots, and three short research briefs. One chart compared the out-of-sample RMSE of the four predictive approaches. Another summarized the 12 developed-country topics and the 11 developing-country topics. The briefs translated those outputs into communication questions: how technical should a post be, which topics fit each market, and which emotional signals require careful interpretation?
This work exposed weak explanations quickly. If I could not explain why Random Forest was preferred to XGBoost, or what a topic coefficient meant relative to the reference topic, the figure was not ready for another reader.
The internship required Python, natural language processing, regression, machine learning, and model evaluation. My economics courses helped me define variables and separate association from causation. My AI and machine-learning coursework helped me build the pipeline and compare models.
The work also required careful records. I kept track of cleaning rules, exclusions, topic specifications, model versions, and evaluation results. I treated the code as shared work. A co-author should be able to see what changed and why.
The project developed into the article Drivers of Social Media Engagement on Organizational Communication on Sustainable Technological Innovation: Insights from Developed and Developing Countries, published in 2026 in IEEE Transactions on Engineering Management.
The manuscript went through several rounds of revision. I learned to treat a changed result as information rather than as a setback. If a coefficient, topic, or model ranking changed after a reasonable specification check, we needed to understand the reason before keeping the claim.
The question comes before the model.
I now write the research question in plain language before opening a notebook. In this project, the question was not simply which algorithm predicts engagement best. We also needed to understand which message characteristics were associated with engagement and whether the relationships differed across country groups.
Small decisions need a record.
A stop-word choice can change the meaning of a sentence. A different number of LDA topics can change the labels used in the regression. Weeks later, those decisions are difficult to reconstruct from memory, so I keep a short log of exclusions, transformations, and model changes.
Prediction and explanation are different tasks.
Poisson regression helped us interpret relationships between the features and interaction counts. Random Forest was more useful for prediction. Neither result replaced the other. A strong research design needed both an interpretable explanation and an honest test on unseen data.
Attention is scarce in financial markets as well as on social media. Investors cannot read every earnings call, filing, news article, and management post. Topic and engagement analysis can help researchers study which firms or themes receive attention and whether that attention is associated with trading activity.
Earnings calls, annual reports, and management commentary contain language that can be measured systematically. The internship taught me to inspect how a sentiment variable was built before treating it as a signal. A financial sentence with negation or technical terminology can confuse the same dictionary methods used for social-media text.
The XGBoost result offered a direct example of model risk. A model can fit the training data closely and perform poorly on new observations. Out-of-sample tests, sensitivity checks, and readable documentation are essential when a model may influence an investment or business decision.
In my current work as an AI Growth Intern at Pocket FM, I apply the same method to a different problem: a GenAI workflow that converts Hindi audio scripts into Marathi. I work with editors to define quality criteria for emotional beats, narrative hooks, and regional expressions. A fluent translation is not enough if it changes the scene’s intent.
The research internship taught me to make those criteria explicit, inspect intermediate outputs, and document recurring errors. The tools have changed, but the habit of testing each transformation remains useful.
A published paper can make research look linear. My experience was less tidy. A cleaning rule affected the topics, the topics affected the models, and the models sometimes sent us back to an earlier decision. The practical work was in those links.
For students interested in finance, data science, or AI, the project shows how researchers can turn raw language into evidence. Technical skill is necessary, but the result is easier to trust when the examples, assumptions, and out-of-sample performance are visible.
Looking for an internship or a research experience? You may find useful information in other posts where contributors describe their professional work:
▶ All posts about Professional experiences
▶ Haiyuan XU My professional experience as a financial research assistant in a green finance institute
▶ Anant JAIN My internship experience at Deloitte
Chatterjee, S., Ghatak, A., Meena, A. K. and Meena, P. (2026). Drivers of Social Media Engagement on Organizational Communication on Sustainable Technological Innovation: Insights from Developed and Developing Countries. IEEE Transactions on Engineering Management, 73. DOI: 10.1109/TEM.2026.3665711.
Blei, D. M., Ng, A. Y. and Jordan, M. I. (2003). Latent Dirichlet Allocation. Journal of Machine Learning Research, 3, 993-1022.
NLTK Project. Natural Language Toolkit documentation.
Explosion. spaCy usage documentation.
The article was written in July 2026 by Anirudh KUMAR, who completed a B.S. in Economics at IIT Kanpur with a minor in Artificial Intelligence and Machine Learning. He is an AI Growth Intern at Pocket FM and previously worked in product management at AryaVastra and machine-learning research under Professor Swagato Chatterjee. His interests include behavioral finance, empirical research, and AI applications in business.
▶ Discover all articles by Anirudh KUMAR.
In this article, Isaac FAINSTEIN, Director at Petrini Valores and Visiting Lecturer at IESEG School of Management (Lille), shares his professional experience as a trader in illiquid fixed income and emerging markets — and what practitioners know that most finance courses never cover.
Petrini Valores is an Argentine broker-dealer specializing in fixed income, equities, derivatives, and financing. The firm operates as a market maker in illiquid corporate and provincial bonds, across multiple asset classes: peso-denominated, USD-denominated, inflation-linked, and dollar-linked instruments. It also participates as a member of underwriting syndicates in primary bond issuances.
As Director of the trading desk, I am responsible for pricing, execution, and risk management across these asset classes on a daily basis.
Logo of Petrini Valores.
Source: Petrini Valores.
I have been trading fixed income and foreign exchange in Argentine markets for over fifteen years. Over that same period, I have taught applied finance courses at IESEG School of Management in Lille — courses built around the situations I encounter at the desk every week. What follows is an attempt to bridge those two worlds.
Most finance programs teach students how to price securities. Fewer teach them what it actually feels like to put the firm’s capital at risk to make a market. The distinction between agency trading, intermediation, and principal trading is more consequential than most courses suggest.
In agency trading, you act on behalf of a client — executing their order in the market, taking no position yourself, earning a fee for the service. The client bears the market risk. You are their agent.
Intermediation — what practitioners often call riskless principal — is already a form of proprietary trading, technically speaking. You act as principal on both legs: you buy from one counterparty and simultaneously sell to another, earning the bid-ask spread. Because both legs close at the same time, your market exposure is minimal. You are not an agent of either side. You are a counterparty to both, just briefly, and without meaningful inventory risk.
Principal trading with inventory risk is something else entirely. The firm puts its own capital on the line with no guaranteed exit. You buy a bond from a client with no buyer lined up on the other side. You sell from your own inventory because a client needs to buy. You absorb the spread — and the full market risk that comes with holding the position until you can unwind it. The longer you hold, the more exposure you carry. This is the mode that no simulation fully replicates, and the one this article is about.
A large portion of my daily activity involves corporate and provincial bonds that do not trade on a liquid exchange. There is no visible order book. There is no Bloomberg mid-price that everyone agrees on. There is a fragmented OTC market where each dealer forms their own view of value.
When a client calls and asks for a bid or offer on one of these bonds, I have to produce a price — quickly, without full information. I know what I think the bond is worth. What I do not know is whether the client is a buyer or a seller.
This asymmetry is at the heart of market-making in illiquid securities. If I quote too tight a spread, I may find myself on the wrong side of a pre-arranged trade. A client may call five dealers simultaneously, collect our offers, and hit the best one — while already having a buyer on the other side paying more than my offer. In that case, I have sold bonds below what the market was willing to pay, and the client has effectively traded through me.
I use this scenario in class regularly. Students are always surprised. They assume that being a good trader means knowing what something is worth. It does — but it also means understanding the information game you are playing with the person on the other side of the phone.
Then there is the moment that every trader knows: you have priced the trade, the client has everything they need to decide, and then — nothing. They go to lunch. They are in a meeting. They are closing another trade. You are sitting there holding a price in a moving market, watching the bid shift while you wait for a response that may or may not come. No simulation I have seen fully replicates the specific discomfort of that moment.
Beyond secondary market activity, I participate as an underwriter in primary bond issuances for Argentine corporates, as part of the underwriting syndicate organized around each deal. This is a different kind of principal risk — one that is taken on before the bond even exists.
When a company decides to issue a bond, I commit to underwriting a portion of the deal. This is a real financial commitment: if investor demand is insufficient to cover the full issuance, I absorb the remainder onto my own book. In Argentina, primary markets typically use a Dutch auction format — investors submit bids specifying the coupon rate they are willing to accept and the quantity they want. The issuer then sets a clearing rate that satisfies the target issuance amount.
On auction day, I am simultaneously placing bonds with my own client base, managing my underwriting exposure, and monitoring where the clearing rate is likely to land. If I have covered my commitment with investor demand, I am in good shape. If not, the unsold portion of my underwriting commitment ends up on my balance sheet at the clearing rate — and I work that position off over the following days or weeks, offering it into a market that may or may not be ready to absorb it.
This is textbook principal risk. It is also something that very few students have any mental model for before entering the industry.
Argentina has operated with capital controls for years. At their peak, the gap between the official exchange rate and the blue-chip swap rate — a market-implied rate derived from the implicit FX embedded in cross-market bond transactions — reached several hundred percent. Today the gap has narrowed significantly, but the structure remains.
This creates situations that no standard finance course addresses. A bond denominated in dollars can be bought and sold in different currencies. If I buy a USD bond paying dollars and sell it against pesos, I receive pesos for an asset I paid for in dollars. I now have a currency mismatch on my book: I am effectively long pesos, short dollars. I can hedge that exposure immediately by buying back the dollars in the FX market, or — if I have a view that the implied exchange rate will move in my favor — I can hold the position and let it run.
The decision is not mechanical. It depends on my reading of the regulatory environment, the direction of the blue-chip swap rate, and how much currency risk I am willing to carry on the book at that moment. This is daily life on the desk. And it is very difficult to teach without the context that produces it.
In April 2020, front-month WTI crude oil futures briefly traded at negative prices. Physical storage constraints had overwhelmed the market’s mechanics, and sellers were willing to pay counterparties to take delivery of crude oil they had nowhere to store.
I watched it happen from the desk in real time. What struck me was not the price itself — it was the reaction across the industry. Many traders assumed it was a glitch. Some platforms were simply not built to display or process negative prices, and brokers whose systems could not show the quotes found themselves liable to clients who could not see — let alone act on — what was happening in the market. Several firms had to absorb losses because their technology had never contemplated the possibility.
I use this episode as an opening in class — not to explain futures mechanics, which students can read in any textbook — but to ask a different question: what do you do when the model produces an answer that the real world seems to reject? What is your decision framework when your screen shows something that looks impossible? The answer is that you need to understand the why behind the price before you can act on it. That understanding is not something you can look up in real time. Either you have built it, or you have not.
Every trader makes mistakes. A wrong-way position, a misread signal, a fat-finger entry. What separates good trading culture from bad is not the absence of errors — it is what happens in the first thirty seconds after one occurs.
The worst thing a trader can do is wait. Hiding a mistake, even briefly, turns a manageable problem into a serious one. A position that could have been closed at a small loss will compound. The bid-ask spread you avoided paying once will have widened by the time you are forced to act.
The most important rule on any trading desk is this: when you make a mistake, communicate it immediately. No fear of consequences should outweigh the cost of silence. A well-run desk creates an environment where immediate transparency is rewarded — because the alternative is invariably more expensive. This is not a financial concept. It is a cultural one. And it may be the most practically useful thing I can tell any student before they sit down at a real trading desk for the first time.
With a World Cup recently concluded — and Argentina’s performance still fresh in everyone’s memory — there is a useful analogy worth making. Argentina’s best players did not all come through polished academies with perfect pitches and controlled conditions. Many learned on uneven surfaces, in chaotic environments, where improvisation and resilience were not optional. Those conditions, more often than not, produced technically complete and mentally durable players.
The same logic applies to trading in an environment like Argentina. Multiple asset classes, multiple yield curves, structural illiquidity, capital controls, and macroeconomic volatility — all simultaneously, all the time. Traders who come through this market and move to larger ones — Brazil, Mexico, or developed markets — typically find the transition smoother than expected. They have already navigated conditions that most traders in more liquid markets never face. When you learn to trade in the mud, the rest feels like solid ground.
I present below four financial concepts central to my daily work as a trader in illiquid and emerging markets.
In principal trading, the broker-dealer buys or sells securities using its own capital, taking market risk onto its own balance sheet. This contrasts with agency trading, where the firm executes on behalf of a client and earns a fee, or with intermediation (riskless principal), where the firm matches both sides simultaneously and earns the bid-ask spread without holding inventory risk. The critical difference is time: in principal trading, the firm holds a position that may not be unwound immediately, and the longer it is held, the greater the market exposure.
In a primary bond issuance, several broker-dealers form an underwriting syndicate, each committing to place a portion of the deal with investors. The book runner is the lead of this syndicate — it manages the investor order book, coordinates pricing with the issuer, and oversees the allocation process. Other syndicate members, such as Petrini Valores in many Argentine corporate issuances, commit to their own underwriting tranche and are responsible for placing it with their client base. If a syndicate member cannot fully place its portion, the unsold bonds remain on its balance sheet at the clearing rate.
A Dutch auction is a price-discovery mechanism in which investors submit bids specifying both quantity and the coupon rate they are willing to accept. The issuer sets a single clearing rate that satisfies the target issuance amount. All successful bidders receive bonds at the clearing rate, regardless of their individual bids. This format is widely used in Argentine primary markets for corporate bond issuances.
In markets with capital controls, such as Argentina, the blue-chip swap rate (also known as the contado con liquidación or CCL rate) is an implied exchange rate embedded in cross-market bond transactions. It reflects the market’s assessment of currency value in the absence of free convertibility and can diverge significantly from the official rate. Managing positions across currencies in this environment requires an understanding of the regulatory framework and a constant read on the gap between official and market-implied rates.
If you are a finance student planning to work in sales and trading, fixed income, or any market-facing role, the situations described here are among the ones you will encounter earliest — and none of them are fully captured in a simulation or a pricing model.
The gap between finance education and market reality is not about knowledge. Most graduates know their bond math. The gap is about judgment: knowing how to act when information is incomplete, the counterparty is not responding, and the market is moving. Understanding how principal risk, illiquidity, and currency mismatches interact in real time is the difference between arriving prepared and arriving surprised.
▶ All posts about Professional experiences
▶ Abel ARAYA Inside the Markets COO Office at HSBC: Understanding How Trading Floors Are Managed
▶ David GONZALEZ Discovering the Secrets of a Bank Trading Room
▶ Mickael RUFFIN My Internship Experience as a Structured Finance Analyst at Société Générale
▶ All posts about Financial techniques
Gkillas K. and Longin, F. (2018) Financial market activity under capital controls: lessons from extreme events, Economics Letters, 171, 10-13.
Martellini, L., Priaulet, P., Priaulet, S. (2003) Fixed-Income Securities: Valuation, Risk Management and Portfolio Strategies, John Wiley & Sons.
Hull, J. C. (2021) Options, Futures, and Other Derivatives, 11th edition, Pearson.
Petrini Valores — Argentine broker-dealer specializing in fixed income, equities, derivatives, and financing.
FINRA Tools and Calculators — public source for US bond transaction data and pricing context.
This article was written in July 2026 by Isaac FAINSTEIN, Director at Petrini Valores and Visiting Lecturer at IESEG School of Management (Lille), where he has taught applied finance and trading courses for over 10 years.
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In this article, Abel ARAYA (ESSEC Business School, Master in Finance, 2025) takes a detailed and practical look at delta hedging, a core concept in options trading. Far from being just a mathematical tool, delta hedging is a real-world technique that allows traders to manage market risk dynamically and stay focused on what they actually want to trade: volatility and relative value.
When a trader buys or sells options, they are taking a view not only on the direction of the market but also on how much the market might move. The value of an option changes constantly, influenced by multiple factors such as the price of the underlying asset, time decay, volatility, and interest rates. Without active management, these continuous changes can make an options book risky and unpredictable.
Delta hedging addresses this by adjusting the position in the underlying asset to neutralize the impact of small price movements. The trader aims for a portfolio that reacts as little as possible to minor changes in the underlying price so that risk is concentrated on the variables they intend to trade, such as volatility.
Delta is one of the key Greeks that measure how the price of an option responds to changes in market variables. It tracks the sensitivity of the option value to the underlying price.
Δ = ∂V / ∂S
Here, V is the option value and S is the price of the underlying asset. For example, if a call option has a delta of 0.60, a rise of 1 euro in the underlying increases the option value by about 0.60 euro. A delta close to 1 behaves like the underlying, while a delta near 0 barely reacts to price moves.
Suppose a trader has sold call options on a stock. If the stock rises, the calls gain value and the short position loses money. To offset this exposure, the trader buys shares of the underlying. The goal is to hold a number of shares that compensates the option’s delta so that small moves in the stock price do not change the portfolio value.
Δportfolio = Δoption + Δhedge = 0
The trader dynamically buys or sells the underlying to keep the combined delta close to zero.
Delta changes as markets evolve, time passes, or volatility shifts. The speed of this change is captured by gamma, which measures the curvature of the option value with respect to the underlying price.
Γ = ∂²V / ∂S²
A high gamma means delta changes quickly, which forces more frequent rebalancing. Large options books can require several adjustments per day during volatile periods. In episodes of market stress, hedging flows can increase significantly as deltas move rapidly.
So far we have described delta hedging for a single option, but a trader on a derivatives desk rarely manages one position in isolation. In practice, the desk holds a book of hundreds or thousands of options across many underlyings, strikes and maturities. Hedging each one separately would be inefficient and very costly in transaction fees.
Instead, the trader looks at the net risk of the entire book. All the individual deltas are aggregated into a single net delta for the portfolio, and only that net exposure is hedged in the underlying. Long and short positions offset each other, so the desk usually needs far fewer hedging trades than the number of options it holds. The same logic applies to the other Greeks: the book is managed at the level of its net delta, gamma and vega rather than option by option. This portfolio approach is what makes running a large options book possible, and it is one of the core skills of a derivatives trader.
Each rebalancing operation generates transaction costs, including bid-ask spreads and slippage. Skilled traders balance precision and efficiency, deciding when to rebalance and when to tolerate a small residual exposure. Frequent re-hedging reduces risk but can erode profits through costs. The optimal approach depends on liquidity, volatility, and position size.
