Measures and statistics of business activity in global derivative markets

February 1, 2026February 1, 2026 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 how the business of derivatives markets has evolved over time and the pivotal role of the Black–Scholes–Merton option pricing model in their development.

Introduction

The derivatives market is among the most dynamic segments of global finance, serving as a tool for risk management, speculation, and price discovery across diverse asset classes. Spanning from bespoke over-the-counter contracts to standardized exchange-traded instruments, derivatives have become indispensable for investors, institutions, and corporations alike.

This post explores the derivatives landscape, examining market structures, contract types, underlying assets, and key statistics of business activity. It also highlights the pivotal role of the Black–Scholes–Merton model, which provided a theoretical framework for options pricing and catalysed the growth of derivatives markets.

Types of derivatives markets

The derivatives market can be categorized according to their market structure (over-the-counter derivatives and exchange-traded derivatives), the types of derivatives contracts traded (futures/forward, options, swaps), and the underlying asset classes involved (equities, interest rates, foreign exchange, commodities, and credit), as outlined below.

Market structure: over-the-counter derivatives and exchange-traded derivatives

Over-the-counter derivatives are privately negotiated, customized contracts between counterparties like banks, corporates, and hedge funds, traded via phone or electronic networks. OTC derivatives offer high flexibility in terms (price, maturity, quantity, delivery) but are less regulated, with decentralized credit risk management, no central clearing, low price transparency, and higher counterparty risk. They suit specialized or low-volume trades and often incubate new products.

Exchange-traded derivatives are standardized contracts traded on organized exchanges with publicly reported prices. Trades are cleared through a central clearing house that guarantees settlement, with daily marking-to-market and margining to reduce counterparty risk. ETDs are more regulated, transparent, and liquid, making them ideal for high-volume, widely traded instruments, though less flexible than OTC contracts.

Types of derivatives contracts

A derivative contract is a financial instrument that derives its values from an underlying asset. The four major types of such instruments are explained below.

A forward contract is a private agreement to buy or sell an asset at a fixed future date and price. It is traded over the counter between two counterparties (e.g., banks or clients). One party takes a long position (agrees to buy), the other a short position (agrees to sell). Settlement happens only at maturity, and contracts are customized, unregulated, and expose parties to direct counterparty risk.

A futures contract has the same economic purpose as a forward, future delivery at a fixed price, but is traded on an exchange with standardized terms. A clearing house stands between buyers and sellers and guarantees performance. Futures are marked to market daily so gains and losses are realized continuously. They are regulated, more transparent, and carry lower counterparty risk than forwards.

Options are contracts that give the holder the right but not the obligation to buy (call) or sell (put) an asset at a fixed strike price by a given expiration date. The buyer pays an upfront premium to the writer. If the option expires unexercised, the buyer loses only the premium. If exercised, the writer bears the payoff. Options can be American (exercise anytime) or European (exercise only at expiry) and are traded both on exchanges (standardized) and OTC (customized).

Swaps are bilateral contracts to exchange streams of cash flows over time, typically based on fixed versus floating interest rates or other reference indices. Payments are calculated on a notional principal that is not exchanged. Swaps are core OTC instruments for managing interest rate and financial risk.

Types of underlying asset classes

Underlying assets are the products on which a derivative instrument or contract derives its value. The most commonly traded underlying assets are explained below.

Equity derivatives include futures and options on stock indices, such as the S&P 500 Index. These instruments offer capital-efficient ways to manage market risk and enhance returns. Through index futures, institutional investors can achieve cost-effective hedging by locking in prices, while index options provide a non-linear, asymmetric payoff structure that protects against tail risk. Furthermore, equity swaps allow for the seamless exchange of total stock returns for floating interest rates, providing exposure to specific market segments without the capital requirements of direct physical ownership.

Interest rate derivatives include swaps and futures that help manage interest rate risk. Interest rate swaps involve exchanging fixed and floating payments, protecting banks against mismatches between loan income and deposit costs. Interest rate futures allow investors to lock in future borrowing or investment rates and provide insight into market expectations of monetary policy.

Commodity derivatives hedge price risk arising from storage, delivery, and seasonal supply-demand fluctuations. Forwards and futures on crude oil, natural gas, and power are widely used.

Foreign exchange derivatives include forward contracts and cross-currency swaps, allowing firms to hedge currency risk. Cross-currency swaps also support local currency bond markets by enabling hedging of interest and exchange rate risk.

Credit derivatives transfer the risk of default between counterparties. The most widely used is the credit default swap (CDS), which acts like insurance: the buyer pays a premium to receive compensation if a reference entity default.

Quantitative measures of derivatives market activity and size

This section presents the principal measures or statistics used to evaluate the size of the derivatives markets, covering both over-the-counter and exchange-traded instruments, the different derivatives products, and asset classes.

Notional outstanding and gross market value are the primary measures used to assess the size and economic exposure of OTC derivatives markets, while ETDs are typically evaluated using indicators such as open interest and trading volume.

Notional amount

Notional amount, or notional outstanding, is the total principal or reference value of all outstanding derivatives contracts. It captures the overall scale of positions in the derivatives market without reflecting actual market risk or cash exchanged.

For example let us consider a FX forward contract in which two parties agree to exchange $50 for euros in three months at a predetermined exchange rate. The notional amount is $50, because all cash flows (and gains or losses) from the contract are calculated with reference to this amount. No money is exchanged when the contract is initiated, and at maturity only the difference between the agreed exchange rate and the prevailing market rate determines the gain or loss computed on the $50 notional.

Now consider a call option on a stock with a strike price of $50. The notional amount is $50. The option buyer pays only an upfront premium, which is much smaller than $50, but the payoff of the option at maturity depends on how the market price of the stock compares to this $50 reference value.

When measuring notional outstanding in the derivatives market, the notional amounts of all individual contracts are simply added together. For example, one FX forward with a notional of $50 and two option contracts each with a notional of $50 result in a total notional outstanding of $150. This aggregated figure indicates the overall scale of derivatives activity, but it typically overstates actual economic risk because contracts may offset each other and only a fraction of the notional is ever exchanged.

Gross market value

Gross market value is the sum of the absolute values of all outstanding derivatives contracts with either positive or negative replacement (mark-to-market) values, evaluated at market prices prevailing on the reporting date. It reflects the potential scale of market risk and financial risk transfer, showing the economic exposure of a dealer’s derivatives positions in a way that is comparable across markets and products.

To continue the previous FX forward example, suppose a dealer has two outstanding FX forward contracts, each with a notional amount of $50. Due to movements in exchange rates, the first contract has a positive replacement value of $0.50 (the dealer would gain $0.50 if the contract were replaced at current market prices), while the second contract has a negative replacement value of –$0.40. The gross market value is calculated as the sum of the absolute values of these replacement values: |0.50| + |−0.40| = $0.90. Although the total notional outstanding of the two contracts is $100, the gross market value is only $0.90. This measure therefore reflects the dealer’s actual economic exposure to market movements at current prices, rather than the contractual size of the positions.

When this concept is extended to the entire derivatives market, the same distinction becomes apparent at a global scale. While the global derivatives market is often described as having hundreds of trillions of dollars in notional outstanding (approximately USD 850 trillion for OTC derivatives), the economically meaningful exposure is an order of magnitude smaller when measured using gross market value. Unlike notional amounts, gross market value aggregates current mark-to-market exposures, making it a more meaningful and comparable indicator of market risk and financial risk transfer across products and markets.

Open Interest

Open interest refers to the total number of outstanding derivative contracts that have not been closed, expired, or settled. It is calculated by adding the contracts from newly opened trades and subtracting those from closed trades. Open interest serves as an important indicator of market activity and liquidity, particularly in exchange-traded derivatives, as it reflects the level of active positions in the market. Measured at the end of each trading day, open interest is widely used as an indicator of market sentiment and the strength behind price trends.

For example on an exchange, a total of 100 futures contracts on crude oil are opened today. Meanwhile, 30 existing contracts are closed. The open interest at the end of the day would be: 100 (new contracts) − 30 (closed contracts) = 70 contracts. This indicates that 70 contracts remain active in the market, representing the total number of positions that traders are holding.

Trading Volume

Trading volume measures the total number of contracts traded over a specific period, such as daily, monthly, or annually. It provides insight into market liquidity and activity, reflecting how actively derivatives contracts are bought and sold. For OTC markets, trading volume is often estimated through surveys, while for exchange-traded derivatives, it is directly reported.

Consider the same crude oil futures market. If during a single trading day, 50 contracts are bought and 50 contracts are sold (including both new and existing positions), the trading volume for the day would be: 50 + 50 = 100 contracts

Here, trading volume shows how active the market is on that day (flow), while open interest shows how many contracts remain open at the end of the day (stock). High trading volume with low open interest may indicate rapid turnover, whereas high open interest with rising prices can signal strong bullish sentiment.

Key sources of statistics on global derivatives markets

Bank for International Settlements (BIS)

The Bank for International Settlements (BIS) provides quarterly statistics on exchange-traded derivatives (open interest and turnover in contracts, and notional amounts) and semiannual data on OTC derivatives outstanding (notional amounts and gross market values across risk categories like interest rates, FX, equity, commodities, and credit). All the data used in this post has been sourced from the BIS database.

Data are collected from over 80 exchanges for ETDs and via surveys of major dealers in 12 financial centers for OTC derivatives. BIS ensures comparability by standardizing definitions, consolidating country-level data, halving inter-dealer positions to avoid double counting, and converting figures into USD. Interpolations are used to fill gaps between triennial surveys, ensuring consistent time series for analysis.

International Swaps and Derivatives Association (ISDA)

ISDA develops and maintains standardized reference data and contractual frameworks that underpin global OTC derivatives markets. This includes machine-readable definitions and value lists for core market terms such as benchmark rates, floating rate options, currencies, business centers, and calendars, primarily derived from ISDA documentation (notably the ISDA Interest Rate Derivatives Definitions). The data are distributed via the ISDA Library and increasingly designed for automated, straight-through processing.

ISDA’s standards are created and updated through industry working groups and are widely used to support trade documentation, confirmation, clearing, and regulatory reporting. Initiatives such as the Common Domain Model (CDM) and Digital Regulatory Reporting (DRR) translate market conventions and regulatory requirements across multiple jurisdictions into consistent, machine-executable logic. While ISDA does not publish comprehensive market volume statistics, its frameworks play a central role in harmonizing OTC derivatives markets and enabling reliable post-trade transparency.

Futures Industry Association (FIA)

Futures Industry Association (FIA), via FIA Tech, provides comprehensive derivatives data including position limits, exchange fees, contract specifications, and trading volumes for futures/options across global products.

Sources aggregate from exchanges, indices (1,800+ products, 100,000+ constituents), and regulators for reference data like symbologist and corporate actions. The process involves standardizing data into consolidated formats with 500+ attributes, automating regulatory reporting (e.g., CFTC ownership/control), and ensuring compliance via databanks.

How to get the data

The data discussed in this article is drawn from the BIS, FIA and Visual Capitalist. For comprehensive statistics on global derivatives markets (both over-the-counter (OTC) and exchange-traded derivatives (ETDs)), the data are available at https://data.bis.org/ and for exchange-traded derivatives specifically, detailed data are provided by the Futures Industry Association (FIA) through its ETD volume reports, accessible at https://www.fia.org/etd-volume-reports. Data on equity spot market and real economy sectors are sourced from Visual Capitalist.

Derivatives market business statistics

Global derivatives market

In this section, we focus on two core measures of derivatives market activity and size: the notional amount outstanding and the gross market value, which together provide complementary perspectives on the scale of contracts and the associated economic exposure.

As of 30^th July 2025, the global derivatives market is estimated to have an outstanding notional value of approximately USD 964 trillion, according to the Bank for International Settlements (BIS). As illustrated in the figure below, the market is largely dominated by over-the-counter (OTC) derivatives, which account for nearly 88% of total notional amounts, whereas exchange-traded derivatives (ETDs) represent a comparatively smaller share of about USD 118 trillion.

Figure 1. Derivatives Markets: OTC versus ETD (2025)
Derivatives Markets: OTC and ETD (2025)
Source: computation by the author (BIS data of 2025).

Figure 2 below compares the scale of the global equity derivatives market with that of the underlying equity spot market as of mid-2025. The figure shows that, although equity derivatives represent a sizeable market in notional terms, they are still much smaller than the equity spot market measured by market capitalization. This suggests that the primary locus of economic value in equities remains in the spot market, while the derivatives market mainly represents contingent claims written on that underlying value rather than a comparable pool of market wealth. The relatively small gross market value of equity derivatives further indicates that only a limited portion of derivative notional translates into actual market exposure.