Delta hedging allows traders to isolate the risks they want to trade. By neutralizing directional exposure, they can focus on volatility, time decay or interest rate sensitivity. For instance, a volatility trader may be long options but delta-hedged, seeking to profit if realized volatility exceeds implied volatility rather than from market direction.
For a bank acting as a market-maker, delta hedging is not only a risk-management tool: it is also what allows the desk to secure its margin. When the bank sells an option to a client, it charges a price that is slightly above the option’s theoretical, or fair, value given by its pricing models. That difference between the price paid by the client and the theoretical value is the bank’s margin.
By delta hedging the position dynamically, the bank replicates the payoff of the option at a cost close to its theoretical value while neutralizing the impact of market direction. If the hedging is done well, the bank is no longer betting on whether the underlying goes up or down: it has locked in that initial margin regardless of how the market moves. In other words, delta hedging turns a directional exposure into a controlled activity whose objective is to capture and protect the spread between the price sold to the client and the theoretical value of the option.
Consider a trader who sells 5 million euros of call options on the EuroStoxx 50 with an average delta of 0.4. To stay neutral, the trader buys 2 million euros of EuroStoxx 50 futures, which offsets the option delta. If the index rises and delta increases, the trader buys more futures. If it falls, they reduce the hedge. The objective is to end the day with minimal unhedged exposure despite continuous fluctuations.
Although the mechanics look simple, judgment matters. On quiet days, fewer adjustments are needed. In unsettled markets, hedging becomes more frequent. Delta hedging therefore blends quantitative discipline with trader intuition.
Delta hedging does not eliminate all risk. It removes first-order sensitivity to small price moves. Sudden jumps in price or volatility introduce residual risks captured by higher-order Greeks such as gamma and vega. Effective options risk management considers these dimensions together.
Delta hedging is a practical cornerstone of modern options trading. By continuously adjusting exposure, traders can focus on pricing, volatility and liquidity rather than guessing direction. Understanding delta hedging provides a clearer view of market neutrality in practice.
If you are a student in finance interested in derivatives, trading, or risk management, delta hedging is one of the most fundamental concepts you will encounter in practice. It bridges the gap between option pricing theory and what traders actually do every day on the floor. Understanding delta hedging will give you a concrete language for discussing risk with traders and structurers in interviews, and it directly underpins roles in equity derivatives, rates options, and exotic products desks.
More broadly, the logic of delta hedging, isolating a risk you want to trade from one you do not, applies far beyond options. It is central to how banks and hedge funds manage their books across all asset classes. Whether you are targeting a front-office internship, a quant role, or a risk management position, mastering this concept will give you a genuine edge.
▶ Jayati WALIA Black-Scholes-Merton option pricing model
▶ Akshit GUPTA Option Greeks: Delta
▶ Akshit GUPTA Option Greeks: Gamma
▶ Akshit GUPTA Option Greeks: Vega
▶ Jayati WALIA Implied Volatility
▶ Saral BINDAL Implied Volatility and Option Prices
Hull J.C. (2022) Options, Futures, and Other Derivatives, Pearson, 11th Edition.
Black F. and Scholes M. (1973) The Pricing of Options and Corporate Liabilities”, Journal of Political Economy, 81(3), 637-654.
Merton R.C. (1973) Theory of Rational Option Pricing Bell Journal of Economics, 4, 141–183.
Bank for International Settlements — OTC Derivatives Statistics
The article was written in July 2026 by Abel ARAYA (ESSEC Business School, Master in Finance, 2025).
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In this article, Abel ARAYA (ESSEC Business School, Master in Finance, 2025) offers an inside look at the Markets COO Office at HSBC Continental Europe. Through his one-year apprenticeship, he shares how this central function coordinates trading activities, manages budgets and risks, and ensures that the Markets division operates with efficiency and strategic discipline.
HSBC was founded in 1865 as the Hongkong and Shanghai Banking Corporation to finance trade between Europe and Asia, and has since grown into one of the world’s leading financial institutions. Headquartered in London and listed in London, Hong Kong, New York, Paris and Bermuda, it held around 3.2 trillion US dollars in assets at the end of 2025, employed roughly 211,000 people across some 56 countries and territories, and served more than 40 million customers.
On the wholesale side, its corporate and institutional clients are covered by the Corporate and Institutional Banking (CIB) division, which generated around 27.6 billion US dollars in revenue in 2025. Within CIB, the Markets and Securities Services teams provide liquidity, financing and risk-management solutions across fixed income, credit, FX, equities and securities services to large corporates, financial institutions, asset managers, hedge funds and governments. In this business HSBC competes with the other major global markets houses, such as JPMorgan, Citi, Bank of America and Goldman Sachs in the United States, and Deutsche Bank, Barclays, BNP Paribas and Société Générale in Europe, differentiating itself through its international network and its historical strength in Asia.
I worked at HSBC Continental Europe, the group’s Paris-headquartered subsidiary covering continental Europe. Since the sale of its French retail banking business on 1 January 2024, it has refocused on corporate and institutional clients, with a consolidated balance sheet of 251 billion euros in total assets at the end of 2025.
Logo of HSBC.
Source: the company.
During my apprenticeship, the Markets division was in the process of being integrated into the broader Corporate and Institutional Banking (CIB) structure. This reorganization involved significant changes to how the division was managed, reported, and resourced, which made my experience at the COO Office particularly rich in terms of exposure to strategic and operational transformation.
As a Business Manager Assistant within the Markets COO team, my work covered a wide range of financial and operational responsibilities. I contributed to the production of internal reports and presentations for senior management, summarizing the performance, expenses, and headcount of the Markets division. These documents were used in management meetings, financial steering committees, and due diligence reviews conducted during the restructuring process.
I was closely involved in cost forecasting and budget follow-up, helping the team anticipate upcoming expenses and identify deviations from plan. One of my key projects was the annual broker review, which required consolidating trading flow data across all asset classes to assess the efficiency, transparency, and compliance of relationships with external counterparties. This involved close coordination with traders, operations, and compliance teams across Paris, Germany, and India.
I also supported the preparation of headcount reports and organizational charts used by senior management to steer the restructuring of the division. These deliverables required precision and a thorough understanding of how each desk contributed to the overall structure of the Markets business.
This role required a combination of analytical and interpersonal skills. On the technical side, strong proficiency in Excel was essential for building budget models, consolidating large datasets, and producing financial summaries. Familiarity with the structure of a markets division, including the roles of front office, operations, compliance, and finance, was also important to contextualize the data I was working with.
Soft skills mattered just as much. Coordinating with stakeholders across multiple countries and hierarchies required clear written and oral communication, the ability to manage competing priorities, and a high level of attention to detail. The pace of the environment also demanded adaptability: priorities shifted quickly, and producing reliable output under time pressure was a daily reality.
This experience gave me a deep understanding of how financial institutions manage their operations behind the scenes. I learned how budgets are built, how costs are tracked and challenged, and how strategic decisions made at senior level translate into concrete actions on the trading floor. I also developed a much clearer picture of how risk is monitored and how compliance frameworks shape the day-to-day behaviour of a markets division.
Working across teams in Paris, Frankfurt, and India gave me direct exposure to how global coordination works in practice. I learned the importance of data quality and rigour: a single inconsistency in a report could lead to misunderstandings or delayed decisions at the highest level. This reinforced my attention to detail and my commitment to producing work that is both accurate and clearly communicated.
I present below three financial concepts related to my internship: cost and budget management, change management, and due diligence.
Cost and budget management refers to the process by which an organization plans, monitors, and controls its financial resources to ensure that spending remains aligned with strategic objectives. In a markets division, this involves tracking a wide range of costs: staff compensation, technology infrastructure, external service providers, and regulatory compliance expenses. The budget is typically set at the beginning of the year based on business forecasts and strategic priorities, and then monitored on a monthly basis against actual expenditure.
In my role at the Markets COO team, cost and budget management was one of my primary responsibilities. I contributed to the monthly budget follow-up by consolidating cost data from different desks and entities, identifying variances between forecasts and actual figures, and preparing summary reports for senior management. When a desk was running significantly above or below budget, the COO team would investigate the drivers and, if necessary, escalate to management for a decision. I learned that in a large institution like HSBC, even small deviations in cost forecasts can have a significant impact on the division’s overall financial performance, particularly during a period of restructuring where cost targets were closely scrutinized.
Change management is the structured approach through which an organization transitions from its current state to a desired future state while minimizing disruption to operations and people. In the context of financial institutions, it often involves reorganizations, mergers of business lines, technology migrations, or regulatory-driven transformations. Effective change management requires clear communication, stakeholder alignment, and careful sequencing of decisions to ensure continuity of service during the transition.
During my apprenticeship, HSBC’s Markets division was undergoing a major strategic restructuring: the Markets and Securities Services unit was being integrated into the broader Corporate and Institutional Banking (CIB) structure. This was not a minor adjustment, but a fundamental reorganization of how the division was governed, resourced, and reported. I observed the effects of this transformation directly through my work: headcount reports were being revised regularly, cost allocation frameworks were changing, and the responsibilities of the COO team were evolving to reflect the new organizational model. I worked closely with the COO based in Germany, who was managing part of this transition, and I saw first-hand how much coordination and precision are required to keep a large division functioning smoothly while simultaneously reshaping it. Change management, in that context, was not an abstract concept: it was a daily operational reality.
Due diligence refers to the comprehensive process of investigating and verifying information before making a significant business decision. In investment banking and financial services, it is most associated with mergers and acquisitions, where a buyer conducts a thorough review of the target company’s finances, legal situation, and operations. However, the concept applies equally to other contexts, including the assessment of external service providers, the validation of financial data before it is presented to management, and the review of counterparty relationships.
In my role, due diligence took the form of the annual broker review process. This involved systematically reviewing the trading flows directed to each external broker, verifying the accuracy of the data, and assessing whether the allocation of business to each counterparty was justified by objective performance criteria. The process required gathering data from multiple sources, reconciling inconsistencies, and presenting findings to senior management with clear supporting evidence. I also contributed to due diligence exercises conducted during the restructuring process, where the COO team was asked to validate headcount and cost data before it was presented to the executive committee. These experiences taught me that rigorous due diligence is not just about finding problems: it is about building the trust and confidence that allow organizations to make well-informed decisions.
If you are a student in business and finance considering a career in financial markets, this post offers a perspective that is rarely covered in mainstream discussions about finance careers: the operational and strategic backbone of a trading floor. Most students aspire to front-office roles in trading or sales, and rightly so. But understanding how a markets division is actually run, how its costs are managed, how its risks are monitored, and how major transformations are navigated, is an invaluable foundation for any finance career.
A role in a Markets COO or Business Management team is particularly well-suited for students who want to develop a transversal understanding of markets while building strong analytical and organizational skills. It is also increasingly recognized as a credible path toward front-office positions: many senior traders and sales managers have spent time in COO or control functions early in their careers, and this experience gives them a level of business awareness that pure front-office profiles often lack. Whether you are targeting trading, sales, risk, or corporate finance, the skills and perspective gained in this type of role will give you a genuine advantage.
▶ Tanguy TONEL My experience as a trading floor intern at CIC Market Solutions
▶ David GONZALEZ Discovering the Secrets of a Bank Trading Room
▶ Louis DETALLE A quick review of an Analyst in Transaction Services’ job
▶ Mickael RUFFIN My Internship Experience as a Structured Finance Analyst at Société Générale
HSBC — Corporate and Institutional Banking
Basel Committee on Banking Supervision — Sound Practices for Operational Risk Management
The article was written in July 2026 by Abel ARAYA (ESSEC Business School, Master in Finance, 2025).
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In this article, Abel ARAYA (ESSEC Business School, Master in Finance, 2025) shares his experience on HSBC’s trading floor in Paris. He explains how this opportunity helped him understand how a global markets division operates, how teams interact, and what makes the environment of a trading floor so unique.
Before joining HSBC, I was working in private banking at Milleis Banque Privée. It was a good introduction to finance, but I wanted to understand how the markets worked behind the scenes. Joining HSBC Continental Europe as a Business Manager Assistant within the Markets COO (Chief Operating Officer) team gave me the chance to discover that world for the first time. The COO function within a markets division is responsible for the operational and financial oversight of the trading floor: it sits at the intersection of strategy, finance, and day-to-day management, supporting the front office without being directly involved in trading itself.
HSBC was founded in 1865 as the Hongkong and Shanghai Banking Corporation, created to finance trade between Europe and Asia. More than a century and a half later, it has become one of the largest banking and financial services groups in the world. Headquartered in London and listed in London, Hong Kong, New York, Paris and Bermuda, the group held around 3.2 trillion US dollars in assets at the end of 2025, employed roughly 211,000 people across some 56 countries and territories, and served more than 40 million customers.
On the wholesale side, where I worked, HSBC brings together its corporate and institutional clients under the Corporate and Institutional Banking (CIB) division, which generated around 27.6 billion US dollars in revenue in 2025. CIB was created on 1 January 2025 by combining the former Global Banking and Markets business with commercial banking activities outside the UK and Hong Kong, with the ambition of ranking among the top three global wholesale banks. Within CIB, the Markets and Securities Services teams provide liquidity, financing and risk-management solutions across fixed income, credit, foreign exchange, equities and securities services.
Its clients are large corporates, financial institutions, asset managers, hedge funds and governments that rely on the bank to trade, hedge and finance their activities across the world. In this space HSBC competes with the other large global markets houses, such as JPMorgan, Citi, Bank of America and Goldman Sachs in the United States, and Deutsche Bank, Barclays, BNP Paribas and Société Générale in Europe. Its main differentiator remains its international network and its historical strength across Asia and emerging markets.
My apprenticeship took place at HSBC Continental Europe, the group’s Paris-headquartered subsidiary covering continental Europe. Since the sale of its French retail banking business on 1 January 2024, HSBC Continental Europe has refocused on corporate and institutional clients, with a consolidated balance sheet of 251 billion euros in total assets at the end of 2025.
During my apprenticeship, this reorganization was still under way, which made it a particularly interesting time to observe how such a large organization adapts its structure while continuing to run its business day to day.
Logo of HSBC.
Source: the company.
Within the Markets COO team, my work focused on the financial and organizational aspects of the trading floor. I contributed to budget monitoring, forecasts of upcoming expenses, and internal reports related to costs and resources. I was involved in the broker review process and in the preparation of financial summaries presented to management.
This position gave me a transversal view of the Markets division and helped me understand how each team contributes to the overall structure. I interacted with many different stakeholders: the COO in Germany, who was managing a restructuring process, teams in India working on operational data, and senior managers in Paris overseeing the desks. These collaborations taught me how coordination and communication are essential to keep such a large platform running efficiently.
Working so close to the trading floor was one of the most rewarding parts of my experience. Even though my role was on the management side, I was constantly in contact with the desks. I often visited traders, salespeople, and structurers to better understand their activities and the financial implications of their operations. One moment that stayed with me was a conversation with a rates trader during a period of elevated volatility in the European bond market. He explained how the sudden widening of spreads between Italian BTPs and German Bunds was forcing him to adjust his hedging positions in real time, something I had only ever read about in textbooks. These interactions helped me connect the numbers I was analysing to the real market dynamics they represented.
The atmosphere on the floor was intense and collaborative at the same time. Information flowed constantly between desks, from rates to credit to repo, and decisions were made quickly. Observing this rhythm every day helped me understand how interconnected market teams are, and how much relies on clear communication and mutual trust.
This experience gave me a real understanding of how a trading floor operates, both economically and humanly. I learned how a large institution like HSBC manages its costs, allocates resources, and balances strategic priorities with budget realities. I also saw how economic pressures, regulatory changes, and internal dynamics influence decisions at every level of the organization.
Spending time close to the Fixed Income desks gave me a concrete sense of how sales, traders, and support teams work together. I realized that beyond products and numbers, markets are built on relationships, coordination, and constant adaptation.
Most importantly, this experience taught me the value of curiosity and initiative. By going to speak directly with teams, asking questions, and trying to understand their world, I gained insights that no spreadsheet could have given me. It made me appreciate both the complexity and the humanity of financial markets.
This one-year apprenticeship was a very strong first step into the world of markets. It helped me confirm that the natural next step for me would be a front-office internship as a Sales in Fixed Income, where I could build on what I learned and continue to grow within a trading environment.
I present below three financial concepts related to my internship: market liquidity, collusion risk, and profit and loss (P&L).
Market liquidity refers to the ease with which a financial instrument can be bought or sold in the market without significantly moving its price. A liquid market has many buyers and sellers, tight bid-ask spreads, and the ability to execute large transactions quickly. An illiquid market, by contrast, forces participants to accept worse prices or wait, which can turn a theoretically profitable position into a loss once execution costs are taken into account.
In fixed income markets, liquidity is not uniform: it varies by product, by maturity, and by the time of day. Sovereign bonds such as French OATs or German Bunds are among the most liquid instruments in the world, with spreads of just a few basis points. Corporate bonds, by contrast, trade far less frequently and can see spreads widen dramatically in periods of stress. Structured products and exotic rates instruments can be even harder to unwind quickly.
One of the things I discovered at HSBC is the central role brokers play in providing liquidity. Not all brokers are equal: some are specialists on particular products or market segments. For example, inter-dealer brokers such as TP ICAP or Tradition are well known for their activity in rates and repo markets, where they connect banks anonymously to facilitate large transactions. During the annual broker review process that I participated in, traders would assess which brokers had provided the best liquidity, the most reliable pricing, and the fastest execution across different products. This review directly influenced how trading flows were allocated across brokers the following year. It made me understand that liquidity is not just a market property: it is also a relationship, built and maintained between institutions over time.
In financial markets, collusion risk between traders and brokers refers to a specific form of conflict of interest: a trader systematically routing a disproportionate volume of transactions to a particular broker, not because that broker offers the best execution, but because of a personal relationship, reciprocal favours, or informal arrangements. This behaviour is harmful to clients, who are entitled under regulation to receive the best available price and execution, a principle known as best execution, enshrined in the MiFID II directive in Europe.
The risk is subtle and not always easy to detect. A trader may genuinely believe that their preferred broker is the best, when in reality they are simply more comfortable with them. Over time, this can result in a concentration of flows toward one or two brokers that is not justified by objective performance criteria such as pricing quality, speed of execution, or market access. In the worst cases, the relationship can involve gifts, entertainment, or the sharing of confidential information, all of which are strictly regulated.
This is exactly what the annual broker review process at HSBC was designed to monitor and prevent. As part of my role in the Markets COO team, I contributed to this review, which involved analysing the distribution of trading flows across brokers and comparing it against objective performance metrics. If a trader was sending a significantly higher share of their volume to one broker without a clear justification, that anomaly would be flagged and discussed. The process ensured that broker relationships remained grounded in performance rather than personal preference, protecting both the bank and its clients. Working on this review gave me a direct understanding of how compliance and governance function in practice on a trading floor, and why they matter.
Profit and Loss (P&L) is the daily measure of how much money a trading desk has made or lost. It captures the combined effect of market movements, trading activity, and fees. In my role within the Markets COO team, the P&L was one of the most important indicators I worked with. Each morning, the desks produced a flash P&L report, and my team consolidated these figures to produce management summaries that were reviewed by senior leadership. I also contributed to the analysis of P&L trends over time, identifying which desks were performing above or below forecast and understanding the drivers behind deviations. I learned that P&L is not just a financial result: it is a real-time signal of how well a desk is managing its positions, its risks, and its client relationships. Monitoring P&L every day gave me a concrete and dynamic view of how financial markets translate into business performance.
If you are a student in business or finance thinking about a career in financial markets, this post can help you understand what to expect from a first experience on a trading floor. Many students have a strong theoretical background in finance but are uncertain about how these concepts translate into day-to-day work. Through my experience at HSBC, I discovered that even a non-front-office role offers an exceptional vantage point: by working within the Markets COO team, I was exposed to P&L reporting, liquidity management, broker reviews, and budget processes that are central to how a bank manages its markets activities.
This post is also relevant if you are considering roles in Markets COO, Business Management, or Finance Control within a bank. These positions are often overlooked by students who focus exclusively on trading or sales, yet they offer direct exposure to the full scope of a markets division and are increasingly valued as a stepping stone toward front-office responsibilities. Whatever your target role, understanding how a trading floor operates, its rhythms, its pressures, and its culture, will give you a real advantage in interviews and on the job.
▶ Max ODEN Leveraged Finance: My Experience as an Analyst Intern at Haitong Bank
▶ Praduman AGRAWAL My Professional Experience as a Quantitative Analyst Intern at Findoc Financial Services
▶ Michel Henry VERHASSELT Trading strategies based on market profiles and volume profiles
▶ Mickael RUFFIN My Internship Experience as a Structured Finance Analyst at Société Générale
HSBC — Corporate and Institutional Banking (including Markets and Securities Services)
ESMA — European Securities and Markets Authority
BIS — OTC Derivatives Statistics
The article was written in July 2026 by Abel ARAYA (ESSEC Business School, Master in Finance, 2025).
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In this article, Axel RUDLOFF (founder and President of Koragence, and a student at ESSEC Business School, Grande Ecole Program – Master in Management (MiM), 2025–2029) shares observations drawn from building and managing software engineering projects involving system integration, artificial intelligence (AI), DevOps and custom business applications. The article explains why a software project is not only a technical investment: it is also a managerial decision about productivity, risk, operating processes and capital allocation.
Companies rarely decide to develop custom software when existing systems no longer support the way the business actually operates. An enterprise resource planning system (ERP), such as SAP, may need to exchange data with a customer relationship management system (CRM), supplier application programming interfaces (APIs), document-management tools or applications developed internally. Difficulties arise when these systems must share data reliably, enforce specific business rules and support critical processes without repeated manual entry.
At Koragence, the projects I supervise often involve synchronising ERPs, integrating partner APIs, automating document processing with AI, rebuilding business workflows, or implementing DevOps architectures. DevOps refers to the practices and tools used to automate, secure and monitor the development and operation of software. Depending on the business criticality of a project, the architecture may target 99.95% availability, or up to 99.99% when the infrastructure supports multi-region redundancy, load balancing and advanced incident-recovery mechanisms.
Koragence is a French digital-services company specialising in custom software, web engineering, and process automation. The company designs business applications, software-as-a-service (SaaS) platforms, internal tools, customer portals and technical integrations for companies, startups, associations and public organisations.
Koragence has delivered more than 20 projects for clients located in more than six countries and works with a network of more than 50 active collaborators and partner companies. This model makes it possible to assemble a team according to the specific needs of each assignment, including software development, user experience and user interface design (UX/UI), cloud infrastructure, cybersecurity, accessibility and data engineering.
Logo of Koragence.
Source: the company.
As founder and President, I am responsible for turning a business problem into a project that can be delivered economically and technically. This includes business development, qualification of client needs, project scoping, pricing, team selection, contractual discussions, delivery governance and long-term client relationships.
I also remain directly involved in product and technical decisions. On a typical assignment, I help identify the critical workflow, define the minimum useful scope, choose which components should be standard and which should be custom, coordinate the specialists involved, monitor delivery and manage relationships with clients.
My work can be divided into five areas: identifying operational problems with measurable business consequences; translating those problems into functional and technical features; building the right project team; controlling scope, budget, quality and delivery risk; and ensuring that the software creates value after deployment rather than becoming an additional tool that employees do not use.
This role requires both technical and managerial skills. Technical knowledge is necessary to assess architecture, security, integrations, databases and infrastructure. Business knowledge is equally important because the best technical solution is not always the best investment. A founder must also understand margins, cash flow, pricing, negotiation, contractual risk and the opportunity cost of allocating a team to one project rather than another.
The most important soft skills are active listening, synthesis, communication and decision-making under uncertainty. Clients rarely describe their problem in technical terms. It is therefore necessary to distinguish symptoms from root causes, challenge assumptions with diplomacy, and explain trade-offs to both technical and non-technical stakeholders.
The main lesson is that the quality of the initial diagnosis, the clarity of responsibilities and the realism of the scope have a major impact on project cost and delivery.
One of the most frequent problems is the repeated entry of the same information into several systems. In one project for an industrial group with several hundred employees, a team spent more than 1,600 hours per year re-entering information from supplier catalogues into an internal database. Beyond the labour cost, this process created entry errors, inconsistencies between reference systems and disputes caused by contradictory information.
The software project removed this break in the information flow. Supplier data were collected automatically through standardized AI document extraction, checked for consistency and integrated into the internal SAP system. The economic value came from reducing labour and disputes with suppliers.
Suppliers increasingly expose APIs, ERPs provide connectors, CRMs publish webhooks and most business software can exchange data automatically. This creates opportunities, but every external connection also becomes a dependency. A partner API may change version, become unavailable or modify its behaviour. Reliable software therefore requires monitoring, logging, error recovery, security controls and ongoing maintenance.
In another project, Koragence developed a platform capable of supervising more than 15 industrial machines through supplier APIs while centralising more than 1,100 alerts and distributing notifications across several channels. The architecture was designed to scale to 100 machines without replacing the underlying technical model.
This type of project illustrates why integration is a strategic issue. The system must not only work on launch day; it must continue to work when the number of users, machines, documents or external dependencies increases. Maintenance is therefore part of the investment decision from the beginning.
In the past, automating supplier catalogues, technical data sheets, invoices and other documents required highly standardised formats and rules written separately for each source. In practice, these standards were often incomplete or inconsistently applied. AI models can now interpret a much wider variety of documents, extract useful information and feed databases or business software. This makes some automations faster and less expensive to implement.
However, AI does not remove the need for architecture, control and human judgment. A 2026 study of more than 100,000 software developers found that autonomous coding agents produced very large gains in coding activity, but smaller gains at the level of completed projects and actual releases. The authors interpret this difference as evidence that human and organisational bottlenecks still limit final output. In other words, writing code faster does not automatically mean shipping reliable software faster.
This distinction matters for managers. AI can lower the marginal cost of implementation, but the investment remains rational only when the company has correctly defined the workflow, data model, responsibilities, controls and expected return. The decision is therefore not “AI or no AI”; it is how AI can be integrated into a dependable operating system.
I present below three concepts that are directly connected to my work at Koragence: transaction costs and the make-or-buy decision, return on investment and payback period, and operating leverage through reusable software assets.
A company can buy standard software, adapt an existing product, outsource a custom development or build internally. The licence price is only one part of the decision. Managers must also consider transaction costs: integration work, manual reconciliation, training, vendor coordination, contract management, switching costs and the risk created by dependence on a supplier.
Standard software is usually preferable when the process is common and the product already satisfies the essential requirements. Custom software becomes more rational when the workflow is strategically important, highly specific, poorly served by standard tools or expensive to operate manually. My role is to help clients compare these alternatives rather than assume that custom development is always the correct answer.
Return on investment (ROI) compares the economic gains generated by a project with its total cost. For software, the benefits may include labour hours saved, fewer errors, faster sales cycles, reduced downtime, better compliance or additional revenue. The cost must include not only development, but also hosting, maintenance, training and change management.
The payback period measures how long it takes for cumulative benefits to recover the initial investment. For example, if an automation costs €40,000 and creates €5,000 of measurable monthly savings, its simple payback period is eight months. This calculation is not sufficient on its own, but it creates a common language between operational teams, finance teams and technical providers.
Software can create operating leverage because the same technical system can support a higher volume of transactions without a proportional increase in labour. A platform designed for 15 machines and capable of supporting 100 machines illustrates this principle: the client can grow while avoiding the need to multiply manual supervision at the same rate.
The same logic applies to Koragence. Reusable components, documented deployment processes, quality controls and specialist partnerships reduce the cost and risk of future projects. Nevertheless, reuse must not become generic copy-and-paste delivery. The objective is to standardise the reliable foundations while preserving the business-specific layer that creates value for each client.
This topic is relevant to students interested in entrepreneurship, corporate finance, consulting, operations, private equity or digital transformation. Software investment decisions increasingly affect company valuation, operating margins, scalability and risk. Understanding these projects therefore requires more than technical knowledge.
For a finance student, custom software provides a concrete example of capital allocation: management commits resources today in exchange for expected future cash flows, cost savings or strategic flexibility. For a future consultant or entrepreneur, the article also shows why technology projects must be framed around measurable business outcomes rather than features alone.
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Brynjolfsson, E., Li, D., & Raymond, L. R. (2025). Generative AI at Work , The Quarterly Journal of Economics, 140(2), 889–942.
Demirer, M., Musolff, L., & Yang, L. (2026). Writing Code vs. Shipping Code: Productivity Effects Across Generations of AI Coding Tools , NBER Working Paper No. 35275.
Williamson, O. E. (1989). Transaction Cost Economics , in Handbook of Industrial Organization, Volume 1, 135–182.
Lacity, M. C., Khan, S. A., & Willcocks, L. P. (2016). The role of Transaction Cost Economics in Information Technology Outsourcing research: A meta-analysis of the choice of contract type , The Journal of Strategic Information Systems, 25(1), 32–48.
Koller, T., Goedhart, M., & Wessels, D. (2020). Valuation: Measuring and Managing the Value of Companies, Seventh Edition, Hoboken (NJ), John Wiley & Sons.
Koragence — When custom software still makes sense in 2026
National Bureau of Economic Research — AI coding tools and software delivery productivity
The article was written in July 2026 by Axel RUDLOFF, founder and President of Koragence and a student at ESSEC Business School, Grande Ecole Program – Master in Management (MiM), 2025–2029.
▶ Discover all articles by Axel RUDLOFF.
In this article, Hubert RODARIE (Honorary President of the French Association of Institutional Investors — Af2i) introduces his latest book, Europe Confronting Trump, published by ESKA in May 2026.
This question lies at the heart of the analyses circulating in the media. Should Donald Trump’s election be viewed as an anomaly? Are all his decisions destined to be overturned by a new president, who will then be portrayed as responsible and serious?
Since 2016, the debate regarding Donald Trump has focused primarily on his personality, his provocations, and his excessive behavior. Most of his actions are described as erratic, questionable, and, above all, ineffective. Yet a fundamental question arises: how can we explain the bewilderment currently gripping Democrats in the United States, as well as leaders in the European Union and Washington’s Asian allies? A second question follows: how can we explain that, despite the Supreme Court’s overturning of the tariffs, the agreements reached in exchange for their adjustment or elimination have not been called into question but, on the contrary, have been confirmed (see, in particular, the European Parliament’s recent decision ratifying the July 2025 agreements)?
On May 14, 2026, L’Europe face à Trump (Europe Facing Trump) was published by ESKA. This book aims precisely to move beyond a superficial interpretation of events.
L’Europe face à Trump by Hubert Rodarie.
Source : ESKA Editions.
The author first demonstrates that Trump is neither an anomaly nor merely a media phenomenon. He derives his power from his ability to rally a majority of Americans who are currently dissatisfied with their living conditions. This ability is characteristic of demagogues, one of the most famous of whom was Pericles, yet he is considered one of the fathers of democracy. Trump and Pericles, the United States and Athens, do indeed share many similarities. Moreover, Trump relies heavily, in his actions, on a long-standing trend toward the concentration of executive powers at the federal level in the hands of the president. Yet this trend is common to both Democratic and Republican administrations. Trump thus acts by both mobilizing popular support and exploiting the underlying logic of the United States’ political organization.
Next, the author analyzes the foundations of a strategic project driven by a single ambition: to rebuild an autonomous American power. Unlike previous administrations, which continued the policies pursued over several decades, this strategy rests on three pillars that can be described as innovative:
This therefore represents a historic attempt to overturn not only the principles that have governed the U.S. economy and its relations with the rest of the world for the past forty years, but also the balance of power within the U.S. executive branch, by further strengthening federal power relative to that of the states. In short, a radical transformation of American society is underway.
In parallel, this book examines the European Union’s capacity to respond, its vulnerabilities, and the essential transformation it must undertake in the face of this upheaval. Faced with a strategy that is as clear as it is unapologetic, the European Union must assert itself. But is it capable of doing so? Do its leaders truly have the will to do so?
An incisive essay that helps us understand, beyond the turmoil, the logic behind a major transformation.
This article was written in July 2026 by Hubert RODARIE (Honorary President of the French Association of Institutional Investors — Af2i).
▶ Discover all articles by Hubert RODARIE.
Dans cet article, Hubert RODARIE (président d’honneur de l’Association française des investisseurs institutionnels — Af2i) présente son dernier ouvrage, L’Europe face à Trump, paru en mai 2026 aux éditions ESKA.
Trump sera-t-il un accident de l’Histoire ? Cette question est au cœur des analyses véhiculées par les médias. Faut-il considérer l’élection de Donald Trump comme une anomalie ? Toutes ses décisions sont-elles vouées à être abrogées par un nouveau président, présenté alors comme responsable et sérieux ?
Depuis 2016, le débat autour de Donald Trump s’est principalement focalisé sur sa personnalité, ses provocations et ses excès. La plupart de ses actions sont décrites comme erratiques, contestables et, surtout, inefficaces. Pourtant, une première interrogation s’impose : comment expliquer la forme de sidération qui frappe aujourd’hui les démocrates aux États-Unis, ainsi que les dirigeants de l’Union européenne et des pays asiatiques alliés de Washington ? Une seconde question en découle : comment expliquer que, malgré l’annulation des droits de douane par la Cour suprême, les accords obtenus en contrepartie de leur modulation ou de leur suppression ne soient pas remis en cause, mais au contraire confirmés — voir notamment la récente décision du Parlement européen entérinant les accords de juillet 2025 ?
Le 14 mai 2026 est paru L’Europe face à Trump aux éditions ESKA. Cet ouvrage se propose précisément de rompre avec une lecture trop superficielle des événements.
L’Europe face à Trump, par Hubert Rodarie.
Source : Éditions ESKA.
L’auteur montre d’abord que Trump n’est ni une anomalie ni un simple phénomène médiatique. Son pouvoir, il le tient de sa capacité à rallier une majorité d’Américains aujourd’hui insatisfaits de leurs conditions de vie. Cette capacité est celle des démagogues, dont l’un des plus célèbres fut Périclès, pourtant considéré comme l’un des pères de la démocratie. Trump et Périclès, les États-Unis et Athènes présentent effectivement de nombreux points communs. Plus encore, Trump s’appuie largement, pour agir, sur une tendance séculaire à la concentration des pouvoirs exécutifs au niveau fédéral entre les mains du président. Or, cette évolution est commune aux administrations démocrates comme républicaines. Trump agit donc à la fois en mobilisant le soutien populaire et en exploitant les logiques profondes de l’organisation politique des États-Unis.
Dans un second temps, l’auteur analyse les fondements d’un projet stratégique porté par une ambition : reconstruire une puissance américaine autonome. Contrairement aux mandats précédents, qui s’inscrivaient dans la continuité des politiques menées depuis plusieurs décennies, cette stratégie repose sur trois piliers que l’on peut qualifier d’innovants :
Il s’agit donc d’une tentative historique de renverser non seulement les principes qui régissent depuis quarante ans l’économie américaine et ses relations avec le reste du monde, mais aussi l’équilibre du pouvoir exécutif américain, en renforçant encore le pouvoir fédéral face à celui des États fédérés. En somme, une transformation radicale de la société américaine est engagée.
En miroir, l’ouvrage interroge la capacité de réaction de l’Union européenne, ses fragilités et la mutation indispensable qu’elle doit entreprendre face à cette rupture. Face à une stratégie aussi claire qu’assumée, l’Union européenne doit s’affirmer. Mais en est-elle capable ? Ses dirigeants en ont-ils réellement la volonté ?
Un essai incisif pour comprendre, au-delà du tumulte, la logique d’une transformation majeure.
Cet article a été écrit en juillet 2026 par Hubert RODARIE (président d’honneur de l’Association française des investisseurs institutionnels — Af2i).
▶ Découvrir tous les articles de Hubert RODARIE
In this article, Frédéric VALOGNES, lecturer, author and Certified European Financial Analyst (CEFA®), examines whether the dynamics of the implied volatility surface may provide a decision-support framework for systematic cash-secured put strategies.