Figure 2. Equity Markets: Spot versus Derivatives (2025)

Source: computation by the author (BIS and Visual Capitalist data of 2025).

Data sources: global derivatives notional outstanding as of mid-2025 BIS OTC and exchange traded data; global equity spot market capitalization as of 2025 (Visual Capitalist).

Figure 3 below juxtaposes the global derivatives market with selected real-economy sectors to provide an intuitive comparison of scale. Values are reported in USD trillions and plotted on a logarithmic axis, such that equal distances along the horizontal scale correspond to ten-fold (×10) changes in magnitude rather than linear increments. This representation allows quantities that differ by several orders of magnitude to be meaningfully displayed within a single chart.

Interpreted in this manner, the figure illustrates that the notional size of derivatives markets far exceeds the market capitalization of major real-economy sectors, including technology, financials, energy, fast moving consumer goods (FMCG), and luxury. The comparison is illustrative rather than like-for-like, and is intended to contextualize the scale of financial contract exposure rather than to imply equivalent economic value or direct risk.

Figure 3. Scale of Global Derivatives Relative to Major Real-Economy Sectors (2025)

Source: computation by the author (BIS and Visual Capitalist data).

Data sources: BIS OTC derivatives statistics (June 2025) for notional outstanding; Visual Capitalist global stock market sector data (2025) for sector market capitalizations; companies market cap / Visual Capitalist for luxury company market caps.

OTC derivatives market

Figures 4 and 5 below illustrate the evolution of the OTC derivatives market from 1998 to 2025 using the two measures discussed above: outstanding notional amounts (Figure 4) and gross market value (Figure 5). As the data show, notional outstanding tends to overstate the effective economic size of the market, as it reflects contractual face values rather than actual risk exposure. By contrast, gross market value provides a more economically meaningful measure by capturing the current cost of replacing outstanding contracts at prevailing market prices.

Figure 4. Size of the OTC Derivatives Market (Notional amount)
Size of the OTC derivative market (Notional amount)
Source: computation by the author (BIS data).

Figure 5. Size of the OTC Derivatives Market (Gross market value)
Size of the OTC derivative market (Gross market value)
Source: computation by the author (BIS data).

The figure below illustrates the OTC derivatives market data as of 30^th July 2025 based on the two metrics discussed above: outstanding notional amounts and gross market value. As the data show, Gross market value (GMV) represents only about 2.6% of total notional outstanding, highlighting the large gap between contractual face values and economically meaningful exposure.

Figure 6. Size measure of the OTC derivatives market (2025)
Size of the OTC derivative market (2025)
Source: computation by the author (BIS data).

Exchange-traded derivatives market

Figure 7 below illustrates the growth of the exchange-traded derivatives market from 1993 to 2025, based on outstanding notional amounts (open interest) and turnover notional amounts (trading volume). For comparability across contracts and exchanges, open interest is expressed in notional terms by multiplying the number of open contracts by their contract size, yielding US dollar equivalents. Turnover is defined as the notional value of all futures and options traded during the period, with each trade counted once.

Figure 7. Size of the Exchange-Traded Derivatives Market
Size of the exchange traded derivatives market
Source: computation by the author (BIS data).

The figure below illustrates the exchange-traded derivatives market data as of 30^th July 2025 based on the two metrics discussed above: open interest and turnover (trading volume). The chart shows that only about 12%, of the open positions is actively traded, highlighting the difference between market size and the trading activity.

Figure 8. Size of the Exchange traded derivatives market (2025)

Source: computation by the author (BIS data).

Figure 9 below illustrates the evolution of the global exchange-traded derivatives market from 1993 to 2025, measured by outstanding notional amounts across major regions. The figure reveals a pronounced concentration of activity in North America and Europe, which drives most of the market’s expansion over time, while Asia-Pacific and other regions play a more modest role. Despite cyclical fluctuations, the overall trajectory is one of sustained long-run growth, underscoring the increasing importance of exchange-traded derivatives in global risk management and price discovery.

Figure 9. Size of the Exchange-Traded Derivatives Market by geographical locations
Size of the exchange traded derivatives market by geographic location
Source: computation by the author (BIS data).

Underlying asset classes

This section analyzes underlying asset-class statistics for derivatives traded in exchange-traded (ETD) and over-the-counter (OTC) markets.

Figure 10 below presents the distribution of exchange-traded derivatives (ETDs) activity across major underlying asset classes. When measured by the number of contracts traded (volume), the market is highly concentrated, with Equity derivatives dominating and accounting for the vast majority of activity. This is followed at a significant distance by Interest Rate and Commodity derivatives. However, this distribution reverses when measured by the notional value of outstanding contracts, where Interest Rate derivatives represent the largest share of the market due to the high underlying value of each contract.

Figure 10. Size of the exchange-traded derivatives market by asset classes
Size of the exchange traded derivatives market
Source: computation by the author (FIA data).

Figure 11 below presents the distribution of OTC derivatives activity across major underlying asset classes, measured by the outstanding notional amounts and displayed on a logarithmic scale. Read in this way, the chart shows that OTC activity is broadly diversified across interest rates, equity indices, commodities, foreign exchange, and credit, with interest rate and foreign exchange derivatives accounting for the largest contract volumes.

Figure 11. Size of the OTC derivatives market by asset classes
Size of the exchange traded derivatives market
Source: computation by the author (BIS data).

Role of the Black–Scholes–Merton (BSM) model

The Black–Scholes–Merton (BSM) model played a role in financial markets that extended well beyond option pricing. As argued by MacKenzie and Millo (2003), once adopted by traders and exchanges, it actively shaped how options markets were organized, priced, and operated rather than merely describing pre-existing price behaviour. Its use at the Chicago Board Options Exchange (CBOE) helped standardize quoting practices, enabled model-based hedging, and supported the rapid growth of liquidity in listed options markets.

At a broader level, MacKenzie (2006) shows that BSM contributed to a transformation in financial culture by embedding theoretical assumptions about risk, volatility, and rational pricing into everyday market practice. In this sense, BSM acted as an “engine” that reshaped markets and economic behaviour, not simply a “camera” recording them.

Beyond markets and firms, the diffusion of the BSM model also had wider societal implications. By formalizing risk as something that could be quantified, priced, and hedged, BSM contributed to a broader cultural shift in how uncertainty was perceived and managed in modern economies (MacKenzie, 2006). This reframing reinforced the view that complex economic risks could be controlled through mathematical models, with public perceptions of financial stability.

Why should you be interested in this post?

For anyone aiming for a career in finance, understanding the derivatives market is essential, as it is currently one of the most actively traded markets and is expected to grow further. Studying the statistics and business impact of derivatives provides valuable context on past challenges and the solutions developed to manage risks, offering a solid foundation for analyzing and navigating modern financial markets.

Useful resources

Academic research on option pricing

Black F. and M. Scholes (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. The Bell Journal of Economics and Management Science, 4(1), 141–183.

Hull J.C. (2022) Options, Futures, and Other Derivatives, 11th Global Edition, Chapter 15 – The Black–Scholes–Merton model, 338–365.

Academic research on the role of models

MacKenzie, D., & Millo, Y. (2003). Constructing a Market, Performing Theory: The Historical Sociology of a Financial Derivatives Exchange. American Journal of Sociology, 109(1), 107–145.

MacKenzie, D. (2006). An Engine, not a Camera: How Financial Models Shape Markets. MIT Press.

Data

Bank for International Settlements (BIS). Retrieved from BIS Statistics Explorer.

Futures Industry Association (FIA). Retrieved from ETD Volume Reports.

Visual Capitalist. Retrieved from The Global Stock Market by Sector.

Visual Capitalist. Retrieved from Piecing Together the $127 Trillion Global Stock Market.

About the author

The article was written in February 2026 by Saral BINDAL (Indian Institute of Technology Kharagpur, Metallurgical and Materials Engineering, 2024-2028 & Research assistant at ESSEC Business School).

▶ Discover all articles written by Saral BINDAL

Volatility curves: smiles and smirks

January 31, 2026January 31, 2026 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) analyzes the various shapes of volatility curves observed in financial markets and explains how they reveal market participants’ beliefs about future asset price distributions as implied by option prices.

Introduction

In financial markets characterized by uncertainty, volatility is a fundamental factor shaping the dynamics of the prices of financial instruments. Implied volatility stands out as a key metric as a forward-looking measure that captures the market’s expectations of future price fluctuations, as reflected in current market prices of options.

Implied volatility is inherently a two-dimensional object, as it is indexed by strike K and maturity T. The collection of these implied volatilities across all strikes and maturities constitutes the volatility surface. Under the Black–Scholes–Merton (BSM) framework, volatility is assumed to be constant across strikes and maturities, in which case the volatility surface would degenerate into a flat plane. Empirically, however, the volatility surface is highly structured and varies significantly across both strike and maturity.

Accordingly, this post focuses on implied volatility curves across moneyness for a fixed maturity (i.e. cross-sections of the volatility surface), examining their canonical shapes, economic interpretation, and the insights they reveal about market beliefs and risk preferences.

Option pricing

Option pricing aims to determine the fair value of options (calls and puts). One of the most widely used frameworks for this purpose is the Black–Scholes–Merton (BSM) model, which expresses the option value as a function of five key inputs: the underlying asset price S, the strike price K, time to maturity T, the risk-free interest rate r, and volatility σ. Given these parameters, the model yields the theoretical value of the option under specific market assumptions. The details of the BSM option pricing formulas along with variable definitions can be found in the article Black-Scholes-Merton option pricing model.

Implied volatility

In the Black–Scholes–Merton (BSM) model, volatility is an unobservable parameter, representing the expected future variability of the underlying asset over the option’s remaining life. In practice, implied volatility is obtained by inverting the BSM pricing formula (using numerical methods such as the Newton–Raphson algorithm) to find the specific volatility that equates the BSM theoretical price to the observed market price. The details for the mathematical process of calculation of implied volatility can be found in Implied Volatility and Option Prices.

Moneyness

Moneyness describes the relative position of an option’s strike price K with respect to the current underlying asset price S. It indicates whether the option would have a positive intrinsic value if exercised at the current moment. Moneyness is typically parameterized using ratios such as K/S or its logarithmic transform.

Moneyness formula

In practice, moneyness classifies an option based on its intrinsic value. An option is said to be in-the-money (ITM) if it has positive intrinsic value, at-the-money (ATM) if its intrinsic value is zero, and out-of-the-money (OTM) if its intrinsic value is zero and immediate exercise would not be optimal. In terms of the relationship between the underlying asset price (S) and the strike price (K), a call option is ITM when S > K, ATM when S = K, and OTM when S < K. Conversely, a put option is ITM when S < K, ATM when S = K, and OTM when S > K.

The payoff, that is the cash flow realized upon exercising the option at maturity T, is given for call and put options by:

Payoff formula for call and put options

where S_T is the underlying asset price at the time the option is exercised.

Figure 1 below illustrates the payoff of a call option, that is the call option value at maturity as a function of its underlying asset price. The call option’s strike price is assumed to be equal to $4,600. For an underlying price of $3,000, the call option is out-of-the-money (OTM); for a price of $4,600, the call option is at-the-money (ATM); and for a price of $7,000, the call option is in-the-money (ITM) and worth $2,400.

Figure 1. Payoff for a call option and its moneyness (OTM, ATM and ITM)

Source: computation by the author.

Similarly, Figure 2 below illustrates the payoff of a put option, that is the put option value at maturity as a function of its underlying asset price. The put option’s strike price is assumed to be equal to $4,600. For an underlying price of $3,000, the put option is in-the-money (ITM) and worth $1,600; for a price of $4,600, the put option is at-the-money (ATM); and for a price of $7,000, the put option is out-of-the-money (OTM).

Figure 2. Payoff for a put option and its moneyness (OTM, ATM and ITM)

Source: computation by the author.

Figure 3 below illustrates the temporal dynamics of moneyness for a European call option with a strike price of $4,600, showing how the option transitions between out-of-the-money, at-the-money, and in-the-money states as the underlying asset price moves relative to the strike over its lifetime.

Figure 3. Evolution of a call option moneyness

Source: computation by the author.

Similarly, Figure 4 below illustrates the temporal dynamics of moneyness for a European put option with a strike price of $4,600, showing how the option transitions between out-of-the-money, at-the-money, and in-the-money states as the underlying asset price moves relative to the strike over its lifetime.

Figure 4. Evolution of a put option moneyness

Source: computation by the author.

You can download the Excel file below for the computation of moneyness of call and put options as discussed in the above figures.

Empirical observation: implied volatility depends on moneyness

Smiles and smirks

Volatility curves refer to plots of implied volatility across different strikes for options with the same maturity. Two distinct shapes are commonly observed: the “volatility smile” and the “volatility smirk”.

A volatility smile is a symmetric pattern commonly observed in options markets. For a given underlying asset and expiration date, it is defined as the mapping of option strike prices to their Black–Scholes–Merton implied volatilities. The term “smile” refers to the distinctive shape of the curve: implied volatility is lowest near the at-the-money (ATM) strike and rises for both lower in-the-money (ITM) strikes and higher out-of-the-money (OTM) strikes.

A volatility smirk (also called skew) is an asymmetric pattern in the implied volatility curve and is mainly observed in the equity markets. It is characterized by high implied volatilities at lower strikes and progressively lower implied volatilities as the strike increases, resulting in a downward-sloping profile. This shape reflects the uneven distribution of implied volatility across strikes and stands in contrast to the more symmetric volatility smile observed in other markets.

Stylized facts about the implied volatility curve across markets

Stylized facts characterizing implied volatility curves are persistent and statistically robust empirical regularities observed across financial markets. Below, I discuss the key stylized facts for major asset classes, including equities, foreign exchange, interest rates, commodities, and cryptocurrencies.

Equity market: For major equity indices, the implied volatility curve at a given maturity is typically a negatively sloped smirk: IV is highest for out of the money puts and declines as the strike moves up, rather than forming a symmetric smile (Zhang & Xiang, 2008). This left skew is persistent across maturities and provides useful signals at the individual stock level, where steeper smirks (higher OTM put vs ATM IV) forecast lower subsequent returns, consistent with markets pricing crash risk into downside options (Xing, Zhang & Zhao, 2010).

FX market: For foreign currency options, implied volatility curves most often display a U shaped smile: IV is lowest near at the money and higher for deep in or out of the money strikes, especially for major FX pairs (Daglish, Hull & Suo, 2007). The degree of symmetry depends on the correlation between the FX rate and its volatility, so near zero correlation gives a roughly symmetric smile, while non zero correlations generate skews or smirks that have been empirically documented in options on EUR/USD, GBP/USD and AUD/USD (Choi, 2021).

Commodity market: For commodity options, cross market evidence shows that implied volatility curves are generally negatively skewed with positive curvature, meaning they exhibit smirks rather than flat surfaces, with higher IV for downside strikes but still some smile like curvature (Jia, 2021). Studies on crude oil and related commodities also find pronounced smiles and smirks whose strength varies with fundamentals such as inventories and hedging pressure, reinforcing it is a core stylized fact in commodity derivatives (Soini, 2018; Vasseng & Tangen, 2018).

Fixed income market: Swaption markets display smiles and skews on their volatility curves: for a given expiry and tenor, implied volatility typically curves in moneyness and may tilt up or down depending on the correlation between the underlying rate and volatility (Daglish, Hull & Suo, 2007). Empirical work on the swaption volatility cube shows that simple one factor or SABR lifted constructions do not capture the full observed smile, indicating that a rich, strike and maturity dependent IV surface is itself a stylized feature of interest rate options (Samuelsson, 2021).

Crypto market: Bitcoin options exhibit a non flat implied volatility smile with a forward skew, and short dated options can reach very high levels of implied volatility, reflecting heavy tails and strong demand for certain strikes (Zulfiqar & Gulzar, 2021). Because of this forward skew, the paper concludes that Bitcoin options “belong to the commodity class of assets,” although later studies show that the Bitcoin smile can change shape across regimes and is often flatter than equity index skew (Alexander, Kapraun & Korovilas, 2023).

Summary of stylized facts about implied volatility

An Empirical Analysis of S&P 500 Implied Volatility

This section describes the data, methodology, and empirical considerations for the analysis of implied volatility of put options written on the S&P 500 index. We begin by highlighting a classical challenge in cross-sectional option data: asynchronous trading.

Asynchronous trading and measurement error

In empirical option pricing, the non-synchronous observation of option prices and the underlying asset price generates measurement errors in implied volatility estimation, as the building of the volatility curve based on an option pricing model relies on option prices with the underlying price observed at the same point of time.

Formally, let the option price C be observed at time t_c, while the underlying asset price S is observed at time t_s with t_s ≠ t_c. The observed option price therefore satisfies

Asynchronous call option price and underlying asset price

Since the option price at time t_c depends on the latent spot price S(t_c), rather than the asynchronously observed price S(t_s), this mismatch introduces measurement error in the underlying price variable and implied volatility at the end.

Various standard filters including no-arbitrage, liquidity, moneyness, maturity, and implied-volatility sanity checks are typically applied to mitigate errors-in-variables arising from asynchronous observations of option prices and their underlying assets.

Example: options on the S&P 500 index

Consider the following sample of option data written on the S&P 500 index. Data can be obtained from FirstRate Data.

Figure 5 below illustrates the volatility smirk (or skew) for an option chain (a series of option prices for the same maturity) written on the S&P 500 index traded on 3rd July 2023 with time to maturity of 2 days after filtering it out from the above data.

Figure 5. Volatility smirk for put option prices on the S&P 500 index
Volatility smirk computed for put option on the S&P 500 index
Source: computation by the author.

You can download the Excel file below to compute the volatility curve for put options on the S&P 500 index.

Economic Insights

This section explains how the shape of the implied volatility curve reveals key economic forces in options markets, including demand for crash protection, leverage-driven volatility feedback effects, and the role of market frictions and limits to arbitrage.

Demand for crash protection:

Out-of-the-money put options serve as insurance against market crashes and hedge tail risk. Because this demand is persistent and largely one-sided, put options become expensive relative to their Black–Scholes-Merton values, resulting in elevated implied volatilities at low strikes. This excess pricing reflects the market’s willingness to pay a premium to insure against rare but severe losses.

Leverage and volatility feedback effects:

When equity prices fall, firms become more leveraged because the value of equity declines relative to debt. Higher leverage makes equity riskier, increasing expected future volatility. Anticipating this effect, markets assign higher implied volatility to downside scenarios than to upside moves. This endogenous feedback between price declines, leverage, and volatility naturally produces a negative volatility skew, even in the absence of crash-risk preferences.

Market frictions and limits to arbitrage:

In practice, option writers are subject to capital constraints, margin requirements, and exposure to jump and tail risk. These constraints limit their capacity to supply downside protection at low prices. As a result, downside options embed not only compensation for fundamental crash risk, but also a risk premium reflecting the balance-sheet costs and risk-bearing capacity of intermediaries. The observed volatility skew therefore arises endogenously from limits to arbitrage rather than purely from differences in underlying return distributions.

Conclusion

The dependence of implied volatility on moneyness is neither an anomaly nor a technical artifact. It reflects market expectations, risk preferences, and the perceived probability of extreme outcomes. For both pedagogical and investment applications, the implied volatility curve is a central tool for understanding how markets price tail and downside risk.

Why should I be interested in this post?

Understanding implied volatility and its relationship with moneyness extends beyond option pricing, offering insights into how markets perceive risk and assess the likelihood of extreme events. Patterns such as volatility smiles and skews reflect investor behavior, the demand for protection, and the asymmetric emphasis on potential losses over gains, providing a clearer view of both pricing anomalies and the economic forces that shape financial markets.

Useful resources

Academic research on Option pricing

Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities, Journal of Political Economy, 81(3), 637–654.

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 15 – The Black-Scholes-Merton model, 343-375.

Merton, R.C. (1973). Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4(1), 141–183.

Academic research on Stylized facts

Alexander, C., Kapraun, J. & Korovilas, D. (2023) Delta hedging bitcoin options with a smile, Quantitative Finance, 23(5), 799–817.

Bakshi, G., Cao, C., & Chen, Z. (1997). Empirical performance of alternative option pricing models, The Journal of Finance, 52(5), 2003–2049.

Bates, D. S. (1991). The crash of ’87: Was it expected? The evidence from options markets, The Journal of Finance, 46(5), 1777–1819.

Bates, D. S. (2000). Post-’87 crash fears in the S&P 500 futures option market, Journal of Econometrics, 94(1–2), 181–238.

Choi, K. (2021) Foreign exchange rate volatility smiles and smirks, Applied Stochastic Models in Business and Industry, 37(3), 405–425.

Daglish, T., Hull, J. & Suo, W. (2007) Volatility surfaces: theory, rules of thumb, and empirical evidence, Quantitative Finance, 7(5), 507–524.

Jia, G. (2021) The implied volatility smirk of commodity options, Journal of Futures Markets, 41(1), 72–104.

Samuelsson, A. (2021) Empirical study of methods to complete the swaption volatility cube. Master’s thesis, Uppsala University.

Soini, E. (2018) Determinants of volatility smile: The case of crude oil options. Master’s thesis, University of Vaasa.

Xing, Y., Zhang, X. & Zhao, R. (2010) What does individual option volatility smirk tell us about future equity returns? Review of Financial Studies, 23(5), 1979–2017.

Zhang, J.E. & Xiang, Y. (2008) The implied volatility smirk, Quantitative Finance, 8(3), 263–284.

Zulfiqar, N. & Gulzar, S. (2021) Implied volatility estimation of bitcoin options and the stylized facts of option pricing, Financial Innovation, 7(1), 67.

About the author

The article was written in January 2026 by Saral BINDAL (Indian Institute of Technology Kharagpur, Metallurgical and Materials Engineering, 2024-2028 & Research assistant at ESSEC Business School).

▶ Discover all articles written by Saral BINDAL

Implied Volatility and Option Prices

January 30, 2026December 23, 2025 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 how implied volatility is calculated or extracted from option prices using an option pricing model.

Introduction

In financial markets characterized by uncertainty, volatility is a fundamental factor shaping the pricing and dynamics of financial instruments. Implied volatility stands out as a key metric as a forward-looking measure that captures the market’s expectations of future price fluctuations, as reflected in current market prices of options.

The Black-Scholes-Merton model

In the early 1970s, Fischer Black and Myron Scholes jointly developed an option pricing formula, while Robert Merton, working in parallel and in close contact with them, provided an alternative and more general derivation of the same formula.

Together, their work produced what is now called the Black Scholes Merton (BSM) model, which revolutionized investing and led to the award of 1997 Nobel Prize in Economic Sciences in Memory of Alfred Nobel to Myron Scholes and Robert Merton “for a new method to determine the value of derivatives,” developed in close collaboration with the late Fischer Black.

The Black-Scholes-Merton model provides a theoretical framework for options pricing and catalyzed the growth of derivatives markets. It led to development of sophisticated trading strategies (hedging of options) that transformed risk management practices and financial markets.

The model is built on several key assumptions such as, the stock price follows a geometric Brownian motion with constant drift and volatility, no arbitrage opportunities, constant risk-free interest rate and options are European-style (options that can only be exercised at maturity).

Key Parameters

In the BSM model, there are five essential parameters to compute the theoretical value of a European-style option is calculated are:

Strike price (K): fixed price specified in an option contract at which the option holder can buy (for a call) or sell (for a put) the underlying asset if the option is exercised.
Time to expiration (T): time left until the option expires.
Current underlying price (S₀): the market price of underlying asset (commodities, precious metals like gold, currencies, bonds, etc.).
Risk-free interest rate (r): the theoretical rate of return on an investment that is continuously compounded per annum.
Volatility (σ): standard deviation of the returns of the underlying asset.

The strike price (exercise price) and time to expiration (maturity) correspond to characteristics of the option while the current underlying asset price, the risk-free interest rate, and volatility reflect market conditions.

Option payoff

The payoff for a call option gives the value of the option at the moment it expires (T) and is given by the expression below:

Payoff formula for call option

Where C_T is the call option value at expiration, S_T the price of the underlying asset at expiration, and K is the strike price (exercise price) of the option.

Figure 1 below illustrates the payoff function described above for a European-style call option. The example considers a European call written on the S&P 500 index, with a strike price of $5,000 and a time to maturity of 30 days.

Figure 1. Payoff value as a function of the underlying asset price.
Payoff function
Source: computation by the author.