The Black-Scholes-Merton model remains one of the most influential developments in modern financial economics. Whilst its mathematical formulation continues to provide the benchmark for pricing European options, one of its central assumptions — namely that volatility remains constant throughout the life of an option — is persistently contradicted by observed market prices.
Rather than constituting a weakness of the model, these discrepancies reveal valuable information regarding investors’ expectations, market sentiment and the pricing of downside risk. The resulting volatility skews and smiles have therefore become essential components of both academic research and professional option trading.
This paper argues that the implied volatility surface should not be viewed solely as a pricing adjustment. Its geometry and, more importantly, its evolution over time may provide additional information capable of assisting investment decisions. Attention is devoted to cash-secured short put strategies, for which the level of implied volatility alone frequently proves insufficient.
Drawing upon preliminary observations obtained from listed CAC 40 index options across several maturities, the article explores whether the dynamics of the implied volatility surface may constitute a useful decision-support indicator. Rather than proposing a predictive pricing model, the objective is to examine whether changes in the shape, slope and term structure of implied volatility can contribute to a more disciplined framework for identifying favourable market environments in which to initiate systematic cash-secured put strategies.
Within option markets, implied volatility occupies a rather singular position. Originally introduced as the unknown parameter required to reconcile observed option prices with the Black-Scholes-Merton valuation model, it has progressively evolved from a purely technical pricing input into one of the most closely monitored indicators in financial markets. Today, implied volatility is commonly interpreted not simply as a pricing parameter, but as a market-based measure of uncertainty, reflecting the aggregate expectations of thousands of market participants.
For investors employing cash-secured short put strategies, that is, selling put options while maintaining sufficient cash reserves to purchase the underlying asset if assignment occurs, implied volatility plays an obvious practical role. Higher implied volatility generally translates into higher option premiums, thereby increasing the potential income associated with selling options. This simple observation has encouraged many practitioners to associate elevated implied volatility with favourable selling opportunities.
Experience, however, suggests that such a conclusion is frequently incomplete. Periods characterised by exceptionally high implied volatility often coincide with episodes of considerable financial stress, during which uncertainty continues to increase and option premiums expand further. Entering short option positions solely because implied volatility appears elevated may therefore expose investors to significant mark-to-market losses before market conditions eventually stabilise.
The question addressed in this article is therefore slightly different.
Rather than asking whether implied volatility is high, it may be more appropriate to ask whether the behaviour of the implied volatility surface itself contains additional information capable of assisting investment decisions.
More specifically, can the dynamics of the implied volatility surface, particularly the evolution of the downside volatility skew, provide useful information regarding changing market conditions? Since deep out-of-the-money put options typically incorporate a substantial premium reflecting institutional demand for portfolio insurance, does a progressive flattening of the skew signal that market stress is easing while option premiums remain comparatively attractive?
If such behaviour can be observed consistently, the volatility surface ceases to be merely an output of an option pricing model. Instead, it becomes a potential decision-support framework, capable of complementing more traditional criteria such as premium level, strike selection or time to maturity.
The purpose of the present article is not to challenge the theoretical foundations of the Black-Scholes-Merton model. On the contrary, the model remains indispensable, since implied volatility itself is extracted from its pricing equation. The objective is rather to investigate whether the systematic departures observed between theoretical assumptions and market prices may themselves convey exploitable information through the dynamics of the implied volatility surface, thereby supporting decisions regarding option selection, strike prices, market conditions and the implementation of systematic cash-secured put strategies.
Figure 1. Options transfer market risk between two counterparties with fundamentally different expectations. Whilst the buyer acquires protection against adverse price movements, the seller receives an option premium in exchange for assuming the corresponding contingent obligation. This transfer of risk constitutes the economic foundation upon which option markets operate and explains the central role played by option premiums in systematic short-put strategies.
The following sections revisit the theoretical foundations of implied volatility before examining why market observations systematically depart from the assumptions of constant volatility. Attention is subsequently devoted to the informational content embedded within volatility skews and smiles, leading to the introduction of a practical analytical framework intended to investigate whether changes in the implied volatility surface may contribute to the identification of favourable environments for systematic cash-secured put-selling.
Since its publication in 1973, the Black-Scholes-Merton model has become one of the most influential achievements in financial economics. Beyond providing a closed-form solution for the valuation of European options, it established a rigorous mathematical framework linking derivative prices to the stochastic behaviour of the underlying asset. More than half a century later, despite the emergence of increasingly sophisticated numerical models, Black-Scholes remains the common language of option markets.
Its enduring success stems from the remarkable intuition underlying the model. Rather than attempting to forecast future prices directly, Black-Scholes demonstrates that an option may be replicated through a continuously adjusted portfolio combining the underlying asset and a risk-free investment. Under a specific set of assumptions, this replication argument leads to a unique theoretical option value independent of investors’ individual expectations.
These assumptions are well known. Asset prices are assumed to follow a geometric Brownian motion with constant volatility. Markets are perfectly liquid and frictionless, allowing continuous trading without transaction costs or taxes. Interest rates remain constant throughout the life of the contract, whilst European options can only be exercised at maturity. Finally, market participants are assumed to behave rationally and possess homogeneous expectations.
From a practical perspective, few of these assumptions are fully satisfied in real financial markets. Transaction costs exist, volatility varies continuously, liquidity fluctuates and investors frequently react in heterogeneous ways to new information. Nevertheless, the model remains extraordinarily useful because it provides a coherent reference framework from which market observations may subsequently be interpreted.
One of its most significant contributions lies in the concept of implied volatility. Rather than treating volatility as an observable market variable, the Black-Scholes equation can be solved inversely. By inserting the observed option premium together with the remaining market parameters, it becomes possible to determine the level of volatility required for the theoretical model to reproduce the market price exactly. This inferred quantity is known as implied volatility.
Implied volatility therefore represents considerably more than a simple mathematical parameter. It embodies the level of uncertainty collectively embedded within option prices by market participants. Every quoted option premium implicitly reflects the market’s assessment of future price variability, making implied volatility one of the most informative indicators available to option traders.
Yet an important observation immediately follows. If the assumptions of the Black-Scholes model were perfectly satisfied, every option sharing the same maturity would exhibit the same implied volatility, irrespective of its strike price. Reality tells a rather different story.
Figure 2. A call option grants its holder the right, but not the obligation, to purchase the underlying asset at a predetermined strike price. Conversely, a put option grants the right to sell the underlying asset under identical contractual conditions. In both cases, the buyer acquires a right by paying an option premium, whilst the seller receives that premium in exchange for assuming the corresponding contingent obligation.
The original formulation of Black-Scholes implicitly assumes that volatility constitutes a characteristic of the underlying asset itself. If this were strictly true, every option written on the same asset and sharing an identical maturity would produce the same implied volatility once observed market prices are introduced into the valuation equation.
Empirical evidence has demonstrated otherwise. When implied volatilities are computed across a range of strike prices, they rarely remain constant. Instead, they exhibit systematic patterns whose shape varies according to both the underlying asset and prevailing market conditions. These observations, initially regarded as anomalies, have gradually become recognised as fundamental characteristics of option markets. The discrepancy is not accidental. It reflects the collective behaviour of investors rather than any mathematical imperfection within the pricing equation itself.
Institutional investors, pension funds and asset managers frequently purchase out-of-the-money put options to protect equity portfolios against severe market declines. This persistent demand for downside insurance increases put premiums relative to those predicted under constant volatility assumptions. Consequently, implied volatilities extracted from these option prices become progressively higher as strike prices decrease.
The resulting asymmetry gives rise to what practitioners commonly describe as the volatility skew. Rather than representing a flaw in Black-Scholes, the skew reveals how financial markets collectively price extreme downside events. It therefore provides direct insight into investors’ perception of risk, their appetite for protection and the relative scarcity of option sellers willing to assume such exposure.
Viewed from this perspective, implied volatility ceases to be merely an intermediate calculation. It becomes a market variable, capable of conveying valuable information regarding the balance between fear and confidence prevailing amongst market participants.
When implied volatilities are calculated across a range of strike prices for a given maturity, the resulting profile rarely corresponds to the horizontal line predicted by the Black-Scholes-Merton model. Instead, distinct empirical patterns emerge according to both the underlying asset and prevailing market conditions.
The earliest observations concerned currency and commodity options, where implied volatility frequently followed a symmetrical U-shaped profile. Deep in-the-money and deep out-of-the-money options exhibited higher implied volatilities than contracts whose strike prices were close to the prevailing market price. This phenomenon rapidly became known as the volatility smile, reflecting the characteristic curvature obtained when implied volatilities were plotted against strike prices.
The market crash of October 1987 marked a decisive turning point in option pricing. Following the unprecedented decline in global equity markets, practitioners observed that the Black-Scholes-Merton assumption of constant volatility no longer matched market prices. Implied volatilities began to differ substantially across strike prices, particularly for downside put options, reflecting investors’ increased demand for protection against extreme losses. Rather than attempting to force market prices into a single volatility parameter, traders progressively adopted the implied volatility surface itself as the practical input for option valuation. Since then, the smile and, even more prominently, the volatility skew have become standard features of option markets and indispensable tools for pricing, hedging and risk management.
Although initially regarded as an anomaly, the volatility smile gradually became recognised as a natural consequence of market behaviour rather than a failure of financial theory. Financial returns do not follow the perfectly lognormal distribution assumed by the Black-Scholes-Merton framework. Instead, empirical distributions exhibit heavier tails, occasional jumps and varying degrees of asymmetry, all of which contribute to systematic differences in implied volatility across strike prices.
Equity index options, however, generally display a markedly different pattern. Rather than producing a symmetrical smile, implied volatility typically increases as strike prices decrease. Conversely, call options with higher strike prices tend to exhibit progressively lower implied volatilities. The resulting profile no longer resembles a smile but rather a downward-sloping curve commonly referred to as the volatility skew.
This asymmetry is far from accidental. It reflects the structural demand for downside protection that characterises modern equity markets. Pension funds, insurance companies, institutional asset managers and other long-term investors regularly purchase out-of-the-money put options to protect diversified equity portfolios against severe market downturns. Such contracts effectively operate as insurance policies against extreme market events.
As demand for these protective puts increases, their market prices rise beyond the levels predicted by constant-volatility models. Once these prices are translated back into implied volatilities through the Black-Scholes equation, lower strike prices systematically exhibit higher implied volatility. The volatility skew therefore represents considerably more than a graphical curiosity. It provides a direct visual representation of how financial markets collectively price downside risk.
Rather than indicating that the Black-Scholes model has failed, the skew demonstrates that investors attribute different probabilities to upward and downward market movements. In practice, the cost of insuring against a sharp decline is significantly greater than the cost of participating in an equally pronounced upward movement. For option sellers, this distinction is of particular importance.
The additional premium associated with out-of-the-money put options constitutes the primary source of return for many systematic short-put strategies. Yet this additional premium simultaneously reflects the market’s perception of elevated downside risk. The option seller is therefore continuously confronted with a fundamental trade-off: richer premiums are generally accompanied by greater uncertainty.
Understanding this relationship represents the first step towards interpreting implied volatility not merely as a pricing parameter, but as a genuine source of market information.
Figure 3. Under the Black-Scholes assumption of constant volatility, implied volatility should remain identical across strike prices. Empirical observations reveal two distinct market structures: the volatility smile, historically observed in several currency option markets, and the downward volatility skew that characterises most equity index options.
Traditional option pricing theory treats implied volatility as a parameter required to value derivative contracts. Market practitioners increasingly adopt a rather different perspective. For many traders, implied volatility has progressively become an observable market variable.
Its level reflects the price investors collectively assign to uncertainty, whilst its distribution across strike prices reveals how that uncertainty is allocated between favourable and unfavourable market scenarios. This distinction is fundamental.
If all future price movements were regarded as equally probable, the volatility surface would remain broadly symmetrical. The persistent existence of a downward skew instead demonstrates that investors consistently attribute a greater economic significance to adverse market movements than to equivalent upward fluctuations. In this respect, the volatility skew may be interpreted as a continuously updated measure of collective risk aversion.
Unlike conventional market indicators, which frequently rely upon historical observations, implied volatility incorporates forward-looking expectations embedded directly within option prices. Every transaction reflects the judgement of buyers and sellers regarding future uncertainty. The resulting volatility surface therefore aggregates thousands of independent market assessments into a single observable structure. From the perspective of a systematic put seller, the implications are immediate.
Periods during which the skew becomes exceptionally steep frequently coincide with heightened demand for downside protection. Conversely, a gradual flattening of the skew may indicate that the market is beginning to reassess the likelihood of extreme adverse scenarios.
The central hypothesis explored throughout the remainder of this article is based precisely upon this observation. Rather than considering implied volatility in isolation, greater attention may usefully be devoted to the evolution of the entire volatility surface.
For most option practitioners, implied volatility is primarily regarded as a pricing variable. Whether calculated directly from market quotations or displayed by professional trading platforms, it is generally interpreted as a measure of the market’s expectation of future uncertainty. Consequently, trading decisions often rely upon a relatively simple observation: higher implied volatility produces higher option premiums.
For investors writing cash-secured puts, this relationship is naturally attractive. Selling options during periods of elevated implied volatility allows the collection of larger premiums whilst maintaining identical contractual obligations. Yet this apparent advantage immediately raises a practical difficulty.
Periods characterised by elevated implied volatility rarely occur in isolation. They are frequently associated with deteriorating market sentiment, increasing downside risk and heightened investor demand for protection. In such circumstances, high option premiums merely compensate sellers for assuming substantially greater uncertainty. The absolute level of implied volatility therefore provides only a partial description of market conditions. A more informative question may instead concern the behaviour of implied volatility itself.
Is the volatility surface continuing to steepen? Has it reached a plateau? Or has it begun to return progressively towards more stable market conditions?
These questions introduce an important distinction between two different approaches to option selling. The first consists simply of identifying expensive options based on their implied volatility. The second seeks to determine whether market conditions themselves have begun to evolve in favour of the option seller. The distinction is subtle but potentially significant.
A market characterised by high implied volatility, and an increasingly steep volatility skew reflects persistent demand for downside protection. Under such circumstances, option premiums may continue to increase despite already appearing historically elevated.
Conversely, if implied volatility remains relatively high whilst the overall structure of the volatility surface begins to normalise, market expectations may be undergoing a gradual transition. Although uncertainty remains elevated, the balance between buyers and sellers of protection may already be changing.
From the perspective of a systematic option seller, such an environment appears fundamentally different. The option premium remains attractive, yet the dynamics of market expectations may already be evolving towards greater stability. This observation forms the central hypothesis explored in the present work.
Rather than evaluating implied volatility solely through its absolute level, the proposed approach investigates whether the progressive normalisation of the implied volatility surface may itself constitute useful information capable of assisting the timing of cash-secured short put strategies.
Importantly, this hypothesis should not be interpreted as an attempt to forecast future market prices. No volatility model can predict future market movements with certainty. Instead, the objective is considerably more modest.
The purpose is to investigate whether the collective information continuously embedded within option prices can be organised into a coherent analytical framework capable of improving the selection of favourable option-selling environments.
Figure 4. The proposed framework focuses less on the absolute level of implied volatility than on the evolution of the volatility surface itself. A gradual normalisation of the skew whilst option premiums remain comparatively elevated may provide a more favourable environment for initiating systematic cash-secured put positions.
The preceding discussion naturally raises a practical question: if the geometry of the implied volatility surface reflects the collective assessment of market risk, can its evolution also provide useful information regarding the timing of option-selling strategies?
This question forms the starting point of the present investigation. Rather than considering implied volatility as a static variable observed at a single point in time, the proposed framework examines the volatility surface as a dynamic structure whose characteristics evolve continuously in response to changing market expectations. The distinction is important.
Most market participants focus primarily on the absolute level of implied volatility. Elevated implied volatility is generally interpreted as an opportunity to collect richer option premiums, whilst low implied volatility often discourages option-selling strategies. Such reasoning, however, overlooks an essential aspect of market behaviour.
Two market environments may exhibit comparable average implied volatilities whilst reflecting fundamentally different underlying conditions.
In the first case, implied volatility may still be increasing, accompanied by a progressively steeper volatility skew and a persistent demand for downside protection. In the second, implied volatility may remain elevated, but the volatility surface itself may already be beginning to stabilise, suggesting that market participants are gradually reassessing the probability of extreme downside events.
From the perspective of a systematic put seller, these two situations should not necessarily be regarded as equivalent. Although option premiums may appear imilarly attractive, the evolution of collective market expectations differs substantially.
The working hypothesis explored throughout this study is therefore deliberately modest. Rather than attempting to predict future market prices, the objective is to determine whether the progressive normalisation of the implied volatility surface may provide additional information capable of assisting the selection of favourable market environments for initiating cash-secured short put positions.
In this respect, the volatility surface is not viewed as a forecasting instrument. Instead, it is interpreted as a continuously updated representation of market sentiment whose evolution may contribute to a more disciplined investment process.
Figure 5. General workflow of the proposed analytical framework. Market option prices are first converted into implied volatilities using the Black-Scholes-Merton model. The resulting volatility surface is subsequently analysed through a series of descriptive indicators before being interpreted within a decision-support framework for systematic cash-secured put strategies.
The methodology developed in this work follows a sequence of analytical steps intended to transform raw market quotations into interpretable market indicators.
The process begins with the systematic collection of listed option prices for a given underlying asset and maturity. Preference is given to highly liquid option contracts to minimise distortions resulting from wide bid-ask spreads or infrequent trading activity.
Observed market premiums are then converted into implied volatilities through the inverse application of the Black-Scholes-Merton pricing equation. Once computed across the available strike prices, these implied volatilities collectively define the observed volatility surface for the selected maturity.
Rather than analysing each implied volatility independently, several global characteristics of the surface are examined simultaneously.
Attention is devoted to:
The purpose of this multidimensional approach is to characterise market conditions more comprehensively than would be possible through the observation of implied volatility alone. Naturally, not all option markets exhibit comparable behaviour.
The preliminary investigations presented in this article suggest that market liquidity and option maturity play a decisive role in determining the regularity of the resulting volatility surface. Highly liquid equity index options with medium- to long-term maturities appear particularly well suited to this type of analysis, whereas shorter maturities or less actively traded underlying assets may generate substantially noisier implied volatility structures.
These observations should not be interpreted as definitive conclusions. Rather, they provide an empirical motivation for the exploratory analyses presented in the following section.
Figure 6. Illustrative workflow describing the successive stages of the proposed methodology: market data acquisition, implied volatility computation, volatility surface construction, statistical charac-terisation and decision-support interpretation.
The analytical framework presented above was subsequently applied to listed option data to examine whether the proposed interpretation of the implied volatility surface could be observed under actual market conditions.
At this stage, the objective was not to perform an exhaustive statistical validation of the methodology. Rather, the purpose was to investigate whether the dynamics of the implied volatility surface exhibited sufficiently regular behaviour to justify further quantitative analysis.
Several option chains were therefore examined, covering different underlying assets and maturities.
Attention was devoted to the CAC 40 index, whose option market offers a high level of liquidity across a broad range of strike prices. Additional observations were conducted on selected individual equities to assess the robustness of the approach under different market conditions.
The first observation concerns the influence of option maturity.
Short-dated options, particularly those approaching expiration, frequently generated irregular implied volatility profiles. Individual quotations occasionally produced local distortions, whilst relatively small pricing discrepancies resulted in disproportionately large variations in calculated implied volatility. Such behaviour appears consistent with the increasing influence of time decay and the reduced amount of remaining time value as maturity approaches.
Consequently, short maturities should be interpreted with caution when constructing continuous volatility surfaces. A markedly different picture emerged for longer maturities.
Options with approximately six months to one year remaining until expiration generally produced substantially smoother implied volatility structures. The resulting volatility skews exhibited the regular downward slope commonly described in the empirical literature, with only limited local distortions across neighbouring strike prices.
These observations proved particularly apparent for the CAC 40 index.
The high liquidity of the option market appeared to facilitate a more stable estimation of implied volatility, thereby providing a significantly more coherent representation of the underlying volatility surface. An equally important observation concerns the distinction between index options and individual equity options.
Whilst the CAC 40 generated relatively stable and interpretable volatility structures, several individual equities produced substantially noisier results. In certain cases, isolated market quotations generated implausibly high or even negative implied volatility estimates, suggesting either temporary pricing inconsistencies or insufficient market liquidity.
Such observations reinforce an important practical consideration.
The proposed methodology appears particularly well suited to highly liquid option markets where quoted premiums reflect continuous interaction between buyers and sellers. Conversely, less liquid markets may introduce local pricing distortions capable of obscuring the global characteristics of the volatility surface.
These preliminary observations do not constitute definitive statistical conclusions.
Nevertheless, they suggest that both liquidity and maturity represent essential prerequisites when analysing implied volatility surfaces for decision-support purposes.
Figure 7. Comparison of implied volatility curves obtained for different maturities. Short-dated maturities frequently exhibit irregular local behaviour owing to limited time value and increased pricing sensitivity. Longer maturities generally produce smoother volatility skews, thereby facilitating the interpretation of surface dynamics.
A further observation emerged during the analysis: although several volatility surfaces displayed the expected downward skew, not all of them generated identical decision-support signals.
Certain maturities exhibited a progressive flattening of the skew whilst implied volatility remained at comparatively elevated levels. Others retained a persistent steep slope despite similar average volatility levels. This distinction proved particularly informative. If confirmed through broader empirical investigation, it suggests that the overall geometry of the volatility surface may contain additional information beyond the absolute level of implied volatility alone. From the perspective of systematic option selling, this observation may prove significant.
A market characterised by elevated implied volatility, and a progressively normalising volatility surface appears fundamentally different from one in which both implied volatility and downside protection demand continue to increase simultaneously.
The former may correspond to a market gradually returning towards equilibrium. The latter may still reflect an environment dominated by uncertainty.
Consequently, analysing the dynamics of the volatility surface rather than its static characteristics alone may provide a richer description of prevailing market conditions.
The following section illustrates how these observations may be translated into a practical decision-support framework for systematic cash-secured put strategies.
The preliminary observations presented above suggest that the practical usefulness of the implied volatility surface depends upon two essential conditions: the quality of market data and the maturity of the option contracts under consideration.
The first point appears relatively intuitive.
Implied volatility is not directly observable. It is inferred from quoted option prices through the inverse application of the Black-Scholes-Merton model. Consequently, any inconsistency in market quotations is immediately reflected in the calculated implied volatilities.
This phenomenon proved particularly evident during the exploratory analyses conducted on individual equities.
Whilst certain option chains generated coherent volatility structures, others produced isolated implied volatility values that were incompatible with neighbouring strike prices. In a limited number of cases, implausible or unstable implied volatility estimates were obtained despite apparently valid market quotations. Such behaviour most likely reflects temporary liquidity deficiencies, unusually wide bid-ask spreads or isolated transactions executed outside normal market conditions.
These observations underline an important methodological requirement.
The proposed framework should preferably be applied to option markets characterised by sufficient liquidity and a broad distribution of actively traded strike prices. Under such conditions, quoted premiums are more likely to represent the consensus valuation of market participants rather than isolated transactions.
The second observation concerns option maturity.
Short-dated contracts frequently produced irregular volatility profiles whose local fluctuations appeared dominated by pricing noise rather than genuine changes in market expectations. As expiration approaches, the remaining time value becomes progressively smaller, and option prices exhibit increasing sensitivity to relatively minor changes in the underlying asset. Consequently, the resulting implied volatility estimates become substantially less stable.
Conversely, medium- and long-dated maturities generally generated considerably smoother volatility structures.
The downward skew remained clearly identifiable whilst local distortions became significantly less pronounced. This regularity considerably facilitated the interpretation of the surface and its evolution over successive market observations.
Among the datasets examined, listed CAC 40 index options consistently provided the most coherent results. Their combination of high liquidity, narrow bid-ask spreads and broad strike availability produced volatility surfaces whose overall geometry remained remarkably stable. This characteristic makes such instruments particularly well suited to exploratory research concerning the dynamics of implied volatility.
An additional observation deserves particular attention: not every regular volatility surface generated the same analytical conclusion.
Certain maturities displayed a progressive flattening of the volatility skew whilst implied volatility remained comparatively elevated. Others retained a persistent and pronounced downward slope despite exhibiting similar average volatility levels. This distinction appears especially interesting.
If future empirical analyses confirm these preliminary observations, the evolution of the volatility surface may provide information that cannot be obtained from the absolute level of implied volatility alone. Such a conclusion would carry practical implications for systematic option-selling strategies.
Rather than selecting opportunities exclusively according to premium levels or historical volatility, investors may benefit from incorporating the dynamics of the implied volatility surface into their broader decision-making process. Naturally, these findings should be interpreted with appropriate caution.
The present work remains exploratory in nature and does not claim to establish a predictive model. Instead, it proposes an analytical framework intended to organise market information already embedded within option prices into a more coherent decision-support process.
Further empirical investigation involving longer observation periods, multiple market regimes and additional underlying assets will naturally be required before more general conclusions may be drawn.
Figure 8. Evolution of the implied volatility surface across successive market observations. The figure illustrates the conceptual distinction between a market in which the volatility skew continues to steepen and one in which the surface progressively normalises whilst implied volatility remains comparatively elevated.
From a practical perspective, the observations discussed throughout this article suggest that implied volatility should perhaps be interpreted less as an isolated numerical indicator and more as one component of a broader analytical framework.
Option sellers have traditionally focused on premium maximisation. Although this objective remains entirely legitimate, premium alone provides only a partial description of prevailing market conditions.
The same premium may arise under markedly different market environments. One may correspond to an increasingly stressed market characterised by rapidly rising demand for downside protection. Another may reflect a market in which uncertainty remains elevated but has already begun to stabilise.
Distinguishing between these situations may prove particularly valuable when implementing systematic cash-secured put strategies. Rather than attempting to forecast market direction, the proposed framework encourages a more disciplined interpretation of the information continuously embedded within option prices.
In this respect, the implied volatility surface becomes considerably more than a graphical representation of option quotations. It evolves into a dynamic indicator describing the collective perception of risk within financial markets.
The Black-Scholes-Merton model remains the fundamental reference upon which modern option pricing is built. Although one of its central assumptions — constant volatility — is systematically contradicted by market observations, these apparent discrepancies have progressively become one of the richest sources of information available to option practitioners.
The implied volatility surface should therefore not merely be regarded as a technical consequence of option pricing theory. It reflects the collective judgement of market participants regarding future uncertainty, the asymmetrical pricing of downside risk and the continuously evolving balance between buyers and sellers of financial protection. The purpose of the present study has been to explore whether this information may be exploited beyond its traditional pricing function.
Rather than concentrating exclusively on the absolute level of implied volatility, this article has proposed a broader analytical perspective based upon the dynamics of the entire volatility surface. Attention has been devoted to the progressive evolution of the volatility skew, whose gradual normalisation may provide additional insight into changing market conditions.
The preliminary empirical observations presented throughout this paper suggest practical conclusions.
First, market liquidity appears to constitute a fundamental prerequisite for obtaining sufficiently stable implied volatility surfaces. Highly liquid option markets, such as listed CAC 40 index options, produce considerably more coherent structures than many individual equity options, whose implied volatilities may occasionally be distorted by isolated transactions or limited trading activity.
Secondly, option maturity also plays a decisive role. Medium- and long-dated contracts generally generate smoother volatility surfaces that appear more suitable for structural analysis than very short-dated maturities, where the increasing influence of time decay frequently introduces substantial local irregularities.
Finally, and perhaps most importantly, the observations suggest that two markets exhibiting comparable average implied volatility levels may nevertheless convey markedly different information through the geometry of their respective volatility surfaces. This distinction may prove particularly relevant for systematic cash-secured put strategies.
Whilst elevated implied volatility undoubtedly increases option premiums, the progressive normalisation of the volatility surface may provide complementary information regarding the evolution of collective market expectations. The proposed framework should therefore not be interpreted as a predictive model. Financial markets remain inherently uncertain, and no analytical methodology can eliminate investment risk.
Instead, the approach presented here seeks to organise information already embedded within option prices into a structured decision-support framework capable of complementing more traditional valuation techniques. Viewed from this perspective, the implied volatility surface ceases to be merely a graphical representation of option prices. It becomes a dynamic description of market behaviour.
Understanding how this structure evolves through time may ultimately prove as informative as measuring its absolute level at any single observation date.
The present study should be regarded as an exploratory investigation rather than a definitive empirical validation. Several limitations naturally remain.
The observations reported here are based upon a limited number of underlying assets and observation dates. Broader empirical investigations covering multiple market regimes, longer historical periods and additional asset classes will be required before more general conclusions may be established. Future research could also investigate whether quantitative indicators describing the geometry of the implied volatility surface — such as skew slope, local curvature or cross-sectional dispersion — may be systematically incorporated into algorithmic decision-support models for option-selling strategies.
Another promising avenue concerns the comparative behaviour of implied volatility surfaces across different asset classes, including equity indices, individual equities, exchange-traded funds and commodity options.
Finally, machine learning techniques may eventually provide complementary tools capable of identifying recurring patterns within the evolution of volatility surfaces. Such approaches, however, should be viewed as extensions of the present analytical framework rather than substitutes for the economic interpretation of market behaviour. Ultimately, the principal contribution of this work lies less in proposing a new pricing model than in suggesting an alternative way of interpreting information already contained within option markets. If the geometry of the implied volatility surface indeed reflects the collective perception of financial risk, then monitoring its evolution may offer valuable additional insight into the timing of systematic option-selling strategies.
Readers wishing to retain a concise visual summary of the concepts presented throughout this article may download the accompanying high-resolution infographic below.
Download the Summary Infographic (High-Resolution PDF)
▶ Jayati WALIA Brownian Motion in Finance
▶ Jayati WALIA Black-Scholes-Merton option pricing model
▶ Saral BINDAL Implied Volatility and Option Prices
▶ Saral BINDAL Volatility curves: smiles and smirks
▶ Saral BINDAL Implied Volatility Surface: Smiles, Smirks and Term Structure
Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654.
Gatheral, J. (2006). The Volatility Surface: A Practitioner’s Guide. John Wiley & Sons.
Hull, J. C. (2024). Options, Futures and Other Derivatives (11th ed.). Pearson.
Merton, R. C. (1973). Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science, 4(1), 141-183.
Natenberg, S. (2015). Option Volatility and Pricing (2nd ed.). McGraw-Hill Education.
Rebonato, R. (2004). Volatility and Correlation: The Perfect Hedger and the Fox. John Wiley & Sons.
Taleb, N. N. (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. John Wiley & Sons.
Tis article was written in July 2026 by Frédéric VALOGNES , who is a lecturer in corporate finance, financial analysis, financial markets and derivatives, with more than twenty-five years of professional experience spanning financial management, higher education, research administration and executive training. He is a Certified European Financial Analyst (CEFA®), a professional designation awarded by the European Federation of Financial Analysts Societies (EFFAS), Frankfurt.
This article is intended solely for educational and research purposes. It presents the author’s personal reflections on implied volatility, option pricing and systematic option-selling strategies. It should not be construed as investment advice or as a recommendation regarding any financial instrument or trading strategy.
The ideas developed in this article are the result of many years of teaching, professional practice and ongoing research in corporate finance, financial analysis, financial markets and derivatives. They have also been enriched by numerous discussions with academics, finance professionals and market practitioners, whose expertise, critical insights and constructive exchanges have played an important role in shaping the analytical framework presented here.
The author wishes to express his sincere gratitude to all those who have contributed, directly or indirectly, to the development of these ideas. Their encouragement, intellectual generosity and commitment to rigorous financial analysis have been a constant source of inspiration.
In this article, Saral BINDAL (Indian Institute of Technology Kharagpur, Metallurgical and Materials Engineering, 2024-2028 & Research assistant at ESSEC Business School) explains the implied volatility surface: its characteristic shapes, static arbitrage constraints, and application using S&P 500 index options data.
In financial markets characterized by uncertainty, volatility is a crucial factor in the valuation of derivative securities. For options traders, an option price is essentially volatility. Although options are traded at monetary prices, professionals routinely quote and compare them in terms of implied volatility (%), making volatility the common language of options markets. Moreover, in a reverse way, implied volatility occupies a central role as a forward-looking indicator that reflects the market’s expectations of future price fluctuations embedded in option prices.
Under the Black-Scholes-Merton (BSM) model, volatility is assumed to be constant and independent of option characteristics like the strike price (K) and the time to maturity (T). Empirical evidence, however, reveals that implied volatility varies across option contracts and especially depends on both parameters K and T of the option.
Volatility curves represent a cross-sectional view of the implied volatility surface (IVS), depicting the relationship between implied volatility and strike price for a fixed maturity. They are constructed by plotting implied volatility as a function of strike while holding time-to-maturity constant.
The volatility curves are commonly observed in two distinct shapes, most notably the volatility smile and the volatility smirk. A detailed discussion of the empirical relationship between implied volatility and option moneyness, the associated stylized facts, and their economic interpretation can be found in the article Volatility curves: smiles and smirks.
Figure 1 below illustrates the implied volatility smile (1a) and smirk (1b).
Figure 1a and 1b. Implied Volatility Smile and Smirk
Source: computation by the author (with python).
While volatility curves describe the strike dependence of implied volatility, the maturity dimension captures how implied volatility varies across expiration dates for a given strike. This relationship is commonly referred to as the term structure of implied volatility.
Using daily implied volatility data for S&P 100 index options (December 1983 to September 1987), Stein (1989) documented that volatility shocks are transmitted across maturities more strongly than predicted by standard rational expectations theory. Under this standard theoretical framework, because implied volatility is strongly mean-reverting, a near-term shock should naturally decay over time, causing long-dated implied volatilities to change by only a fractional amount. However, following an increase in short-dated implied volatility, long-dated implied volatilities tend to rise by a disproportionately large amount, indicating that changes in near-term uncertainty heavily influence market expectations over a broad range of maturities.
Using data for options on the S&P 500, FTSE 100, DAX 30, CAC 40, and Nikkei 225 stock indexes spanning the period May 9, 1994 to October 12, 2001, Mixon (2007) suggested mean reversion as a key characteristic of the implied volatility term structure. While short-dated implied volatilities exhibit substantial sensitivity to changes in market conditions, long-dated implied volatilities remain comparatively stable, reflecting expectations of convergence toward a long-run volatility level. Consequently, the term structure is generally upward sloping (contango) during periods of low market uncertainty but may become inverted (backwardation) during episodes of market stress, when short-term volatility rises sharply.
Figure 2 illustrates an upward-sloping (2a) and a downward-sloping (2b) implied volatility term structure.
Figure 2a and 2b. Implied Volatility Term Structure
Source: computation by the author (with python).
Christoffersen, Heston, and Jacobs (2009) demonstrate that the volatility term structure is not necessarily monotonic, arguing that capturing its true dynamics requires multifactor stochastic volatility frameworks. Evaluating European S&P 500 call options from 1990 through 2004, their empirical evidence reveals that implied volatility frequently displays significant curvature across maturities. This non-monotonic curvature is difficult to reconcile with traditional single-factor specifications like the benchmark Heston (1993) model, which restricts the term structure of implied volatility because it relies on only a single variance factor to model volatility over time.
The volatility surface provides a three-dimensional representation of implied volatility across strike prices and maturities. It is represented by the function σ(K,T), which assigns an implied volatility to each combination of strike price K and time to maturity T that reproduces the observed market option prices under the Black-Scholes-Merton (BSM) model.
Constructed from a cross-section of traded options, the volatility surface provides a comprehensive description of how the market prices uncertainty across both strike price (K) and maturity (T).
In practice, market option quotes are available only for a discrete set of strikes and maturities. Constructing a continuous volatility surface therefore requires interpolation and smoothing techniques. To ensure economic consistency, we must have σ(K,T) ≥ 0 for all strikes K and expirations T and the resulting surface must satisfy the static no-arbitrage conditions: namely the absence of butterfly arbitrage across strikes and calendar-spread arbitrage across maturities. (see, Breeden & Litzenberger, 1978; Gatheral, 2006)
The absence of butterfly arbitrage requires option prices to remain convex with respect to strike. Equivalently, the risk-neutral probability density implied by option prices must remain non-negative across all strikes, this condition can be expressed as:
A violation of this condition implies a negative risk-neutral probability density over some range of strikes and leads to arbitrage opportunities.