Call option value

While the value of an option is known at maturity (being determined by its payoff function), its value at any earlier time prior to maturity, and in particular at issuance, is not directly observable. Consequently, a valuation model is required to determine the option’s price at those earlier dates.

The Black–Scholes–Merton model is formulated as a stochastic partial differential equation and the solution to the partial differential equation (PDE) gives the BSM formula for the value of the option.

For a European-style call option, the call option value at issuance is given by the following formula:

Formula for the call option value according to the BSM model

with

Formula for the call option value according to the BSM model

Where the notations are as follows:

C₀= Call option value at issuance (time 0) based on the Black-Scholes-Merton model
K = Strike price (exercise price)
T = Time to expiration
S₀ = Current underlying price (time 0)
r = Risk-free interest rate
σ = Volatility of the underlying asset returns
N(·) = Cumulative distribution function of the standard normal distribution

Figure 2 below illustrates the call option value as a function of the underlying asset price. The example considers a European call written on the S&P 500 index, with a strike price of $5,000 and a time to maturity of 30 days. The current price of the underlying index is $6,000, and the risk-free interest rate is set at 3.79% corresponding to the 1-month U.S. Treasury yield, and the volatility is assumed to be 15%.

Figure 2. Call option value as a function of the underlying asset price.

Source: computation by the author (BSM model).

Option and volatility

In the Black–Scholes–Merton model, the value of a European call or put option is a monotonically increasing function of volatility. Higher volatility increases the probability of finishing in-the-money while losses remain limited to the option premium, resulting in a strictly positive vega (the first derivative of the option value with respect to volatility) for both calls and puts.

As volatility approaches zero, the option value converges to its intrinsic value, forming a lower bound. With increasing volatility, option values rise toward a finite upper bound equal to the underlying price for calls (and bounded by the strike for puts). An inflection point occurs where volga (the second derivative of the option value with respect to volatility) changes sign: at this point vega is maximized (at-the-money) and declines as the option becomes deep in- or out-of-the-money or as time to maturity decreases.

The upper limit and the lower limit for the call option value function is given below (Hull, 2015, Chapter 11).

Formula for upper and lower limits of the option price

Figure 3 below illustrates the value of a European call option as a function of the underlying asset’s price volatility. The example considers a European call written on the S&P 500 index, with a strike price of $5,000 and a time to maturity of 30 days. The current price of the underlying index is $6,000, and the risk-free interest rate is set at 3.79% corresponding to the 1-month U.S. Treasury yield. A deliberately wide (and economically unrealistic) range of volatility values is employed in order to highlight the theoretical limits of option prices: as volatility tends to infinity, the option value converges to an upper bound ($6,000 in our example), while as volatility approaches zero, the option value converges to a lower bound $1,015.51).

Figure 3. Call option value as a function of price volatility

Source: computation by the author (BSM model).

Volatility: the unobservable parameter of the model

When we think of options, the basic equation to remember is “Option = Volatility”. Unlike stocks or bonds, options are not primarily quoted in monetary units (dollars or euros), but rather in terms of implied volatility, expressed as a percentage.

Volatility is not directly observable in financial markets. It is an unobservable (latent) parameter of the pricing model, inferred endogenously from observed option prices through an inversion of the valuation formula given by the BSM model. As a result, option markets are best interpreted as markets for volatility rather than markets for prices.

Out of the five essential parameters of the Black-Scholes-Merton model listed above, the volatility parameter is the unobservable parameter as it is the future fluctuation in price of the underlying asset over the remaining life of the option from the time of observation. Since future volatility cannot be directly observed, practitioners use the inverse of the BSM model to estimate the market’s expectation of this volatility from option market prices, referred to as implied volatility.

Implied Volatility

In practice, implied volatility is the volatility parameter that when input into the Black-Scholes-Merton formula yields the market price of the option and represents the market’s expectation of future volatility.

Calculating Implied volatility

The BSM model maps five input variables (S, K, r, T, σ_implied) to a single output variable uniquely: the call option value (Price), such that it’s a bijective function. When the market call option price (C_BSM) is known, we invert this relationship using (S, K, r, T, C_BSM) as inputs to solve for the implied volatility, σ_implied.

Formula for implied volatility

Newton-Raphson Method

As there is no closed form solution to calculate implied volatility from the market price, we need a numerical method such as the Newton–Raphson method to compute it. This involves finding the volatility for which the Black–Scholes–Merton option value C_BSM equals the observed market option price C_Market.

We define the function f as the difference between the call option value given by the BSM model and the observed market price of the call option:

Function for the Newton-Raphson method.

Where x represents the unknown variable (implied volatility) to find and C_Market is considered as a constant in the Newton–Raphson method.

Using the Newton-Raphson method, we can iteratively estimate the root of the function, until the difference between two consecutive estimations is less than the tolerance level (ε).

Formula for the iterations in the Newton-Raphson method

In practice, the inflexion point (Tankov, 2006) is taken as the initial guess, because the function f(x) is monotonic, so for very large or very small initial values, the derivative becomes extremely small (see Figure 3), causing the Newton–Raphson update step to overshoot the root and potentially diverge. Selecting the inflection point also minimizes approximation error, as the second derivative of the function at this point is approximately zero, while the first derivative remains non-zero.

Formula for calculating the volatility at inflexion point.

Where σ_inflection is the volatility at the inflection point.

Figure 4 below illustrates how implied volatility varies with the call option price for different values of the market price (computed using the Newton–Raphson method). As before, the example considers a European call written on the S&P 500 index, with a strike price of $5,000 and a time to maturity of 30 days. The current level of the underlying index is $6,000, and the risk-free interest rate is set at 3.79% corresponding to the 1-month U.S. Treasury yield.

Figure 4. Implied volatility vs. Call Option value
Implied volatility as a function of call option price
Source: computation by the author.

You can download the Excel file provided below, which contains the calculations and charts illustrating the payoff function, the option price as a function of the underlying asset’s price, the option price as a function of volatility, and the implied volatility as a function of the option price.

You can download the Python code provided below, to calculate the price of a European-style call or put option and calculate the implied volatility from the option market price (BSM model). The Python code uses several libraries.

Alternatively, you can download the R code below with the same functionality as in the Python file.

Why should I be interested in this post?

The seminal Black–Scholes–Merton model was originally developed to price European options. Over time, it has been extended to accommodate a wide range of derivatives, including those based on currencies, commodities, and dividend-paying stocks. As a result, the model is of fundamental importance for anyone seeking to understand the derivatives market and to compute implied volatility as a measure of risk.

Useful resources

Academic research

Black F. and M. Scholes (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. The Bell Journal of Economics and Management Science, 4(1), 141–183.

Hull J.C. (2022) Options, Futures, and Other Derivatives, 11th Global Edition, Chapter 15 – The Black–Scholes–Merton model, 338–365.

Cox J.C. and M. Rubinstein (1985) Options Markets, First Edition, Chapter 5 – An Exact Option Pricing Formula, 165-252.

Tankov P. (2006) Calibration de Modèles et Couverture de Produits Dérivés (Model calibration and derivatives hedging), Working Paper, Université Paris-Diderot. Available at https://cel.hal.science/cel-00664993/document.

About the BSM model

The Nobel Prize Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997

Harvard Business School Option Pricing in Theory & Practice: The Nobel Prize Research of Robert C. Merton

Other

NYU Stern Volatility Lab Volatility analysis documentation.

About the author

The article was written in December 2025 by Saral BINDAL (Indian Institute of Technology Kharagpur, Metallurgical and Materials Engineering, 2024-2028 & Research assistant at ESSEC Business School).

▶ Discover all articles written by Saral BINDAL

The Golden Boy: Une immersion dans l’univers des banques d’investissement

January 1, 2026December 22, 2024 by Lucas BAURIANNE

Dans cet article, Lucas BAURIANNE (ESSEC Business School, Programme Grande Ecole – Master in Management, 2024-2027) nous propose de découvrir The Golden Boy, une bande dessinée innovante qui retrace l’aventure d’un étudiant en école de commerce découvrant les rouages des banques d’investissement. Ce récit, autant éducatif que captivant, aborde les concepts fondamentaux de la finance de marchés et les secrets pour réussir dans ce domaine compétitif.

Couverture de la bande dessinée The Golden Boy.
Logo de l’entreprise
Source : Lucas Baurianne.

Une immersion complète dans la finance de marché

The Golden Boy se distingue par son approche unique : intégrer la théorie et la pratique dans une narration inspirante. À travers plus de 110 pages, plus de 40 concepts de finance de marché sont expliqués avec simplicité et profondeur. Vous découvrirez par exemple des notions comme le pricing des options, les mécanismes de trading algorithmique, et les dynamiques des marchés obligataires.

Des insights concrets pour réussir

En plus de la théorie, la BD offre des conseils pratiques sur la préparation aux stages, des astuces pour briller lors des entretiens, et des récits inspirés de la réalité. Les étudiants peuvent se reconnaître dans le parcours du protagoniste, un jeune plein d’ambition qui découvre les codes des banques d’investissement et décroche une opportunité dans une prestigieuse banque américaine à Wall Street.

Cas pratiques et actualité

Un autre aspect fascinant de The Golden Boy est l’intégration de cas pratiques liés à l’actualité présidentielle américaine. Ces exemples permettent de comprendre comment les événements politiques influencent les marchés financiers et les décisions stratégiques des traders.

Trois concepts financiers à découvrir dans la BD

Les produits dérivés

Avec The Golden Boy , vous comprendrez l’utilisation des produits dérivés et leurs objectifs, comme la gestion des risques ou la spéculation. Appréhender leur pricing, leurs payoffs et les facteurs qui influencent leur valeur. Ces produits sont évalués à l’aide de modèles tels que Black-Scholes, prenant en compte des éléments comme la volatilité, la durée jusqu’à l’échéance et les taux d’intérêt.

Les stratégies de couverture

Avec The Golden Boy , vous découvrirez comment les traders et les investisseurs utilisent des instruments dérivés, tels que les options, les futures ou les swaps, pour se protéger contre les risques de marché. Ces stratégies permettent de limiter les pertes potentielles liées à des fluctuations imprévues des actifs sous-jacents, comme les actions, les devises ou les matières premières. La BD illustre ces notions à travers des exemples concrets, comme la protection contre la volatilité des marchés lors d’événements géopolitiques ou économiques majeurs, montrant comment une couverture bien pensée peut sécuriser les portefeuilles tout en maintenant des opportunités de profit.

Les Greeks en finance

Avec The Golden Boy , vous maitriserez les Greeks, des outils fondamentaux en finance pour évaluer et gérer les risques associés aux options. Dans la BD, ces concepts sont illustrés à travers des cas pratiques, tels que l’effet des élections présidentielles américaines sur la volatilité des marchés financiers, offrant un aperçu concret de leur application dans des contextes réels.

Pourquoi devriez-vous lire cette BD ?

Que vous soyez étudiant curieux ou passionné par la finance, cette BD vous permettra de mieux comprendre un univers complexe et captivant. Elle a été réalisée avec l’aide de traders issus des plus grandes banques d’investissement et hedge funds, garantissant une authenticité et une précision rare dans le domaine.

La bande dessinée The Golden Boy est aussi un excellent point de départ pour ceux et celles qui envisagent de postuler dans les banques d’investissement. Elle offre un aperçu réaliste des défis et des opportunités de ce secteur.

Articles du blog SimTrade

Expériences professionnelles

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▶ Aastha DAS My experience as an investment banking analyst intern at G2 Capital Advisors

▶ Mickael RUFFIN My Internship Experience as a Structured Finance Analyst at Société Générale

▶ Ziqian ZONG My experience as a Quantitative Investment Intern in Fortune Sg Fund Management

Techniques financières

▶ Jayati WALIA Black-Scholes-Merton option pricing model

▶ Luis RAMIREZ Understanding Options and Options Trading Strategies

▶ Akshit GUPTA Option Greeks – Delta Gamma Vega Theta

Ressources utiles

LinkedIn Vidéo The Golden Boy

A propos de l’auteur

Cet article a été écrit en décembre 2024 par Lucas BAURIANNE (ESSEC Business School, Programme Grande Ecole – Master in Management, 2024-2027).

Analysis of “The Madoff Affair” documentary

January 28, 2026December 26, 2023 by Raphael TRAEN

In this article, Raphael TRAEN (ESSEC Business School, Global BBA, 2023-2024) analyzes “The Madoff Affair” documentary and explains the key financial concepts related to this documentary.