The absence of calendar-spread arbitrage states that with increase in maturity of an option, it should not result in a reduction of its value, since a longer-dated option provides all the rights of an otherwise identical shorter-dated option together with additional time for favourable price movements to occur. In volatility surface modelling, this condition is typically expressed in terms of the total implied variance
where σBS(K,T) denotes the Black-Scholes-Merton implied volatility for strike K and maturity T. Total implied variance measures the total accumulated expected variance over the entire life of the option
For a fixed strike, total implied variance must be non-decreasing with maturity
A violation would imply that a longer-dated option embeds less cumulative uncertainty than a shorter-dated option at the same strike, resulting in an arbitrage opportunity.
Together, the butterfly-arbitrage and calendar-spread-arbitrage constraints ensure that the interpolated volatility surface produces arbitrage-free option prices and a valid risk-neutral distribution.
In this section, we discuss how an implied volatility surface can be estimated from the S&P 500 index observed market option prices using a parametric model to fit the data and how the parameters can be adjusted to represent different macro-economic stress conditions.
To construct the implied volatility surface, we import the S&P 500 index option chain (set of call and put options across various strikes and maturities) directly from Yahoo! Finance. Because raw market data often contains stale quotes and asynchronous prices, we apply a robust set of filtering techniques to clean the dataset before model estimation.
First, we apply illiquidity filters, removing any option contracts with zero trading volume or zero open interest. Second, we filter the dataset to retain only Out-of-the-Money (OTM) options (OTM puts where K < S0, and OTM calls where K > S0). This is a standard practice, as implied volatility is theoretically independent of option type due to Put-Call Parity, focusing strictly on OTM contracts ensures we utilize the most liquid instruments (liquidity being measured by the bid-ask spread) to minimize pricing noise.
Third, we enforce intrinsic value and no-arbitrage boundary conditions. Any contracts with mispriced or economically impossible quotes are filtered out by verifying the upper and lower price bounds for the option contracts as given below
where:
Finally, we check for vertical (strike) arbitrage to ensure the data adheres to fundamental shape restrictions. We sort the contracts by time-to-maturity and then by strike price in ascending order. We then verify shape monotonicity: for any given maturity, call prices must strictly decrease as the strike price increases, and put prices must strictly increase as the strike price increases. Applying these standard empirical filters ensures a clean, arbitrage-free dataset ready for surface estimation.
To model and plot the implied volatility surface as shown below in Figure 1, we implement a parametric approach originally proposed by Dumas, Fleming, and Whaley (1998). This technique fits a deterministic volatility function (DVF) directly through the observed option market data. Under this framework, the implied volatility function is expressed as a second-order polynomial function across log-moneyness (M) and time-to-maturity (T):
where:
In the polynomial function above, we utilize the log-forward moneyness (M), defined as:
where F0 is the forward price of the underlying asset, calculated as:
This is usually done in practice because it is F0, and not S0, that represents the expected stock price on the option’s maturity date in a risk-neutral world. Consequently, traders often define an “at-the-money” option as a contract where K = F0, rather than an option where K = S0.
To fit this model, we first apply a numerical root-finding algorithm to invert the Black-Scholes-Merton (BSM) pricing model against observed market prices (mid prices defined as the average of bid and ask prices) to extract the market implied volatilities. We restrict our sample to contracts with maturities under one year (T < 1.0) and a log-moneyness of |M| < 0.45. This filters out deep Out-of-the-Money (OTM) options, which typically suffer from low trading volumes and wide bid-ask spreads, as they are primarily held for structural tail-hedging by institutional investors.
Finally, we apply Ordinary Least Squares (OLS) regression to the filtered dataset to solve for the six α parameters simultaneously. Once estimated, these parameters can be used to generate the implied volatility curves, term structures, and 3D surfaces under various macroeconomic stress scenarios, as discussed below.
Figure 3 illustrates the estimated implied volatility surface of the S&P 500 index using options data collected on June 18, 2026. The market environment at the time of collection was defined by an index spot price (S0) of $7496.04, a risk-free interest rate (r) of 3.658%, and a continuous dividend yield (q) of 1.04%. Based on these inputs, the resulting empirical surface is presented below.
Figure 3. Implied Volatility Surface of the S&P 500 index options (June 18, 2026)
Source: computation by the author (with python).
From the surface, we can observe that amid ongoing US-Iran tensions in the Middle East, out-of-the-money (OTM) put options exhibit high implied volatility for short-term maturities. This reflects panic buying of downside protection due to fears of conflict escalation and immediate uncertainty in the market. Toward the far end of the maturity, however, the surface balances out with OTM call options. This indicates that while near-term sentiment is dominated by risk aversion, long-term market expectations are highly speculative, positioning for a potential recovery once the geopolitical uncertainty resolves. To isolate and observe these market dynamics more precisely, the individual implied volatility smiles (by strike) and term structures (by maturity) are plotted below.
Figure 4 illustrates the implied volatility curves for three distinct maturities. As discussed above, we can clearly observe the steep downside skew flattening out and transitioning into a asymmetric smile as maturity increases.
Figure 4. Implied Volatility Curves of the S&P 500 options (June 18, 2026)
Source: computation by the author (with python).
Figure 5 illustrates the implied volatility term structure (up to 1 year) for three different strike prices. For out-of-the-money (OTM) call options, we can observe that the term structure is upward-sloping, indicating a long-term uncertainty alongside expectations of an upward market movement. Conversely, the OTM put option term structure is inverted and reflecting high short-term panic and uncertainty. Over the time horizon, this near-term panic subsides, balancing out with the OTM call options in the long run.
Figure 5. Implied Volatility Term Structure of the S&P 500 index options (June 18, 2026)
Source: computation by the author (with python).
The at-the-money (ATM) option term structure exhibits a shallow, non-monotonic U-shape, characterized by elevated short-dated volatility, a flattened middle region, and higher long-term volatility. This reflects that, in the short-to-medium term, the curve demonstrates the classic mean-reversion, where the immediate geopolitical shock dissipates and flattens out over a 3-to-6-month horizon. Conversely, the upward drift at the longer end of the mature horizon reflects the structural term premium demanded by investors to account for broader, open-ended macroeconomic uncertainties, as discussed above in the stylized facts section.
Figure 6 provides a compelling visual framework for observing how the implied volatility surface structurally shifts under different stress conditions. The surface on the left (a) represents the systemic crash caused by the COVID-19 pandemic (2020), while the surface on the right (b) illustrates the hypothetical impact on index options if the ongoing US-Iran conflict were to escalate significantly. These surfaces are constructed by adjusting the values of the six model parameters estimated in the preceding section; as such, they serve as illustrative examples of structural shifts rather than exact numerical forecasts.
Figures 6a and 6b. Implied Volatility Surface of the S&P 500 options under different stress environments
Source: computation by the author (with python).
Note: In both figures, the lightly shaded surface serves as the baseline, representing the actual market implied volatility surface as of June 18, 2026.
From Figure 6a, we can observe a massive surge in overall implied volatility across the board, driven by widespread panic buying of both out-of-the-money (OTM) puts and calls. This systemic shock resulted in a relatively flatter skew but severe inversion across the maturity spectrum, reflecting the acute, immediate fear of economic collapse as global lockdowns were implemented.
In contrast, Figure 6b models a scenario where the US-Iran conflict escalates into a full-scale regional crisis. Such an event would severely disrupt global oil supply chains, acting as a prolonged macroeconomic drag that hits S&P 500 corporate earnings over many months. Because this represents a lingering economic threat rather than an overnight liquidity freeze, the market’s response is highly asymmetric: demand is heavily concentrated in OTM puts for long-term downside protection and a steady increase in long-term implied volatility across the maturity.
While these stress scenarios represent extreme events, day-to-day movements follow structured patterns. Cont and Da Fonseca (2002) showed that daily dynamic deformations of the S&P 500 volatility surface are not chaotic. Instead, using principal component analysis, they demonstrated that surface movements are driven by just a few common statistical factors: parallel shifts, changes in the strike slope (skew), and twists in the maturity curvature.
You can download the Python code provided below, for the construction of the implied volatility curves, term structures and surfaces under different stress conditions as discussed above.
Alternatively, you can download the R code below with the same functionality as in the Python file.
You can download the cleaned S&P 500 index options data for 18 June 2026, as used in the above Python and R codes to make the plots as discussed before.
A volatility surface determines the risk-neutral distributions implied by option prices (see Option Implied Risk-Neutral Distribution), but it does not uniquely specify the underlying stochastic process governing asset-prices. As highlighted by Cont (2006), this introduces significant model uncertainty: different mathematical frameworks can calibrate perfectly to the exact same market volatility surface today, yet yield wildly divergent prices and hedges for exotic options because they imply different future surface dynamics. Consequently, a substantial body of research has focused on developing models capable of reproducing both the observed shape of the volatility surface and its evolution through time.
The principal modelling approaches include local volatility models, stochastic volatility models, parametric surface models and, more recently, rough volatility models.
In the standard BSM formula, volatility is assumed constant, which however does not correspond to reality, as markets exhibit volatility smile and skews. Local volatility model, extends the BSM, by assuming that volatility is a function of stock price (St) and time (t), and the instantaneous local volatility is given by σt( St,t).
The Dupire (1994) formula that links the instantaneous local volatilities, to the implied volatility surface is given as follows:
Stochastic volatility refers to the modelling of volatility using time-dependent stochastic processes, in contrast to the constant volatility assumption made in the standard BSM model. These models are better able to capture the observed features such as volatility clustering and mean reversion. One of the most widely used stochastic volatility models is the Heston (1993) model. The model describes the dynamics of the underlying asset price and its variance using a system of two coupled stochastic differential equations (SDEs), given by:
Where:
Parametric volatility surface models are used to interpolate and extrapolate implied volatility across strikes and maturities, for sparse or illiquid strikes and ensure the resulting surface is free from static arbitrage (butterfly and calendar arbitrage).
Among the most widely used approaches is the SVI (Stochastic Volatility Inspired) parametrization, developed by Gatheral and Jacquier (2014), which is commonly applied to equity and index volatility surfaces. It models the total implied variance as a function of log-moneyness, providing a parsimonious representation of the volatility smile.
Other important parametric frameworks include the SABR model (Stochastic Alpha, Beta, Rho), which is widely used in interest rate and FX markets, and SSVI (Surface SVI), which extends the SVI framework to ensure arbitrage-free surface dynamics across maturities.
Rough volatility models represent one of the most important recent developments in volatility modelling. Gatheral, Jaisson, and Rosenbaum (2018) provided the empirical evidence that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable timescale.
The Hurst exponent (H) is a statistical parameter that characterises the roughness of a stochastic process: when H = 0.5, the process reduces to a standard Brownian motion with no memory, corresponding to a random walk. Whereas, values of H > 0.5 indicate persistent behaviour, while H < 0.5 imply anti-persistence, where increments tend to reverse direction more frequently, leading to rougher sample paths.
This observation, led to adoption of the fractional stochastic volatility (FSV) model of Comte and Renault (1998). The Rough FSV (RFSV) in contrast to FSV, is remarkably consistent with financial time series data. Compared to classical stochastic volatility models, it better captures the extremely rough nature of volatility paths and enables improved forecasting of realized volatility.
Implied volatility surfaces are among the most important tools in modern quantitative finance. They play a central role in the pricing and hedging of derivatives, particularly exotic options, and are widely used in risk management, stress testing, and scenario analysis. A good understanding of volatility surfaces is therefore essential for students, practitioners, and anyone seeking a career in derivatives, quantitative finance, trading, or risk management.
▶ Saral BINDAL Historical Volatility
▶ Saral BINDAL Implied Volatility and Option Prices
▶ Saral BINDAL Volatility curves: smiles and smirks
▶ Saral BINDAL Option Implied Risk-Neutral Distribution
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Hull J.C. (2015) Options, Futures, and Other Derivatives, Eleventh Edition, Global Edition, Chapter 15 – The Black-Scholes-Merton model, 338-369.
Merton, R.C. (1973). Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4(1), 141-183.
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The article was written in June 2026 by Saral BINDAL (Indian Institute of Technology Kharagpur, Metallurgical and Materials Engineering, 2024-2028 & Research assistant at ESSEC Business School). His interests include tracking geopolitical developments and analysing their direct impact on macroeconomic factors such as inflation, trade balances, and currency volatility, with a focus on using data to quantify these global economic ripple effects.
Discover all posts written by Saral BINDAL.
In this article, Saral BINDAL (Indian Institute of Technology Kharagpur, Metallurgical and Materials Engineering, 2024-2028 & Research assistant at ESSEC Business School) explains the CBOE methodology for the construction of the volatility index or ‘VIX’.
The Chicago Board Options Exchange (CBOE) Volatility Index, or VIX, is a real-time market index designed to measure the market’s expectation of 30-day forward-looking annualized volatility. It is option-based, calculated using the market prices of S&P 500 index options to gauge expected volatility.
In 1993, CBOE Global Markets introduced the CBOE Volatility Index (VIX Index). Originally designed by Robert E. Whaley (1993) to measure the market’s expectation of 30-day volatility, the index was calculated using an option-pricing model to derive the implied volatility of at-the-money S&P 100 index (OEX Index) options. The VIX Index quickly became the premier benchmark for U.S. stock market volatility and is widely referred to as the market’s “fear gauge”.
Ten years later in 2003, CBOE partnered with Goldman Sachs to completely overhaul the index. This update introduced a methodology independent of option-pricing models, adapting the seminal theoretical framework for model-free implied variance established by Britten-Jones and Neuberger (2000) alongside the practical replication insights of Demeterfi et al. (1999). This modern version of the VIX shifted its underlying base to the broader S&P 500 index. Rather than tracking a narrow selection of options, it estimates market expectations by aggregating a heavily weighted cross-section of SPX puts and calls across a wide range of strike prices.
Academic research confirms that this model-free aggregation method captures more information and provides a more efficient forecast of future realized volatility than individual Black-Scholes implied volatilities (Jiang & Tian, 2005).
While VIX is often regarded as the market’s fear index, it might give one a false impression that it moves opposite to the S&P 500. Mathematically it has no directional bias, and only measures the magnitude of expected volatility. Instead, the real-world inverse relationship is driven by corporate capital structures and asymmetric investor behavior. As Black (1976) pointed out, when a stock price drops, a company’s financial leverage automatically increases, making the equity riskier and naturally driving up volatility.
Furthermore, market sell-offs trigger a sudden panic where investors rush to buy portfolio insurance (put options) all at once. Because the supply of this insurance is limited, options market makers must aggressively raise prices to protect themselves. Gârleanu et al. (2009) formalize this mechanism, demonstrating that because market makers cannot perfectly hedge their positions, concentrated investor demand directly drives option pricing and inflates implied volatility premiums. Since the VIX is calculated directly from these option prices, this demand-pressure mechanically forces the index to spike.
This same demand explains why the S&P 500 and the VIX occasionally rise together. During massive market rallies, investors experience FOMO (Fear of Missing Out) and rush to buy upside call options, or quickly buy puts to lock in their rapid gains. Just like during a market crash, this sudden increase in demand for options overwhelms market makers. To protect themselves, they hike option prices, which mechanically forces the VIX up even as the stock market climbs.
The VIX is designed to measure the market’s expectation of volatility over the next 30 calendar days. However, listed S&P 500 options rarely expire exactly 30 days from the calculation date. To address this, the methodology selects two option maturities: a near-term maturity of less than 30 days and a next-term maturity of more than 30 days remaining. Variance estimates are calculated for both maturities and subsequently interpolated to obtain a constant 30-day measure of expected volatility.
In the CBOE volatility index calculation methodology, time to expiration of a constituent option series, is calculated by dividing the number of minutes until expiration (MTime to Expiry) of the selected options (rounded down to the nearest minute) by the number of minutes in a year (M365).
The next step is to estimate the forward index level of the S&P 500 using option markets prices. It represents the market’s expectation of the index value at expiration under the risk-neutral measure and serves as the reference point for selecting the relevant option contracts used in the calculation.
It is calculated using the principle of put-call parity, specifically by finding the unique strike price where the price difference between the call and the put option is at its absolute minimum.
Where:
Once the forward index level has been estimated, we then identify K0, defined as the first strike price equal to or immediately below the forward index level (F). This strike acts as a reference point for the option selection process, separating the out-of-the-money put options from the out-of-the-money call options used in the calculation.
The VIX methodology uses a wide range of out-of-the-money (OTM) put and call options. OTM options are sensitive to changes in expected future volatility and provide information about the market’s expectations across a broad range of potential future outcomes. By incorporating both downside and upside option prices, the methodology captures the entire market-implied distribution of future index values rather than relying on a single option contract.
Each selected option contributes unique information about the market’s expectation of future variance. The weight of this contribution depends on three key factors: the option’s mid-price (Q(Ki)), the strike spacing (ΔKi) between neighbouring contracts, and the inverse square of its strike price (1/(Ki)2). This precise weighting scheme ensures that information from the entire out-of-the-money option chain is integrated into the final variance estimate.
where for:
The option selection and weighting procedure described above is formally represented by the VIX variance formula. Rather than estimating volatility from a single option, the formula aggregates information from all selected option contracts to produce an estimate of expected future annualized variance.
Where:
Applying the variance formula to both the near-term and next-term options produces two separate estimates of expected future variance. The methodology calculates variance first because option portfolios can replicate future variance directly. As demonstrated by Demeterfi et al. (1999), a continuously weighted portfolio of out-of-the-money options across all strikes can replicate the payoff of a log contract, which is a theoretical derivative whose payout is tied to the logarithm of an asset’s price, making its returns purely dependent on variance rather than direction. Because a log contract captures total realized variance regardless of the asset’s price path, this foundational result allows expected future variance to be inferred directly and purely from observable option prices.
The variance estimates obtained from the near-term and next-term option maturities are linearly interpolated to obtain a constant 30-day estimate of annualized variance. Taking the square root converts variance into volatility, while multiplying by 100 expresses the result as a percentage. The resulting value is reported as the VIX index. The formula used in the interpolated CBOE volatility index calculation is as follows:
Where:
For this section, we consider the S&P 500 index options data collected on June 18, 2026, with a spot price of $7,496.04 and a risk-free rate of 3.66%. Excel file with complete data and VIX calculations can be downloaded below.
Our calculations yield a VIX value of 13.69, reflecting the market’s expectation of a ±13.69% movement over the next year. In Figure 1, we map this percentage onto a standard bell curve, where this expected movement in the S&P 500 index prices represent one standard deviation. This allows us to visualize the market’s expected range of price movements under the 68%, 95%, and 99.7% confidence intervals over the next one year.
Figure 1. Market Expected Price Over the Next 1 Year
Source: computation by the author.
To calculate expected movements for shorter time frames, the VIX is scaled by dividing it by the square root of N, where N represents the number of periods in a year. For instance, N equals 12 to find a 1-month expected move, 52 for a 1-week move, and 252 trading days for a 1-day move.
Figure 2. Expected Movements for Shorter Time Frames
Source: computation by the author.
You can download the Python code provided below, for VIX calculation using the modern CBOE methodology.
Alternatively, you can download the R code below with the same functionality as in the Python file.
For anyone interested in finance or a career in trading, understanding how the VIX is constructed is crucial. As one of the most widely used measures of market uncertainty and expected volatility, it serves as an important tool for market analysis, risk assessment and numerous volatility-based trading strategies.
▶ Akshit GUPTA Options
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▶ Youssef LOURAOUI VIX index
Black F. and M. Scholes (1973) The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.
Black, F. (1976), “Studies of Stock Price Volatility Changes”, Proceedings of the Business and Economics Section of the American Statistical Association, 177–181.
Britten-Jones, M. and A. Neuberger (2000) Option prices, implied price processes, and stochastic volatility. The Journal of Finance, 55(2), 839–866.
Demeterfi, K., Derman, E., Kamal, M., & Zou, J. (1999). A guide to volatility and variance swaps. The Journal of Derivatives, 6(4), 9-32.
Gârleanu, N., Pedersen, L. H., & Poteshman, A. M. (2009). Demand-based option pricing. The Review of Financial Studies, 22(11), 4259–4299.
Hull J.C. (2022) Options, Futures, and Other Derivatives, 11th Global Edition, Chapter 15 – The Black-Scholes-Merton model, 338–365.
Jiang, G. J. and Y. S. Tian (2005) The model-free implied volatility and its information content. The Review of Financial Studies, 18(4), 1305–1342.
Merton R.C. (1973) Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4(1), 141–183.
Whaley, R. E. (1993). Derivatives on market volatility: Hedging tools long overdue. The Journal of Derivatives, 1(1), 71-84.
Cboe Global Markets (February 26, 2026) Version 6.0 Cboe Volatility Index (VIX) Methodology.
Cboe Global Markets (February 26, 2026) Version 5.0 Cboe Volatility Index Mathematics Methodology.
The article was written in June 2026 by Saral BINDAL (Indian Institute of Technology Kharagpur, Metallurgical and Materials Engineering, 2024-2028 & Research assistant at ESSEC Business School). His interests include tracking geopolitical developments and analyzing their direct impact on macroeconomic factors such as inflation, trade balances, and currency volatility, with a focus on using data to quantify these global economic ripple effects.
Discover all posts written by Saral BINDAL.
In this article, Marco SIMONETTI (ESSEC Business School, MSc in Finance, 2025-2027) shares his experience as an Off-Cycle Sales & Trading Derivatives Intern at Banca Monte dei Paschi di Siena in Milan. The internship gave me direct exposure to how a corporate and investment banking desk transforms market information into practical hedging and trading solutions for corporate clients.
My role sits at the intersection of markets, corporate finance and client advisory. On one side, I follow macroeconomic and market developments in real time; on the other, I help translate those developments into concrete ideas for clients exposed to commodities, foreign exchange and interest rates.
Banca Monte dei Paschi di Siena (MPS) is one of the major Italian banking groups and is widely known as the world’s oldest bank still in operation, with origins in Siena in 1472. Today, the Group operates across retail banking, corporate banking, wealth management and capital markets activities.
My internship is based in Milan, within Sales & Trading Derivatives. The desk works with corporate clients that are exposed to fluctuations in commodity prices, exchange rates and interest rates. For example, an industrial company may need to hedge the cost of energy or raw materials, while an exporter may need to manage the risk that currencies move against its future revenues.
The value added by this type of desk is not simply to sell a financial product. It is to understand the client’s business model, identify the risk exposure, structure an appropriate solution and coordinate with traders to deliver an executable price. In other words, the role combines technical knowledge, market timing and commercial judgment.
Logo of the company.
Source: Banca Monte dei Paschi di Siena.
Client coverage and needs identification: I supported the coverage of a portfolio of around 20 clients, helping identify hedging and trading needs across commodities, foreign exchange (FX) and interest rates. In practice, this meant understanding what each company buys, sells, imports, exports or finances, and how market volatility can affect its margins, cash flows and planning.
Market intelligence: I prepared real-time market reports and macro-driven trade ideas using Bloomberg. Bloomberg is a professional financial data platform used by banks, asset managers and corporates to monitor market prices, news, analytics and execution tools. My work involved following central-bank decisions, inflation data, interest-rate curves, energy markets and metals prices, then summarizing the implications for clients.
Product structuring: I worked on vanilla and semi-structured products such as forwards, swaps, options, collars and TARNs. A forward locks in a future price or exchange rate; a swap exchanges one stream of cash flows for another; an option gives protection or upside participation; a collar combines options to create a protection band; and a TARN (Target Redemption Note) is a structured product that terminates when a predefined target is reached.
Pricing and execution support: I worked day to day with traders to understand derivative pricing, bid-ask spreads and Greeks. The bid-ask spread is the difference between the price at which a dealer is willing to buy and the price at which it is willing to sell. The Greeks are risk measures used for options: for example, delta measures sensitivity to the underlying price, vega measures sensitivity to volatility and theta measures sensitivity to time decay.
Transaction process: I followed transactions from the client request to trade execution. This made me understand that the job requires both technical precision and process discipline: the client problem must be clearly identified, the structure must be suitable, the price must be executable and the documentation must be aligned with internal and regulatory requirements.
Commercial impact: From a core group of clients, the activity contributed approximately EUR 10k of daily revenues, primarily across oil & gas, energy and metals. This gave me a concrete view of how client relationships, market timing and product structuring can translate into measurable business results.
Hard skills: The internship requires knowledge of derivatives, fixed income, FX, commodities, option pricing, Bloomberg, macroeconomics, financial modeling and risk management. It also requires the ability to understand payoff profiles, compare hedging alternatives and interpret market data quickly.
Soft skills: The role also requires clear communication, attention to detail, speed under pressure and the ability to simplify complex market information. In derivatives sales, technical knowledge is useful only if it can be translated into a clear and relevant message for the client.
The main lesson I learned is that derivatives sales is a bridge between markets and the real economy. A company does not hedge because a model says so; it hedges because volatility in oil, gas, metals, currencies or interest rates can directly affect its margins, debt service or investment plans.
I also learned that the quality of a trade idea depends on three elements: the market view, the client fit and the execution level. A correct macro view is not enough if the product is too complex for the client, too expensive to execute or misaligned with the company’s risk appetite.
Finally, the experience showed me the importance of discipline. Every price, spread, scenario and payoff profile must be checked carefully because derivatives can create both protection and risk. This is why sales and traders must work closely together before a transaction is executed.
I present below three financial concepts related to my internship experience:
Hedging means using financial instruments to reduce exposure to an unwanted risk. In my internship, typical risks include commodity price risk, FX risk and interest-rate risk. A commodity consumer may use swaps or options to stabilize future input costs; an exporter may use FX forwards to lock in an exchange rate; and a borrower may use interest-rate derivatives to reduce uncertainty around future financing costs.
The bid-ask spread is the difference between the price at which the bank can buy and the price at which it can sell a product. In derivatives, this spread compensates the bank for liquidity, hedging costs, market risk and operational complexity. Understanding the spread is important because it affects both the client’s execution level and the bank’s revenue.
The Greeks measure how the value of an option changes when market variables change. Delta measures sensitivity to the underlying price, gamma measures the change in delta, vega measures sensitivity to volatility, theta measures time decay and rho measures sensitivity to interest rates. These measures help traders hedge the risks created by client transactions and manage the desk’s exposure.
This post is relevant for ESSEC MiF students because it shows how financial theory becomes operational in a real banking environment. Courses on derivatives, portfolio management and financial markets provide the analytical foundation, but the internship shows how these tools are used under time pressure, with real clients and real market constraints.
For students interested in sales & trading, corporate banking or risk management, the role demonstrates that technical excellence and commercial understanding must go together. The best solutions are not necessarily the most complex ones, but the ones that are suitable, executable and useful for the client.
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Banca Monte dei Paschi di Siena — Group website
Banca MPS — Commodity derivatives
Banca MPS — Foreign exchange derivatives
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The article was written by Marco SIMONETTI (ESSEC Business School, MSc in Finance, 2025-2027), based on his experience as an Off-Cycle Sales & Trading Derivatives Intern at Banca Monte dei Paschi di Siena in Milan.
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In this article, Marco SIMONETTI (ESSEC Business School, Master in Finance, 2025-2026) shares his founder experience building, scaling, and exiting Cristoforo Travel (2021-2025), a traveltech venture focused on B2B software and analytics for travel agencies and tour operators.
I founded Cristoforo Travel in early 2021 to help travel providers rebound after the pandemic with better technology and analytics. A traveltech company applies digital tools to the travel industry: for example booking engines, payment integrations, inventory management, pricing automation, customer data, and forecasting models. In our case, the objective was to help travel agencies and tour operators sell more efficiently, integrate fragmented systems, and use data to improve margins.
The company combined consulting with custom development to integrate booking and payment rails, automate inventory and pricing, and deliver lightweight forecasting tools. This hybrid model generated revenue quickly while compounding reusable IP. IP, or intellectual property, refers to proprietary assets that a company owns or controls; for Cristoforo Travel, this included connectors, software modules, analytics templates, and technical documentation that could be reused across clients. Reusing this IP reduced implementation time over successive engagements and made each new project easier to scale.
Our clients were mainly travel agencies and tour operators, ranging from independent agencies to larger B2B accounts. Among the most recognizable names, we worked with clients such as Alpitour and Evaneos. The value added was practical and measurable: we helped clients connect booking and payment systems, structure cross-selling flows, improve inventory visibility, and test pricing or demand assumptions with data instead of intuition. Cross-selling is now common across tourism: once a traveler buys a flight, hotel, or package, providers try to add insurance, transfers, activities, excursions, upgrades, or ancillary services. Our role was to make those add-on opportunities easier to manage and monetize for professional travel sellers.
The competitive landscape included traditional booking engines, travel CRM/ERP providers, destination-management software, and larger travel technology platforms used by agencies and tour operators. We competed less on brand size and more on flexibility, speed of integration, and the ability to combine product development with hands-on business consulting. Compared with large off-the-shelf platforms, our added value was the capacity to customize workflows for each client while gradually transforming repeated requests into reusable software modules.
Over time, I built a global partnership footprint – more than 90 partners across six continents – and secured enterprise-level agreements that pressure-tested reliability, security, and scale. Commercially, the business reached approximately €2 million in annual sales. As is typical in B2B travel services, gross margins were relatively low and varied by contract, usually between 5% and 20%, with an average of around 10% over five years. After operational costs and personnel expenses, the business generated approximately €30k-€40k per year of personal income for me, which I used to finance my studies abroad.
In June 2025, I sold my shares through a clean share sale. Due to confidentiality obligations, I cannot disclose the name of the acquiring company. However, I can say that it is listed on a Milan startup/SME stock market segment and operates with a business model very close to ours, which made the strategic fit natural.
Logo of the company.
Source: the company.
As founder and CEO, I led capital raising, product and delivery, sales and partnerships, and financial planning – owning the P&L, forecasting, and investor relations.
Capital & financing: I raised €200k in seed funding from two angel investors to accelerate product and commercial rollout. I built a lean operating plan that linked hiring and product sprints to cash runway. Cash runway is the number of months a company can continue operating before running out of cash, based on its cash balance and monthly burn rate.
Product & delivery: I shipped integrations for booking and payments, pricing automation, and demand-forecasting tools. I balanced bespoke implementations with reusable modules: the first projects were more customized and lower-margin, but each engagement helped us identify features that could later become standardized modules.
Go-to-market: I created partnership playbooks, prospected and closed over 90 global partners, and established enterprise agreements with travel agencies and tour operators. I showcased our solutions at international trade fairs to generate pipeline, validate pain points directly with buyers, and compare our positioning against larger travel technology providers.
Data & strategy: I developed macro leading-indicator models for Southern Europe to guide market sequencing, inventory focus, and pricing experiments. These models helped prioritize which geographies, destinations, and product categories were more likely to convert depending on demand signals and seasonality.
Exit & integration: I negotiated a clean share sale in June 2025. A clean share sale means selling shares through a straightforward transaction with limited unresolved liabilities, clear ownership transfer, and clearly defined post-closing obligations. After the transaction, the technology and client logic were prepared for integration into the acquiring company, whose name I cannot disclose for confidentiality reasons. The acquirer is listed on a Milan startup/SME stock market segment and has a business model very similar to Cristoforo Travel.
Hard skills: financial modeling and runway management, pricing and unit economics, SaaS implementation and systems integration, data analysis for forecasting, and contract structuring, including SLAs, security, and compliance. SaaS means Software as a Service: software delivered online, usually through a subscription or recurring-fee model, instead of being installed and maintained locally by each client. SLA means Service Level Agreement: a contractual commitment that defines expected service quality, such as uptime, response times, support obligations, data protection, and remedies if service levels are not met.
Soft skills: enterprise sales storytelling, stakeholder management with investors, partners, and customers, cross-functional leadership, negotiation, and execution under uncertainty. In a small traveltech company, the founder often has to sell to clients, translate their operational problems into technical specifications, manage developers, and keep cash discipline at the same time.
I learned that in traveltech, the best product ideas often come from concrete client problems. Our clients – travel agencies and tour operators, including accounts such as Alpitour and Evaneos – did not simply want software; they wanted fewer manual operations, better cross-selling, faster integrations, and more reliable data for pricing and inventory decisions. Competitors were often larger platforms or generic booking/CRM systems, but our advantage was speed, customization, and the ability to turn repeated client requests into reusable modules.
I also learned that consulting and development can fund product while accelerating learning. With around €2 million in annual sales, margins in B2B travel remained tight: contracts usually delivered 5%-20% gross margin, with an average around 10% over five years. This forced disciplined capital allocation. After costs and personnel, the company generated around €30k-€40k per year for me personally, enough to finance my studies abroad. That outcome taught me that a startup does not need to become a unicorn to create real value: it can also finance education, build professional credibility, and create strategic exit options.
Finally, selling at the edge of the roadmap validated security and compliance early, while clean interfaces and documentation made future M&A or platform integration smoother. The most important lesson was that sustainable growth depends on linking product decisions to client demand, cash discipline, and unit economics rather than chasing growth for its own sake.
I present below three financial concepts related to my founder experience:
Raising €200k from angel investors required balancing valuation and dilution with the operating runway necessary to reach commercial milestones. I built cash-flow forecasts, set hiring gates, and linked product sprints to liquidity checkpoints to avoid premature scaling. In practice, runway management meant asking: how many months can we finance development, sales, and support before the next cash inflow or funding milestone?
Our hybrid model began with lower margins from custom work but improved contribution as reusable modules, connectors, and templates reduced delivery time. Tracking gross margin by engagement type and CAC payback by partner cohort guided where to standardize and where to remain bespoke. Since B2B travel margins can be low, the key was to increase repeatability: each reusable connector or analytics template improved future unit economics.
For my share sale in June 2025, I evaluated considerations beyond the headline price: representations and warranties, transition obligations, confidentiality, and the strategic value of integration into a listed company with a similar business model. Clean interfaces and documentation lowered integration risk and preserved the long-term value of the technology.
If you are an ESSEC MiF student curious about venture building or fintech-adjacent B2B business models, my story shows how financial discipline can combine with product-market execution to create real optionality. B2B means business-to-business: a company sells products or services to other companies rather than directly to consumers. In my case, Cristoforo Travel sold to travel agencies and tour operators, so success depended on enterprise trust, integrations, contract discipline, and measurable ROI for professional clients.
The broader advice is simple: start from a painful operational problem, sell early, measure margins contract by contract, document everything, and build reusable assets whenever a client request repeats. That combination can support profitable growth, finance personal and academic goals, and make a strategic exit more credible.
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The article was written in June 2026 by Marco SIMONETTI (ESSEC Business School, Master in Finance, 2025-2026).
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In this article, Saral BINDAL (Indian Institute of Technology Kharagpur, Metallurgical and Materials Engineering, 2024-2028 & Research assistant at ESSEC Business School) explains how option prices can be used to build an implied risk-neutral distribution.
Derivative markets provide a rich source of information for market expectations. For example, a futures price is the market’s expectation of the future value of an asset. More interestingly, we can derive the moments of the statistical distribution of future asset values from the market prices of options, like the variance (second moment), the skewness (third moment) and the kurtosis (fourth moment). More generally, we can extract the ex-ante risk-neutral probability distribution of future asset prices at a given date from option market prices with the corresponding maturity date.
A real-world probability measure represents the statistical distribution of asset returns typically estimated using historical data. These measures incorporate risk premia, market frictions, and investor behaviour, and are primarily used for statistical inference and risk modelling.
In contrast, risk-neutral probability measure is a mathematical pricing measure used in no-arbitrage valuation of financial derivatives. Under this framework, asset prices are evaluated as discounted expected payoffs under an equivalent martingale measure. In this setting, the expected return of any risky asset is adjusted to the risk-free rate within the pricing measure, simplifying valuation by transforming uncertain future payoffs into present values computed via expectation (Hull, 2018; Shreve, 2004).