Key characters in the documentary

Bernard Madoff: key person, the admitted mastermind of the Ponzi scheme
Avellino: partner in Avellino and Bienes, advising its clients to invest with Madoff
Bienes: accountant for Madoff’s father-in-law, later partner in Avellino and Bienes, advising its clients to invest with Madoff

Summary of the documentary

Bernard Lawrence Madoff (“Bernie”) was an American stockbroker, market maker and an unofficial investment advisor (because he did not have the necessary license to do so) who operated what has been considered the largest Ponzi scheme in history. He defrauded investors out of billions over a long period.

The Madoff Affair

How did the scheme work?

Madoff’s Ponzi scheme was a classic example of a “pyramid scheme,” in which money from new investors is used to pay returns to earlier investors, creating the illusion of strong returns. Madoff claimed to be investing in a “secret” arbitrage strategy that generated consistent returns, even during periods of market downturn.

In reality, Madoff was simply lying to investors and using the money to pay returns to existing investors and to enrich himself. He kept his scheme going by attracting new investors, who were lured by the promise of high returns and the reputation of Madoff, who was a well-respected figure on Wall Street.

Bernard Madoff was able to maintain his Ponzi scheme for so long in part because he had help from two of his closest associates: Avellino and Bienes. Avellino and Bienes were investment advisors who were responsible for soliciting investments from Madoff’s funds. They were also responsible for creating false account statements that showed investors were making consistently high returns.

Avellino and Bienes first met Madoff and were impressed by his reputation and his consistent track record of high returns. They even approached Madoff about managing their own investments. Madoff agreed, and Avellino and Bienes began to introduce Madoff to their own clients.

Avellino and Bienes were instrumental in helping Madoff build his Ponzi scheme. They were able to attract new investors to Madoff’s funds by touting his track record and his reputation for integrity.

Technical details about the Madoff investment strategy

Bernie Madoff told his investors he was using a legitimate investing strategy called split-strike conversion. This strategy involves buying a stock index and simultaneously purchasing put options to limit the downside potential and selling call options to generate additional income.

Evolution of the Fairfield Sentry fund of Madoff Source: Madoff

Statistical measures of the Fairfield Sentry fund of Madoff Source: Bernard and Boyle (2009)

Should you be more interested in this strategy I definitely recommend watching the following video explaining the strategy with an example:

Bernie Madoff’s infamous split-strike conversion strategy

Theoretically, this strategy aims to provide a steady stream of income while protecting against significant losses. However, Madoff’s claims about his split-strike conversion strategy were entirely fabricated. He was not actually making these trades or generating the reported returns. Instead, he was using money from new investors to pay off existing investors, replicating a classic Ponzi scheme. This is also further confirmed by the picture I added above comparing the different strategies. The Fairfield Sentry fund was one controlled by Madoff. You can immediately see that the return is higher than what it would be according to the strategy and also that the standard deviation is much lower.

The downfall of the scheme

The Madoff Ponzi scheme began to unravel in the fall of 2008, as the global financial crisis took hold. As investors grew increasingly nervous about their investments, they began to withdraw their money from Madoff’s funds. Madoff was unable to meet these withdrawals, and the scheme collapsed.

In December 2008, Madoff’s sons, Mark and Andrew, confronted him about the scheme. Madoff confessed to his sons, and they immediately contacted the FBI.

One important person we should certainly not forget to mention is Markopolos, an American investor who accused Bernard Madoff of running a Ponzi scheme. He warned the SEC multiple times about Madoff’s suspicious investment returns and opaque investment strategy, but the SEC did not take action until after the collapse of Madoff’s Ponzi scheme in 2008. Markopolos was subsequently hailed as a hero for his efforts to expose the fraud.

Markopolos also believed that Madoff was using his position as a market maker to front-run his clients’ trades. This means that Madoff was using his knowledge of his clients’ impending trades to make profitable trades for himself before his clients’ trades were executed. This would allow Madoff to profit from the difference in price between the time his clients’ trades were executed and the time he made his trades.

Investment returns

Madoff’s scheme relied on the promise of consistent, high returns even during periods of market downturn. This was a red flag for many investors, as it is unrealistic for any investment strategy to guarantee such consistent performance.

Greed

Madoff’s scheme was fueled by the greed of both investors and Madoff himself. Investors were willing to overlook red flags because they were attracted to the promise of high returns. Madoff was motivated by his own insatiable desire for wealth and power.

Regulatory oversight

The Securities and Exchange Commission (SEC) failed to detect Madoff’s scheme for many years. This failure allowed Madoff to operate his scheme for many years and highlights the need for stronger enforcement of financial regulations.

What lessons can be learned?

Beware of “too good to be true” opportunities

If an investment opportunity sounds too good to be true, it probably is. Investors should be wary of any investment that promises consistently high returns no matter which market conditions, especially if there is no clear explanation of how those returns are being generated.

Do your own research

Before investing in any fund or product, investors should thoroughly research the company or individual running the investment and understand the risks involved. The Madoff Ponzi scheme is a reminder that even seemingly respectable individuals can commit fraud on a massive scale. It is important for investors to be vigilant and to do their homework before investing their hard-earned money.

Madoff’s cynicism

« In an era of faceless organization owned by other equally faceless organizations, Bernard L. Madoff Investment Securities LLC harks back to an earlier era in the financial world: the owner’s name is on the door. Clients know that Bernard Madoff has a personal interest in maintaining the unblemished record of value, fair-dealing and high ethical standards that has always been the firm’s hallmark. »

Why should I be interested in this post?

As a student pursuing a business or finance degree at ESSEC, I think you will be very fascinated by the Madoff Ponzi scheme for its multifaceted lessons in ethics, financial practices, and regulatory oversight. The scale of the fraud, its longevity, and the involvement of high-profile individuals make it a very interesting case study in the financial world. It is one of the largest financial frauds ever. There are many lessons to be learned.

▶ All posts about Movies and documentaries

▶ Louis DETALLE Quick review of the most famous investments frauds ever

▶ Louis VIALLARD Ponzi scheme

▶ William LONGIN Netflix ‘Billions’ Analysis of characters through CFA Code and Standards

Useful resources

Academic articles

Bernard C. and P.P. Boyle (2009) “Mr. Madoff’s Amazing Returns: An Analysis of the Split-Strike Conversion Strategy” The Journal of Derivatives, 17(1): 62-76.

Monroe H., A. Carvajal and C. Pattillo (2010) “Perils of Ponzis” Finance & development, 47(1).

Videos

FRONTLINE PBS The Madoff Affair (full documentary on YouTube)

TPM TV Roundtable Discussion With Bernard Madoff (YouTube video about regulation by Madoff)

Associated Press Executive: SEC Ignored Warnings About Madoff (YouTube video about the testimony of Harry Markopolos)

TPM TV Roundtable Discussion With Bernard Madoff (YouTube video about the testimony of Harry Markopolos)

About the author

The article was written in December 2023 by Raphael TRAEN (ESSEC Business School, Global BBA, 2023-2024).

Capital Guaranteed Products

March 15, 2023 by Shengyu ZHENG

Capital Guaranteed Products

In this article, Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023) explains how capital guaranteed products are built.

Motivation for investing in capital-guaranteed products

In order to invest the surplus of the firm liquid assets, corporate treasurers take into account the following characteristics of the financial instruments: performance, risk and liquidity. It is a common practice that some corporate investment strategies require that the investment capital should at least be guaranteed. The sacrifice of this no-loss guarantee is limited return in case of appreciation of the underlying asset price.

Capital-guaranteed (or capital-protected) products are one of the most secure forms of investment, usually in the form of certificates. They provide a guarantee that a specified minimum amount (usually 100 per cent of the issuance price) will be repaid at maturity. They are suitable particularly for risk-averse investors who wish to hold the products through to maturity and are not prepared to bear any loss that might exceed the level of the guaranteed repayment.

Performance

Let us consider a capital-guaranteed product with the following characteristics:

Table 1. Characteristics of the capital-guaranteed products

Notional amount	EUR 1,000,000.00
Underlying asset	CAC40 index
Participation rate	40%
Minimum amount guarantee	100% of the initial level
Effective date	February 01, 2022
Maturity date	July 30, 2022

We also have the following information about the market:

Table 2. Market information

Risk-free rate (annual rate)	8%
Implied volatility (annualized)	10%

In case of depreciation of the underlying index, the return of the product remains zero, which means the original capital invested is guaranteed (or protected). In case of appreciation of the underlying index, the product only yields 40% of the return of the underlying index. The following chart is a straightforward illustration of the performance structure of this product.

Performance of the capital guaranteed product

Construction of a capital guaranteed product

We can decompose a capital-guaranteed product into three parts:

Investment in the risk-free asset that would yield the guaranteed capital at maturity
Investment in a call option that guarantees participation in the appreciation of the underlying asset
Margin of the bank

Decomposition of the capital guaranteed product

Investment in the risk-free asset

The essence of the capital guarantee is realized by investing a part of the initial capital in the risk-free asset and obtaining the amount of the guaranteed capital at maturity. Given the amount of the capital to be guaranteed and the risk-free rate, we can calculate the amount to be invested in risk-free asset: 1,000,000/(1+0.08)^0.5 =962,250.45 €

Investment in the call option

To realize the upside exposure, call options are a perfect vehicle. With a notional amount of 1,000,000 € and a maturity of 6 months, an at-the-money call option would cost 41,922.70 € (calculated with the Black-Scholes-Merton formula). Since the participation rate is 40%, the amount to be invested in the call option would be 16,769.08 € (= 40% * 41,922.70 €).

Margin of the bank

The margin of the bank is equal to the difference between the original capital and the two parts of the investment. In this case, the margin is 20,980.47 € (= 1,000,000.00 € – 962,250.45 € – 16,769.08 €)
If we compress the margin, there would be more capital available to invest in the call option, thus increasing the participation rate. In the case of zero margin, we obtain the maximum participation rate. In this scenario, the maximum participation rate would be 90.05% (= (1,000,000.00 € – 962,250.45 €) / 41,922.70 €).

Sensitivity to variations of the marketplace

Considering the two parts of the investment constituting the capital-guaranteed product, we can see that the risk-free rate and the volatility of the underlying asset are the two major factors influencing the pricing of this product. Here let us focus on the maximum participation rate as a proxy of the value of the product to the buyer of the product.

The effect of the risk-free rate could be ambiguous at the first glance. On one hand, if the risk-free rate rises, there needs to be less capital invested in the risk-free asset and there would be therefore more capital to be placed in purchasing the call options. On the other hand, if the risk-free rate rises, the call option value rises as well. With the same amount of capital, fewer call options could be purchased. However, the largest portion of the original capital is invested in the risk-free asset and the impact on this regard is more important. Overall, a rising risk-free rate has a positive impact on the participation rate.

The effect of the volatility of the underlying asset, however, is clear. Rising volatility has no impact on the risk-free investment in the framework of our hypotheses. It, however, raises the value of the call options, which means that fewer options could be purchased with the same amount of capital. Overall, rising volatility has a negative impact on the participation rate.

Statistical distribution of the return

The statistical distribution of the return of the instrument is mixed by two parts: the discrete part equal to 0 corresponding to the case of depreciation of the underlying asset; and the continuous part of positive return. Based on a Gaussian assumption for the statistical distribution, we can calculate the probability mass of the depreciation of the underlying asset is 33.70%. In the continuous part, the return follows a Gaussian statistical distribution, with a mean equal to the periodic return over the participation rate and a standard deviation equal to periodic implied volatility over the participation rate, if the Gaussian assumption prevails.

Statistical distribution of the return of the capital guaranteed product

Risks and constraints

Liquidity risk

Being exotic financial instruments, capital-guaranteed products are not traded in standard exchanges. By construction, these products can normally only be redeemed at maturity and therefore are less liquid. There could be, however, early redemption clauses involved to mitigate the long-term liquidity risks. Investors should be aware of their liquidity needs before entering into a position in this product.

Counterparty risk

Similar to all other over-the-counter (OTC) transactions, there is no mechanism such as a central clearing counterparty (CCP) to ensure the timeliness and integrity of due payments. In case of financial difficulty including the bankruptcy of the issuer, the capital guarantee would be rendered worthless. It is therefore highly recommended to enter into such transactions with issuers of higher ratings.

Limited return

It is worth noting that capital-guaranteed products have weak exposure to the appreciation of the underlying asset. In this case, for a probability of 33.70%, there would be a return of zero, which is lower than investing directly in the risk-free security.

In order to mitigate this limit, the issuer could modify the level of guarantee to a lower level than 100%. This allows the product to have more exposure to the upside movement of the underlying asset with a relatively low risk of capital loss. To realize this involves entering positions of out-of-the-money call options.

Taxation and fees

In many countries, the return of capital-guaranteed products is considered as ordinary income, instead of capital gains or tax-advantaged dividends. For example, in Switzerland, it is not recommended to buy such a product with a long maturity, since the tax burden, in this case, could be higher than the “impaired” return of the product.

Moreover, fees for such products could be higher than exchange-traded funds (ETFs) or mutual funds. This part of investment cost should also be taken into account in making investment decisions.

Download the Excel file to analyze capital-guaranteed products

You can find below an Excel file to analyze capital-guaranteed products.

Why should I be interested in this post?

As a family of investments that is often used in corporate treasury management, it is important to understand the mechanism and structure of capital-guaranteed products. It would be conducive for future asset managers, treasurer managers, or structurers to make the appropriate and optimal investment decisions.

Resources

Books

Cox J. C. & M. Rubinstein (1985) “Options Markets” Prentice Hall.

Hull J. C. (2005) “Options, Futures and Other Derivatives” Prentice Hall, 6th edition.

Articles

Black F. and M. Scholes (1973) The Pricing of Options and Corporate Liabilities Journal of Political Economy, 81(3): 637-654.

Lacoste V. and Longin F. (2003) Term guaranteed fund management: the option method and the cushion method Proceeding of the French Finance Association, Lyon, France.

Merton R. (1974) On the Pricing of Corporate Debt Journal of Finance, 29(2): 449-470.

Websites

longin.fr Pricer for standard equity options – Call and put

Euronext www.euronext.com: website of the Euronext exchange where the historical data of the CAC 40 index can be downloaded

Euronext CAC 40 Index Option: website of the Euronext exchange where the option prices of the CAC 40 index are available

Six General information about capital protection without a cap: website of the Swiss stock exchange where information of various financial products are available.

About the author

The article was written in February 2023 by Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023).

Forex exchange markets

December 2, 2022 by Nakul PANJABI

Forex exchange markets

In this article, Nakul PANJABI (ESSEC Business School, Grande Ecole Program – Master in Management, 2021-2024) explains how the foreign exchange markets work.

Forex Market

Forex trading can be simply defined as exchange of a unit of one currency for a certain unit of another currency. It is the act of buying one currency while simultaneously selling another.

Foreign exchange markets (or Forex) are markets where currencies of different countries are traded. Forex market is a decentralised market in which all trades take place online in an over the counter (OTC) format. By trading volume, the forex market is the largest financial market in the world with a daily turnover of 6.6 trillion dollars in 2019. At present, it is worth 2,409 quadrillion dollars. Major currencies traded are USD, EUR, GBP, JPY, and CHF.

Players

The main players in the market are Central Banks, Commercial banks, Brokers, Traders, Exporters and Importers, Immigrants, Investors and Tourists.

Central banks

Central banks are the most important players in the Forex Markets. They have the monopoly in the supply of currencies and therefore, tremendous influence on the prices. Central Banks’ policies tend to protect aggressive fluctuations in the Forex Markets against the domestic currency.

Commercial banks

The second most important players of the Forex market are the Commercial Banks. By quoting, on a daily basis, the foreign exchange rates for buying and selling they “Make the Market”. They also function as Clearing Houses for the Market.

Brokers

Another important group is that of Brokers. Brokers do not participate in the market but acts as a link between Sellers and Buyers for a commission.

Types of Transactions in Forex Markets

Some of the transactions possible in the Forex Markets are as follows:

Spot transaction

As spot transaction uses the spot rate and the goods (currencies) are exchanges over a two-day period.

Forward transaction

A forward transaction is a future transaction where the currencies are exchanged after 90 days of the deal a fixed exchange rate on a defined date. The exchange rate used is called the Forward rate.

Future transaction

Futures are standardized Forward contracts. They are traded on Exchanges and are settled daily. The parties enter a contract with the exchange rather than with each other.

Swap transaction

The Swap transactions involve a simultaneous Borrowing and Lending of two different currencies between two investors. One investor borrows the currency and lends another currency to the second investor. The obligation to repay the currencies is used as collateral, and the amount is repaid at forward rate.

Option transaction

The Forex Option gives an investor the right, but not the obligation to exchange currencies at an agreed rate and on a pre-defined date.

Peculiarities of Forex Markets

Trading of Forex is not much different from trading of any other asset such as stocks or bonds. However, it might not be as intuitive as trading of stocks or bonds because of its peculiarities. Some peculiarities of the Forex market are as follows:

Going long and short simultaneously

Since the goods traded in the market are currencies themselves, a trade in the Forex market can be considered both long and short position. Buying dollars for euros can be profitable in cases of both dollar appreciation and euro depreciation.

High liquidity and 24-hour market

As mentioned above, the Forex market has the largest daily trading volume. This large volume of trading implies the highly liquid feature of Forex Assets. Moreover, Forex market is open 24 hours 5 days a week for retail traders. This is due to the fact that Forex is exchanged electronically over the world and anyone with an internet connection can exchange currencies in any Forex market of the world. In fact for Central banks and related organisations can trade over the weekends as well. This can cause a change in the price of currencies when the market opens to retail traders again after a gap of 2 days. This risk is known as Gapping risk.

High leverage and high volatility

Extremely high leverage is a common feature of Forex trades. Using high leverage can result in multiple fold returns in favourable conditions. However, because of high trading volume, Forex is very volatile and can go in either upward or downward spiral in a very short time. Since every position in the Forex market is a short and long position, the exposure from one currency to another is very high.

Hedging

Hedging is one of the main reasons for a lot of companies and corporates to enter into a Forex Market. Forex hedging is a strategy to reduce or eliminate risk arising from negative movement in the Exchange rate of a particular currency. If a French wine seller is about to receive 1 million USD for his wine sales then he can enter into a Forex futures contract to receive 900,000 EUR for that 1 million USD. If, at the date of payment, the rate of 1 million USD is 800,000 EUR the French wine seller will still get 900,000 EUR because he hedged his forex risk. However, in doing so, he also gave up any gain on any positive movement in the EUR-USD exchange rate.

Useful resources

Academic resources

Solnik B. (1996) International Investments Addison-Wesley.

Business resources

DailyFX / IG The History of Forex

DailyFX / IG Benefits of forex trading

DailyFX / IG Foreign Exchange Market: Nature, Structure, Types of Transactions

About the author

The article was written in December 2022 by Nakul PANJABI (ESSEC Business School, Grande Ecole Program – Master in Management, 2021-2024).

Reverse Convertibles

August 23, 2022 by Shengyu ZHENG

Reverse Convertibles

In this article, Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023) explains reverse convertibles, which are a structured product with a fixed-rate coupon and downside risk.

Introduction

The financial market has been ever evolving, witnessing the birth and flourish of novel financial instruments to cater to the diverse needs of market participants. On top of plain vanilla derivative products, there are exotic ones (e.g., barrier options, the simplest and most traded exotic derivative product). Even more complex, there are structured products, which are essentially the combination of vanilla or exotic equity instruments and fixed income instruments.

Amongst the structured products, reverse convertible products are one of the most popular choices for investors. Reverse convertible products are non-principal protected products linked to the performance of an underlying asset, usually an individual stock or an index, or a basket of them. Clients can enter into a position of a reverse convertible with the over-the-counter (OTC) trading desks in major investment banks.

In exchange for an above-market coupon payment, the holder of the product gives up the potential upside exposure to the underlying asset. The exposure to the downside risks still remains. Reserve convertibles are therefore appreciated by the investors who are anticipating a stagnation or a slightly upward market trend.

Construction of a reverse convertible

This product could be decomposed in two parts:

On the one hand, the buyer of the structure receives coupons on the principal invested and this could be considered as a “coupon bond”;
On the other hand, the investor is still exposed to the downside risks of the underlying asset and foregoes the upside gains, and this could be achieved by a short position of a put option (either a vanilla put option or a down-and-in barrier put option).

Positions of the parties of the transaction

A reverse convertible involves two parties in the transaction: a market maker (investment bank) and an investor (client). Table 1 below describes the positions of the two parties at different time of the life cycle of the product.

Table 1. Positions of the parties of a reverse convertible transaction

t	Market Maker (Investment Bank)	Investor (Client)
Beginning	Enters into a long position of a put (either a vanilla put or a down-and-in barrier put) Receives the nominal amount for the “coupon” part Invests in the amount (nominal amount plus the premium of the put) in risk-free instruments	Enters into a short position of a put (either a vanilla put or a down-and-in barrier put) Pays the nominal amount for the “coupon” part
Interim	Pays pre-specified interim coupons in respective interim coupon payment dates (if any) Receives interest payment from risk-free investments	Receives the pre-specified interim coupons in respective interim coupon payment dates (if any)
End	Receives the payoff (if any) of the put option component Pays the pre-specified final coupon in the final coupon payment date	Pays the payoff (if any) of the put option component Receives the pre-specified final coupon in the final coupon payment date

Based on the type of the put option incorporated in the product (either plain vanilla put option or down-and-in barrier put option), reserve convertibles could be categorized as plain or barrier reverse convertibles. Given the difference in terms of the composition of the structured product, the payoff and pricing mechanisms diverge as well.

Here is an example of a plain reverse convertible with following product characteristics and market information.

Product characteristics:

Investment amount: USD 1,000,000.00
Underlying asset: S&P 500 index (Bloomberg Code: SPX Index)
Investment period: from August 12, 2022 to November 12, 2022 (3 months)
Coupon rate: 2.50% (quarterly)
Strike level : 100.00% of the initial level

Market data:

Current risk-free rate: 2.00% (annualized)
Volatility of the S&P 500 index: 13.00% (annualized)

Payoff of a plain reverse convertible

As is presented above, a reverse convertible is essentially a combination of a short position of a put option and a long position of a coupon bond. In case of the plain reverse convertible product with the aforementioned characteristics, we have the blow payoff structure:

in case of a rise of the S&P 500 index during the investment period, the return for the reverse convertible remains at 2.50% (the coupon rate);
in case of a drop of the S&P 500 index during the investment period, the return would be equal to 2.50% minus the percentage drop of the underlying asset and it could be negative if the percentage drop is greater than 2.5%.

Figure 1. The payoff of a plain reverse convertible on the S&P 500 index

Source: Computation by author.

Pricing of a plain reverse convertible

Since a reverse convertible is essentially a structured product composed of a put option and a coupon bond, the pricing of this product could also be decomposed into these two parts. In terms of the pricing a vanilla option, the Black–Scholes–Merton model could do the trick (see Black-Scholes-Merton option pricing model) and in terms of pricing a barrier option, two methods, analytical formula method and Monte-Carlo simulation method, could be of help (see Pricing barrier options with analytical formulas; Pricing barrier options with simulations and sensitivity analysis with Greeks).

With the given parameters, we can calculate, as follows, the margin for the bank with respect to this product. The calculated margin could be considered as the theoretical price of this product.

Table 2. Margin for the bank for the plain reverse convertible

Source: Computation by author.

Download the Excel file to analyze reverse convertibles

You can find below an Excel file to analyze reverse convertibles.

Why should I be interested in this post

As one of the most traded structured products, reverse convertibles have been an important instrument used to secure return amid mildly negative market prospect. It is, therefore, helpful to understand the product elements, such as the construction and the payoff of the product and the targeted clients. This could act as a steppingstone to financial product engineering and risk management.

Resources

Academic references

Broadie, M., Glasserman P., Kou S. (1997) A Continuity Correction for Discrete Barrier Option. Mathematical Finance, 7:325-349.

De Bellefroid, M. (2017) Chapter 13 (Barrier) Reverse Convertibles. The Derivatives Academy. Accessible at https://bookdown.org/maxime_debellefroid/MyBook/barrier-reverse-convertibles.html

Haug, E. (1997) The Complete Guide to Option Pricing. London/New York: McGraw-Hill.

Hull, J. (2006) Options, Futures, and Other Derivatives. Upper Saddle River, N.J: Pearson/Prentice Hall.

Merton, R. (1973). Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science, 4:141-183.

Paixao, T. (2012) A Guide to Structured Products – Reverse Convertible on S&P500

Reiner, E. S. (1991) Breaking down the barriers. Risk Magazine, 4(8), 28–35.

Rich, D.R. (1994) The Mathematical Foundations of Barrier Option-Pricing Theory. Advances in Futures and Options Research: A Research Annual, 7, 267-311.