Historical Distributions are constructed from observed past returns under the physical measure (P-measure). They empirically capture the true statistical behaviour of asset prices, including fat tails, skewness, and volatility clustering driven by real market shocks and investor behaviour. These distributions exhibit higher variance and kurtosis, making them particularly valuable for stress testing, Value-at-Risk estimation, and portfolio risk management where realistic loss scenarios matter.
Risk-Neutral Distributions are derived from option market prices rather than historical data, under the implied measure by no-arbitrage pricing (Q-measure). They reflect market-implied expectations of future payoffs discounted at the risk-free rate resulting in smoother, less skewed densities. While highly effective for pricing derivatives and contingent claims, they tend to underestimate tail risk and do not directly represent the actual probabilities investors assign to future market outcomes.
Having distinguished between the physical and risk-neutral probability measures, it is useful to examine the risk-neutral distribution implied by the Black–Scholes–Merton (BSM) model, which is a standard model in quantitative finance. The BSM framework assumes that the underlying asset follows a geometric Brownian motion and provides a simple illustration of how the transition from the physical measure to the risk-neutral measure alters the distribution of future asset prices.
Under the BSM, the standard assumption is that the underlying asset follows a geometric Brownian motion given by the following expressions:
where:
Solving these stochastic differential equations over the interval [0, T] yields the terminal asset price:
Taking logarithms shows that the terminal log-price is normally distributed:
Thus, under the Black–Scholes–Merton framework, the risk-neutral distribution of the terminal asset price is lognormal (as the physical distribution). Relative to the corresponding physical distribution, the volatility remains unchanged, while the drift parameter μ is replaced by the risk-free rate r. This is an important result as the risk-free rate r is known and easily observable while the drift parameter μ has to be estimated and is not directly observable.
To extract a continuous risk-neutral probability distribution from the market, we must first understand how to isolate the market’s view on a specific future asset price. The primary tool for this is a classic option trading strategy: the butterfly spread.
A butterfly spread is an options trading strategy designed to achieve limited profit with strictly bounded risk, typically in market environments where relatively small price movements are anticipated. The strategy may be implemented using either call or put options and can be established in either a long or short configuration. For example, a long call butterfly is constructed by purchasing one call option at a lower strike price, selling two call options at an intermediate strike price, and purchasing one call option at a higher strike price. Depending on the relative spacing between the strike prices, a butterfly spread may be either symmetric or asymmetric.
To understand how option market prices encode the market’s expectations regarding the future distribution of the underlying asset price, we consider a symmetric butterfly. A symmetric butterfly spread is constructed using three European call options with a common maturity T and distinct strike prices. The strategy involves purchasing one call option with strike K – ΔK at a premium of C(K-ΔK,T), selling two call options with strike K at a premium of C(K,T) each, and purchasing one call option with strike K + ΔK at a premium of C(K+ΔK,T).
The price of the resulting butterfly spread is therefore given by
The net cost of the butterfly spread is obtained by summing the premia paid for the two long call positions and subtracting the premiums received from the two short call positions.
The payoff of a symmetric butterfly spread is centred around the strike (K) and can be expressed as
Figure 1 illustrates the payoff profile of a symmetric butterfly spread centred at the strike K = 100 with strike spacing ΔK = 5. The payoff reaches its maximum when the terminal asset price ST equals the strike K and declines to zero as ST moves beyond the adjacent strikes K – ΔK and K + ΔK.
Figure 1. Symmetric Butterfly Spread Payoff at Maturity
Source: computation by the author.
As a result, the butterfly spread effectively isolates a narrow range of terminal asset prices, making it a useful instrument for extracting information about the market-implied probability distribution of the underlying asset price at maturity.
A stack of butterfly spreads refers to a collection of butterfly spreads constructed across a range of strike prices, such that the central strike of each butterfly is equally spaced from the next. The spacing between successive central strikes is equal to the strike spacing ΔK used in the construction of each individual butterfly spread, as discussed above.
Figure 2 illustrates that a collection of butterfly spreads across strikes at a fixed maturity converges to the market-implied probability density of the underlying asset. Each butterfly corresponds to a discrete approximation of the second derivative of option prices with respect to strike, and aggregating these across strikes recovers the risk-neutral density.
We construct seven butterfly spreads centered at strikes K = 85 to K = 115 in increments of 5, with strike spacing ΔK = 5. The weights are specified using a Gaussian distribution with mean μ = 100 and standard deviation σ = 10, reflecting an assumed market belief about the concentration of terminal prices. The payoff profile is scaled by a factor of 200 to improve visual readability, and it is normalized by ΔK2 to remain consistent with the second-order finite-difference interpretation of butterfly spreads as detailed below.
Figure 2. Approximating the Risk-Neutral Density Using Butterfly Spreads
Source: computation by the author.
As the strike spacing ΔK is reduced, additional butterfly spreads can be constructed between existing butterfly spreads. Consequently, the stacked payoff profile becomes increasingly smooth and, in the limit, approaches a continuous representation of the implied probability distribution.
To better understand this limiting behaviour, it is useful to examine the properties of an individual butterfly spread. As the strike spacing ΔK decreases, the payoff of the butterfly spread becomes increasingly concentrated around its central strike. In the limit as ΔK → 0, the butterfly spread approaches an infinitesimally narrow peak centred at K.
Consequently, the value of the butterfly spread decreases as its payoff becomes increasingly concentrated around its central strike. To obtain a meaningful limiting quantity, the butterfly value must therefore be normalized by (ΔK)2. This normalization is motivated by a well-known result from calculus, central finite-difference approximation of the second derivative.
Comparing the two expressions above, reveals that the normalized butterfly value is precisely the finite-difference approximation of the second derivative of the call pricing function with respect to strike.
This observation forms the foundation of the Breeden-Litzenberger (1978) result, which establishes that the second derivative of the call pricing function with respect to strike is directly related to the market-implied risk-neutral probability density embedded in option prices, as demonstrated in the derivation below.
You can download the Excel file provided below to generate and visualize the payoff profiles of the butterfly spread and stacked butterfly spread at maturity, as discussed above.
This section develops the analytical derivation of the risk-neutral distribution using the seminal Breeden-Litzenberger (1978) result. By exploiting the cross-sectional structure of option prices across strikes, we recover the market-implied risk-neutral density embedded in option market prices.
Under the risk-neutral measure, the value of a European call option is given by the present value of its expected payoff at maturity. For a strike price K, continuously compounded risk-free rate r, and time to maturity T, the call pricing function C(K,T) can be expressed as
To obtain a continuous representation of the call price, the expected payoff can be expressed as an integral over the probability density function of the terminal asset price, f(ST).
Note: The integral starts at K because the payoff is zero when St≤K.
Taking the first derivative with respect to K, we get
To obtain the risk-neutral probability density function, as shown by Breeden and Litzenberger (1978), we take an additional derivative with respect to the strike
Rearranging the above formula, we get the risk-neutral distribution
Applying the second-order central difference approximation heuristically developed in the previous section using butterfly spreads, we obtain the following expression:
This expression shows that the risk-neutral probability density can be recovered directly from the second derivative of the call pricing function with respect to strike. In practice, however, option prices are observed only at a finite set of discrete strike prices, requiring numerical methods to approximate the derivatives and extract the implied risk-neutral distribution.
Methods for extracting the risk-neutral distribution can be broadly classified into non-parametric (data-driven with minimal distributional assumptions), semi-parametric (partial structural assumptions, typically imposed on intermediate quantities such as implied volatility), and parametric or structural (explicit assumptions on the distribution or asset price dynamics) approaches. These methodologies differ in the degree of modelling assumptions imposed on the option pricing function and the terminal asset price distribution, leading to different trade-offs between flexibility, numerical stability, and economic interpretability.
Non-parametric methods aim to recover the risk-neutral distribution directly from observed option prices without imposing any specific parametric structure on either the terminal asset price distribution or the stochastic process governing the evolution of the underlying asset price. Consequently, these methods are highly flexible, but they tend to be sensitive to market microstructure noise, sparse strike coverage, and interpolation error in option quotes.
Risk-neutral histograms: the most direct implementation of the Breeden–Litzenberger result constructs a discrete approximation of the implied risk-neutral density using finite differences across traded strikes (Breeden and Litzenberger, 1978; Neuhaus, 1995). Adjacent butterfly spreads may therefore be interpreted as local estimates of state-contingent probabilities.
Because option contracts are quoted only at discrete strike intervals, the recovered distribution resembles a histogram rather than a smooth continuous density, making the approach highly sensitive to strike spacing and pricing noise.
Kernel regression methods: to mitigate the instability of histogram-based estimates, subsequent research introduced non-parametric smoothing techniques that estimate a continuous option pricing function directly from observed market prices. A prominent example is the kernel regression framework of Aït-Sahalia and Lo (1998).
By reducing the influence of local pricing noise, kernel-based methods generally produce smoother and more stable estimates of the implied risk-neutral density.
Spline-based methods: another widely used class of non-parametric methods employs spline interpolation techniques to construct smooth and arbitrage-consistent call pricing functions across strikes (Bates, 1991). Once a sufficiently smooth pricing function has been obtained, the implied risk-neutral density can be recovered through numerical differentiation.
Spline-based approaches offer substantial flexibility but remain sensitive to data quality and sparse observations in the tails of the distribution.
Semi-parametric approaches occupy a middle ground between purely data-driven and fully parametric methodologies. Rather than modelling the risk-neutral density directly, these methods impose structure on intermediate quantities, most commonly the implied volatility smile.
Implied volatility smile methods: in practice, many market participants smooth the implied volatility smile rather than the option prices directly. Observed option prices are first converted into implied volatilities, after which a smooth volatility smile is fitted across strikes using parametric specifications or spline-based interpolation techniques (Shimko, 1993).
The smoothed volatility smile is subsequently mapped back into option prices, allowing the implied risk-neutral density to be recovered through numerical differentiation. These methods generally exhibit greater numerical stability, although tail estimation remains sensitive to extrapolation assumptions in illiquid regions of the smile.
Parametric and structural methodologies recover the implied risk-neutral distribution by imposing explicit assumptions on either the terminal distribution of asset prices or the stochastic process governing their evolution.
Parametric density models: a prominent class of methods assumes that the terminal risk-neutral distribution follows a particular parametric specification. One widely used approach models the distribution as a mixture of lognormal densities calibrated to observed option prices (Bahra, 1997; Melick and Thomas, 1997).
Parametric methods are computationally efficient and often yield economically interpretable measures of skewness, kurtosis, and tail risk. Their flexibility, however, is inherently constrained by the assumed functional form.
Dynamic option pricing models: rather than specifying the terminal distribution directly, structural approaches derive the implied density from an assumed stochastic process governing the evolution of the underlying asset price. Examples include stochastic volatility and jump-diffusion frameworks calibrated to observed option prices (Bates, 1995; Malz, 1995).
Within these models, the risk-neutral density emerges endogenously from the dynamics of the underlying asset under the risk-neutral measure. While theoretically appealing, such models are computationally intensive and sensitive to model misspecification.
Implementing the Breeden and Litzenberger (1978) result in practice requires a continuum of European option prices written on the same underlying asset, all sharing a common maturity and spanning a continuous range of strike prices from zero to infinity. Under such idealized conditions, the risk-neutral density can be recovered directly from the cross-section of option prices (at a given maturity date).
In practice, however, listed option markets provide only a sparse and discrete grid of strike prices, typically concentrated around the at-the-money (ATM) region. The absence of a complete continuum of option strikes, particularly in the deep in-the-money and far out-of-the-money regions, necessitates the use of interpolation across observed strikes and extrapolation into the tails in order to recover a smooth and arbitrage-free implied risk-neutral distribution.
Constructing a risk-neutral distribution requires option chain data (a set of calls and/or puts) for a single maturity, along with the underlying asset price, the prevailing risk-free rate, dividend assumptions, at the exact observation time of the market data.
Such data can be obtained from both free and commercial data providers. One of the most accessible sources is Yahoo! Finance; however, freely available option data is often subject to inconsistencies such as wide bid–ask spreads, stale quotes, and incomplete cross-sectional coverage of strikes, all of which can materially distort empirical estimation of the risk-neutral distribution (RND).
For our application, we employ simulated option data to illustrate the derivation of the implied risk-neutral distribution from an option chain within a controlled and internally consistent setting. This ensures that the resulting distribution remains aligned with the theoretical framework developed above.
From the collected option chain data, we first apply a series of standard filtering procedures designed to remove illiquid and economically inconsistent observations. In empirical applications, this typically includes liquidity screens, moneyness and maturity filters, implied-volatility sanity checks, and no-arbitrage constraints to mitigate errors arising from stale quotes, asynchronous observations, and market microstructure noise. Since the dataset employed here is simulated and internally consistent by construction, these preprocessing steps can be largely omitted.
Figure 3 below presents the implied volatility smile obtained from the simulated European call option chain after numerical inversion of the Black–Scholes–Merton pricing model. The smile is interpolated using a natural cubic spline over a dense strike grid spanning the filtered strike range of 4,000 to 6,000, under the assumptions of an underlying spot price of $5,300, a continuously compounded risk-free interest rate of 5.2%, and a remaining time-to-maturity of 30 days. The resulting smooth volatility curve serves as the key intermediate input for constructing a continuous and differentiable call pricing function required for subsequent risk-neutral density extraction.
Figure 3. Implied Volatility Smile
Source: computation by the author (with python)
The interpolated implied volatility smile is subsequently utilized to reprice European call options across a finely discretized strike grid, thereby constructing a smooth numerical approximation of the cross-sectional call price surface. The option implied risk neutral density is then recovered by applying the Breeden Litzenberger operator, corresponding to the second partial derivative of discounted call prices with respect to strike, to the smoothed pricing function. Figure 4 illustrates the resulting risk neutral density extracted from the simulated European call option chain under an underlying spot level of $5,300, a continuously compounded risk-free interest rate of 5.2%, and a remaining time to maturity of 30 days.
Figure 4. Implied Risk-Neutral Distribution
Source: computation by the author (with python)
You can download the Python code provided below for generating simulated call option chain data and the option-implied risk-neutral distribution, as discussed above.
Alternatively, you can download the R code below with the same functionality as in the Python file.
A primary limitation in empirical recovery of the risk-neutral distribution is the discrete nature of listed option strikes. The Breeden–Litzenberger framework assumes a continuum over strike space, whereas traded options are observed only on a sparse and uneven grid concentrated around the at-the-money region.
A second limitation arises from the unobservability of the distribution tails. Deep in-the-money and far out-of-the-money options are often illiquid or not quoted, implying that tail behaviour of the risk-neutral density must be inferred through extrapolation rather than direct market observation.
A separate issue is asynchronous option quotes. Since option prices across strikes are not necessarily recorded simultaneously, the resulting cross-section may embed timing mismatches, introducing bias in the reconstructed pricing function. This is typically addressed using end-of-day settlement data or synchronized snapshots.
In addition, different levels of market liquidity (due to different levels of bid ask spreads for example) across strikes introduces noise and heterogeneity in observed quotes. Illiquid contracts may exhibit stale or unreliable prices, which can distort the implied volatility surface even after basic filtering.
Finally, the reconstruction procedure does not explicitly impose no-arbitrage conditions or global smoothness constraints across strikes. As a result, when option prices are interpolated to form a continuous surface, the fitted call price function may exhibit local violations of convexity in strike space (e.g., small regions where butterfly spreads imply negative prices or non-monotonic curvature). Such violations are problematic because they imply the possibility of arbitrage and can lead to risk-neutral probability estimates that are not economically consistent.
Despite these limitations, the framework remains a useful reduced-form tool for extracting risk-neutral densities, provided appropriate smoothing and arbitrage constraints are imposed.
Bahra (1997) and Kim (2009) suggest that policymakers extract ex-ante risk-neutral distributions (RNDs) from interest rate, equity, and currency options to assess market-implied expectations and uncertainty around policy decisions. Unlike futures prices, which only reflect the conditional mean, RNDs incorporate higher-order information such as skewness and kurtosis, allowing for a more complete assessment of perceived tail risks and macro-financial stress. For example, during the February 2007 equity sell-off, the European Central Bank (ECB, 2007) used option-implied probability distributions (“fan charts”) to assess whether the move reflected extreme tail risk and to track the evolution of market expectations after stabilization.
Risk management units in investment banks use quantiles derived from implied RNDs to forecast extreme portfolio losses in a forward-looking manner. Compared to traditional historical simulation methods, RND-based approaches incorporate market-implied expectations and have been shown to provide improved performance relative to standard volatility-based models such as GARCH(1,1) (Chang, Chang, Huang, & Hsieh, 2011).
Macroprudential regulators transform option-implied volatility surfaces into arbitrage-consistent risk-neutral distributions to quantify system-wide financial vulnerabilities. By aggregating tail-risk measures across equities, currencies, and interest rates, these distributions can be used to construct time-series indicators of systemic stress and cross-asset fragility (Malz, 2014).
By combining option-implied risk-neutral distributions with empirical (physical) distributions, researchers can infer the market’s implicit risk preferences and aggregate degree of risk aversion (Bliss & Panigirtzoglou, 2004). This allows for the identification of time variation in investor sentiment and risk pricing across different investment horizons (Bliss & Panigirtzoglou, 2004; Gemmill & Saflekos, 2000).
The risk-neutral distribution is one of the few tools in finance that reveals how the market prices uncertainty based on the entire distribution of possible future states implied by option prices. It is widely used in practice to understand how the market is pricing downside risk, fat tails, and asymmetry that is directly used in volatility modelling, pricing, and risk management frameworks. From a practical perspective, it is one of the standard tools used to extract forward-looking information from option prices in both research and industry settings.
▶ Saral BINDAL Historical Volatility
▶ Saral BINDAL Implied Volatility and Option Prices
▶ Saral BINDAL Volatility curves: smiles and smirks
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The article was written in June 2026 by Saral BINDAL (Indian Institute of Technology Kharagpur, Metallurgical and Materials Engineering, 2024-2028 & Research assistant at ESSEC Business School).
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