Business references

Six Structured Products. (2022). Reverse Convertibles et barrier reverse Convertibles

About the author

The article was written in August 2022 by Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023).

Pricing barrier options with simulations and sensitivity analysis with Greeks

February 10, 2026July 10, 2022 by Shengyu ZHENG

In this article, Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023) explains the pricing of barrier options with Monte-Carlo simulations and the sensitivity analysis of barrier options from the perspective of Greeks.

Pricing of discretely monitored barrier options with Monte-Carlo simulations

With the simulation method, only the pricing of discretely monitored barrier options can be handled since it is impossible to simulate continuous price trajectories with no intervals. Here the method is illustrated with a down-and-out put option. The general setup of economic details of the down-and-out put option and related market information are presented as follows:

General setup of simulation for barrier option pricing

Similar to the simulation method for pricing standard vanilla options, Monte Carlo simulations based on Geometric Brownian Motion could also be employed to analyze the pricing of barrier options.

Figure 1. Trajectories of 600 price simulations.

With the R script presented above, we can simulate 6,000 times with the simprice() function from the derivmkts package. Trajectories of 600 price simulations are presented above, with the black line representing the mean of the final prices, the green dashed lines 1x and 2x standard deviation above the mean, the red dashed lines 1x and 2x derivation below the mean, the blue dashed line the strike level and the brown line the knock-out level.

The simprice() function, according to the documentation, computes simulated lognormal price paths with the given parameters.

With this simulation of 6,000 price paths, we arrive at a price of 0.6720201, which is quite close to the one calculated from the formulaic approach from the previous post.

Analysis of Greeks

The Greeks are the measures representing the sensitivity of the price of derivative products including options to a change in parameters such as the price and the volatility of the underlying asset, the risk-free interest rate, the passage of time, etc. Greeks are important elements to look at for risk management and hedging purposes, especially for market makers (dealers) since they do not essentially take these risks for themselves.

In R, with the combination of the greeks() function and a barrier pricing function, putdownout() in this case, we can easily arrive at the Greeks for this option.

Barrier option R code Sensitivity Greeks

Table 1. Greeks of the Down-and-Out Put

Barrier Option Greeks Summary

We can also have a look at the evolutions of the Greeks with the change of one of the parameters. The following R script presents an example of the evolutions of the Greeks along with the changes in the strike price of the down-and-out put option.

Barrier option R code Sensitivity Greeks Evolution

Figure 2. Evolution of Greeks with the change of Strike Price of a Down-and-Out Put

Evolution Greeks Barrier Price

Download R file to price barrier options

You can find below an R file (file with txt format) to price barrier options.

Why should I be interested in this post?

As one of the most traded but the simplest exotic derivative products, barrier options open an avenue for different applications. They are also very often incorporated in structured products, such as reverse convertibles. It is, therefore, important to be equipped with knowledge of this product and to understand the pricing logics if one aspires to work in the domain of market finance.

Simulation methods are very common in pricing derivative products, especially for those without closed-formed pricing formulas. This post only presents a simple example of pricing barrier options and much optimization is needed for pricing more complex products with more rounds of simulations.

Useful resources

Academic articles

Broadie, M., Glasserman P., Kou S. (1997) A Continuity Correction for Discrete Barrier Option. Mathematical Finance, 7:325-349.

Merton, R. (1973) Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science, 4:141-183.

Paixao, T. (2012) A Guide to Structured Products – Reverse Convertible on S&P500

Reiner, E.S., Rubinstein, M. (1991) Breaking down the barriers. Risk Magazine, 4(8), 28–35.

Rich, D. R. (1994) The Mathematical Foundations of Barrier Option-Pricing Theory. Advances in Futures and Options Research: A Research Annual, 7:267-311.

Wang, B., Wang, L. (2011) Pricing Barrier Options using Monte Carlo Methods, Working paper.

Books

Haug, E. (1997) The Complete Guide to Option Pricing. London/New York: McGraw-Hill.

Hull, J. (2006) Options, Futures, and Other Derivatives. Upper Saddle River, N.J: Pearson/Prentice Hall.

About the author

The article was written in June 2022 by Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023).

Pricing barrier options with analytical formulas

February 5, 2024July 10, 2022 by Shengyu ZHENG

Pricing barrier options with analytical formulas

As is mentioned in the previous post, the frequency of monitoring is one of the determinants of the price of a barrier option. The higher the frequency, the more likely a barrier event would take place.

In this article, Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023) explains the pricing of continuously and discretely monitored barrier options with analytical formulas.

Pricing of standard continuously monitored barrier options

For pricing standard barrier options, we cannot simply apply the Black-Sholes-Merton Formula for the particularity of the barrier conditions. There are, however, several models available developed on top of this theoretical basis. Among them, models developed by Merton (1973), Reiner and Rubinstein (1991) and Rich (1994) enabled the pricing of continuously monitored barrier options to be conducted in a formulaic fashion. They are concisely put together by Haug (1997) as follows:

Knock-in and knock-out barrier option pricing formula

Knock-in barrier option pricing formula

Pricing of standard discretely monitored barrier options

For discretely monitored barrier options, Broadie and Glasserman (1997) derived an adjustment that is applicable on top of the pricing formulas of the continuously monitored counterparts.

Let’s denote:

Knock-in barrier option pricing formula

The price of a discretely monitored barrier option of a certain barrier price equals the price of a continuously monitored barrier option of the adjusted price plus an error:

Knock-in barrier option pricing formula

The adjusted barrier price, in this case, would be:

Knock-in barrier option pricing formula

It is also worth noting that the error term o(·) grows prominently when the barrier approaches the strike price. A threshold of 5% from the strike price should be imposed if this approach is employed for pricing discretely monitored barrier options.

Example of pricing a down-and-out put with R with the formulaic approach

The general setup of economic details of the Down-and-Out Put and related market information is presented as follows:

Knock-in barrier option pricing formula

There are built-in functions in the “derivmkts” library that render directly the prices of barrier options of continuous monitoring, such as calldownin(), callupin(), calldownout(), callupout(), putdownin(), putupin(), putdownout(), and putupout (). By incorporating the adjustment proposed by Broadie and Glasserman (1997), all barrier options of both monitoring methods could be priced in a formulaic way with the following function:

Knock-in barrier option pricing formula

For example, for a down-and-out Put option with the aforementioned parameters, we can use this function to calculate the prices.

Knock-in barrier option pricing formula

For continuous monitoring, we get a price of 0.6264298, and for daily discrete monitoring, we get a price of 0.676141. It makes sense that for a down-and-out put option, a lower frequency of barrier monitoring means less probability of a knock-out event, thus less protection for the seller from extreme downside price trajectories. Therefore, the seller would charge a higher premium for this put option.

Download R file to price barrier options

You can find below an R file (file with txt format) to price barrier options.

Why should I be interested in this post?

As one of the most traded but the simplest exotic derivative products, barrier options open an avenue for different applications. They are also very often incorporated in structured products, such as reverse convertibles. It is, therefore, important to understand the elements having an impact on their prices and the closed-form pricing formulas are a good presentation of these elements.

Useful resources

Academic research articles

Broadie, M., Glasserman P., Kou S. (1997) A Continuity Correction for Discrete Barrier Option. Mathematical Finance, 7:325-349.

Merton, R. (1973) Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science, 4:141-183.

Paixao, T. (2012) A Guide to Structured Products – Reverse Convertible on S&P500

Reiner, E.S., Rubinstein, M. (1991) Breaking down the barriers. Risk Magazine, 4(8), 28–35.

Rich, D. R. (1994) The Mathematical Foundations of Barrier Option-Pricing Theory. Advances in Futures and Options Research: A Research Annual, 7:267-311.

Wang, B., Wang, L. (2011) Pricing Barrier Options using Monte Carlo Methods, Working paper.

Books

Haug, E. (1997) The Complete Guide to Option Pricing. London/New York: McGraw-Hill.

Hull, J. (2006) Options, Futures, and Other Derivatives. Upper Saddle River, N.J: Pearson/Prentice Hall.

About the author

The article was written in July 2022 by Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023).

Barrier options

July 6, 2022 by Shengyu ZHENG

Barrier options

In this article, Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023) explains barrier options which are the most traded exotic options in derivatives markets.

Description

Barrier options are path dependent. Their payoffs are not only a function of the price of the underlying asset relative to the option strike, but also depend on whether the price of the underlying asset reached a certain predefined barrier during the life of the option.

The two most common kinds of barrier options are knock-in (KI) and knock-out (KO) options.

Knock-in (KI) barrier options

KI barrier options are options that are activated only if the underlying asset attains a prespecified barrier level (the “knock-in” event). With the absence of this knock-in event, the payoff remains zero regardless of the trajectory of the price of the underlying asset.

Knock-out (KO) barrier options

KO barrier options are options that are deactivated only if the underlying asset attains a prespecified barrier level (the “knock-out” event). In the presence of this knock-out event, the payoff remains zero regardless of the trajectory of the price of the underlying asset.

Observation

The determination of the occurrence of a barrier event (KI or KO conditions) is essential to the ultimate payoff of the barrier option. In practice, the details of the KI or KO conditions are precisely defined in the contract (called “Confirmations” by the International Swaps and Derivatives Association (ISDA) for over-the counter (OTC) traded options).

Observation period

The observation period denotes the period where a barrier event (KI or KO) can be observed, that is to say, when the price of the underlying asset is monitored. There are three styles of observation period: European style, partial-period American style, and full-period American style.

European style: The observation period is only the expiration date of the barrier option.
Partial-period American style: The observation period is part of the lifespan of the barrier option.
Full-period American style: The observation period spans the whole period from the effective date to the expiration date of the barrier option.

Monitoring method

There are two typical types of monitoring methods in terms of the determination of a knock-in/knock-out event: continuous monitoring and discrete monitoring. The monitoring method is one of the key factors in determining the premium of a barrier option.

Continuous monitoring: A knock-in/knock-out event is deemed to take place if, at any time in the observation period, the knock-in/knock-out condition is met.
Discrete monitoring: A knock-in/knock-out event is deemed to occur if, at pre-specific times in the observation period, usually the closing time of each trading day, the knock-in/knock-out condition is met.

Barrier Reference Asset

For the most cases, the Barrier Reference Asset is the underlying asset itself. However, if specified in the contract, it can be another asset or index. It can also be other calculatable properties, such as the volatility of the asset. In this case, the methodology of calculating such properties should be clearly defined in the contract.

Rebate

For knock-out options, there could be a rebate. A rebate is an extra feature and it corresponds to the amount that should be paid to the buyer of the knock-out option in case of the occurrence of a knock-out event.

In-out parity relation for barrier options

Analogous to the call-put parity relation for plain vanilla options, there is an in-out parity relation for barrier options stating that a long position in a knock-in option plus a long position in a knock-out option with identical strikes, barriers, monitoring methods and maturity is equivalent to a long position in a comparable vanilla option. It could be stated as follows:

Knock-in knock-out barrier option parity relation

Where K denotes the strike price, T the maturity, and B the barrier level.

It is worth noting that this parity relation is valid only when the two KI and KO options are identical, and there is no rebate in case of a knock-out option.

Basic barrier options

There are four types of basic barrier options traded in the market: up-and-in option, up-and-out option, down-and-in option, and down-and-out option. “Up” and “down” denotes the direction of surpassing the barrier price. “In” and “out” depict the type of barrier condition, i.e. knock-in or knock-out. These four types of barrier features are available for both call and put options.

Up-and-in option

An up-and-in option is a knock-in option whose barrier condition is achieved if the underlying price arrives higher than the barrier level during the observation period.

Figure 1 illustrates the occurrence of an up-and-in barrier event for a barrier option with full-period American style and discrete monitoring (the closing time of each trading day).

Figure 1. Illustration of an up-and-in barrier option
Example of an up-and-in call option

Up-and-out option

An up-and-out option is a knock-out option whose barrier condition is achieved if the underlying price arrives higher than the barrier level during the observation period.

Figure 2. Illustration of an up-and-out option

Example of an up-and-out call option

Down-and-in option

A down-and-in option is a knock-in option whose barrier condition is achieved if the underlying price arrives lower than the barrier level during the observation period.

Figure 3. Illustration of a down-and-in option
Example of a down-and-in call option

Down-and-out option

A down-and-out option is a knock-out option whose barrier condition is achieved if the underlying price arrives lower than the barrier level during the observation period.

Figure 4. Illustration of a down-and-out option
Example of a down-and-out call option

Download R file to price barrier options

You can find below an R file to price barrier options.

Trading of barrier options

Being the most popular exotic options, barrier options on stocks or indices have been actively traded in the OTC market since the inception of the market. Unavailable in standard exchanges, they are less accessible than their vanilla counterparts. Barrier options are also commonly utilized in structured products.

Why should I be interested in this post?

As one of the most traded but the simplest exotic derivative products, barrier options open an avenue for different applications. They are also very often incorporated in structured products, such as reverse convertibles. Knock-in/knock out conditions are also common features in other types of more complicated exotic derivative products.

It is, therefore, important to be equipped with knowledge of this product and to understand the pricing logics if one aspires to work in financial markets.

References

Academic research articles

Broadie, M., Glasserman P., Kou S. (1997) A Continuity Correction for Discrete Barrier Option. Mathematical Finance, 7:325-349.

Merton, R. (1973) Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science, 4:141-183.

Paixao, T. (2012) A Guide to Structured Products – Reverse Convertible on S&P500

Reiner, E.S., Rubinstein, M. (1991) Breaking down the barriers. Risk Magazine, 4(8), 28–35.

Rich, D. R. (1994) The Mathematical Foundations of Barrier Option-Pricing Theory. Advances in Futures and Options Research: A Research Annual, 7:267-311.

Wang, B., Wang, L. (2011) Pricing Barrier Options using Monte Carlo Methods, Working paper.

Books

Haug, E. (1997) The Complete Guide to Option Pricing. London/New York: McGraw-Hill.

Hull, J. (2006) Options, Futures, and Other Derivatives. Upper Saddle River, N.J: Pearson/Prentice Hall.

About the author

The article was written in July 2022 by Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023).

Implied Volatility

May 15, 2022 by Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains how implied volatility is computed from option market prices and a option pricing model.

Introduction

Volatility is a measure of fluctuations observed in an asset’s returns over a period of time. The standard deviation of historical asset returns is one of the measures of volatility. In option pricing models like the Black-Scholes-Merton model, volatility corresponds to the volatility of the underlying asset’s return. It is a key component of the model because it is not directly observed in the market and cannot be directly computed. Moreover, volatility has a strong impact on the option value.

Mathematically, in a reverse way, implied volatility is the volatility of the underlying asset which gives the theoretical value of an option (as computed by Black-Scholes-Merton model) equal to the market price of that option.

Implied volatility is a forward-looking measure because it is a representation of expected price movements in an underlying asset in the future.

Computation methods for implied volatility

The Black-Scholes-Merton (BSM) model provides an analytical formula for the price of both a call option and a put option.

The value for a call option at time t is given by:

Call option value

The value for a put option at time t is given by:

Put option value

where the parameters d₁ and d₂ are given by:,

call option d1 d2

with the following notations:

S_t : Price of the underlying asset at time t
t: Current date
T: Expiry date of the option
K: Strike price of the option
r: Risk-free interest rate
σ: Volatility of the underlying asset
N(.): Cumulative distribution function for a normal (Gaussian) distribution. It is the probability that a random variable is less or equal to its input (i.e. d₁ and d₂) for a normal distribution. Thus, 0 ≤ N(.) ≤ 1

From the BSM model, both for a call option and a put option, the option price is an increasing function of the volatility of the underlying asset: an increase in volatility will cause an increase in the option price.

Figures 1 and 2 below illustrate the relationship between the value of a call option and a put option and the level of volatility of the underlying asset according to the BSM model.

Figure 1. Call option value as a function of volatility.

Source: computation by the author (BSM model)

Figure 2. Put option value as a function of volatility.

Source: computation by the author (BSM model)

You can download below the Excel file for the computation of the value of a call option and a put option for different levels of volatility of the underlying asset according to the BSM model.

We can observe that the call and put option values are a monotonically increasing function of the volatility of the underlying asset. Then, for a given level of volatility, there is a unique value for the call option and a unique value for the put option. This implies that this function can be reversed; for a given value for the call option, there is a unique level of volatility, and similarly, for a given value for the put option, there is a unique level of volatility.

The BSM formula can be reverse-engineered to compute the implied volatility i.e., if we have the market price of the option, the market price of the underlying asset, the market risk-free rate, and the characteristics of the option (the expiration date and strike price), we can obtain the implied volatility of the underlying asset by inverting the BSM formula.

Example

Consider a call option with a strike price of 50 € and a time to maturity of 0.25 years. The market risk-free interest rate is 2% and the current price of the underlying asset is 50 €. Thus, the call option is ‘at-the-money’. If the market price of the call option is equal to 2 €, then the associated level of volatility (implied volatility) is equal to 18.83%.

You can download below the Excel file below to compute the implied volatility given the market price of a call option. The computation uses the Excel solver.

Volatility smile

Volatility smile is the name given to the plot of implied volatility against different strikes for options with the same time to maturity. According to the BSM model, it is a horizontal straight line as the model assumes that the volatility is constant (it does not depend on the option strike). However, in practice, we do not observe a horizontal straight line. The curve may be in the shape of the alphabet ‘U’ or a ‘smile’ which is the usual term used to refer to the observed function of implied volatility.

Figure 3 below depicts the volatility smile for call options on the Apple stock on May 13, 2022.

Figure 3. Volatility smile for call options on Apple stock.
Apple volatility smile
Source: Computation by author.

We can also observe that the for a specific time to maturity, the implied volatility is minimum when the option is at-the-money.

Volatility surface

An essential assumption of the BSM model is that the returns of the underlying asset follow geometric Brownian motion (corresponding to log-normal distribution for the price at a given point in time) and the volatility of the underlying asset price remains constant over time until the expiration date. Thus theoretically, for a constant time to maturity, the plot of implied volatility and strike price would be a horizontal straight line corresponding to a constant value for volatility.

Volatility surface is obtained when values for implied volatilities are calculated for options with different strike prices and times to maturity.

CBOE Volatility Index

The Chicago Board Options Exchange publishes the renowned Volatility Index (also known as VIX) which is an index based on the implied volatility of 30-day option contracts on the S&P 500 index. It is also called the ‘fear gauge’ and it is a representation of the market outlook for volatility for the next 30 days.

Useful resources

Academic articles

Black F. and M. Scholes (1973) “The Pricing of Options and Corporate Liabilities” The Journal of Political Economy, 81, 637-654.

Dupire B. (1994). “Pricing with a Smile” Risk Magazine 7, 18-20.

Merton R.C. (1973) “Theory of Rational Option Pricing” Bell Journal of Economics, 4, 141–183.

Business

CBOE Volatility Index (VIX)

CBOE VIX tradable products

About the author

The article was written in May 2022 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Black-Scholes-Merton option pricing model

February 4, 2024March 3, 2022 by Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains the Black-Scholes-Merton model to price options.

The Black-Scholes-Merton model (or the BSM model) is the world’s most popular option pricing model. Developed in the beginning of the 1970s, this model introduced to the world, a mathematical way of pricing options. Its success was essentially a starting point for new forms of financial derivatives in the knowledge that they could be priced accurately using the ideas and analyses pioneered by Black, Scholes and Merton and it set the foundation for the flourishing of modern quantitative finance. Myron Scholes and Robert Merton were awarded the Nobel Prize for their work on option pricing in 1997. Unfortunately, Fischer Black had died several years earlier but would certainly have been included in the prize had he been alive, and he was also listed as a contributor by Scholes and Merton.

Today, the Black-Scholes-Merton formula is widely used by traders in investment banks to price and hedge option contracts. Options are used by investors to hedge their portfolios to manage their risks.

Assumptions of the BSM Model

As any model, the BSM model relies on a set of assumptions:

The model considers European options, which we can only be exercised at their expiration date.
The price of the underlying asset follows a geometric Brownian motion (corresponding to log-normal distribution for the price at a given point in time).
The risk-free rate remains constant over time until the expiration date.
The volatility of the underlying asset price remains constant over time until the expiration date.
There are no dividend payments on the underlying asset.
There are no transaction costs on the underlying asset.
There are no arbitrage opportunities.

The BSM equation

The value of an option is a function of the price of the underlying stock and its statistical behavior over the life of the option.

A commonly used model is Geometric Brownian Motion (GBM). GBM assumes that future asset price differences are uncorrelated over time and the probability distribution function of the future prices is a log-normal distribution (or equivalently the probability distribution function of the future returns is a normal distribution). The price movements in a GBM process can be expressed as:

GBM equation

with dS being the change in the underlying asset price in continuous time dt and dX the random variable from the normal distribution (N(0, 1) or Wiener process). σ is the volatility of the underlying asset price (it is assumed to be constant). μdt represents the deterministic return within the time interval with μ representing the growth rate of asset price or the ‘drift’.

Therefore, option price is determined by these parameters that describe the process followed by the asset price over a period of time. The Black-Scholes-Merton equation governs the price evolution of European stock options in financial markets. It is a linear parabolic partial differential equation (PDE) and is expressed as:

BSM model equation

Where V is the value of the option (as a function of two variables: the price of the underlying asset S and time t), r is the risk-free interest rate (think of it as the interest rate which you would receive from a government debt or similar debt securities) and σ is the volatility of the log returns of the underlying security (say stocks).

The key idea behind the equation is to hedge the option and limit exposure to market risk posed by the asset. This is achieved by a strategy known as ‘delta hedging’ and it involves replicating the option through an equivalent portfolio with positions in the underlying asset and a risk-free asset in the right way so as to eliminate risk.

Thus, from the BSM equation we can derive the BSM formulae that describe the price of call and put options over their life time.

The BSM formulae

Note that the type of option we are valuing (call or put), the strike price and the maturity date do not appear in the above BSM equation. These elements only appear in the ‘final condition’ i.e., the option value at maturity, called the payoff function.

For a call option, the payoff C is given by:

C_T = max⁡(S_T – K; 0)

For a put option, the payoff is given by:

P_T = max⁡(K – S_T; 0)

The BSM formula is a solution to the BSM equation, given the boundary conditions (given by the payoff equations above). It calculates the price at time t for both a call and a put option.

The value for a call option at time t is given by:

Call option value equation

The value for a put option at time t is given by:

Put option value equation

where

With the notations:
S_t: Price of the underlying asset at time t
t: Current date
T: Expiry date of the option
K: Strike price of the option
r: Risk-free interest rate
σ: Volatility (the standard deviation of the return on the underlying asset)
N(.): Cumulative distribution function for a normal (Gaussian) distribution. It is the probability that a random variable is less or equal to its input (i.e. d₁ and d₂) for a normal distribution. Thus, 0 ≤ N(.) ≤ 1

Figure 1 gives the graphical representation of the value of a call option at time t as a function of the price of the underlying asset at time t as given by the BSM formula. The strike price for the call option is 50€ with a maturity of 0.25 years and volatility of 50% in the underlying.

Figure 1. Call option value

Source: computation by author.

Figure 2 gives the graphical representation of the value of a put option at time t as a function of the price of the underlying asset at time t as given by the BSM formula. The strike price for the put option is 50€ with a maturity of 0.25 years and volatility of 50% in the underlying.

Figure 2. Put option value
Source: computation by author.

You can download below the Excel file for option pricing with the BSM Model.

Some Criticisms and Limitations

American options

The Black-Scholes-Merton model was initially developed for European options. This is a limitation of the equation for American options which can be exercised at any time before the expiry date. The BSM model would then not accurately determine the option value (an important case when the underlying asset pays a discrete dividend).

Stocks paying dividends

Also, in reality, most stocks pay dividends, and no dividends was an assumption in the initial BSM model, which analysts now eliminated by accommodating the dividend yield in the formula if required.

Constant volatility

Another limitation is the use of constant volatility. Volatility is the measure of risk based on the standard deviation of the return on the underlying asset. In reality the value of an asset will change randomly, not with a specific constant pattern regarding the way it can change.

Finally, the assumption of no transaction cost neglects the liquidity risk in the market since transaction costs are clearly incurred in the real world and there exists a bid-offer spread on most underlying assets. For the most heavily traded stocks, this cost may be low but for others it may lead to an inaccuracy.

Useful resources

Black F. and M. Scholes (1973) The Pricing of Options and Corporate Liabilities The Journal of Political Economy 81, 637-654.

Merton R.C. (1973) Theory of Rational Option Pricing Bell Journal of Economics 4, 141–183.

About the author

The article was written in March 2022 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Protective Put

December 29, 2025January 12, 2022 by Akshit GUPTA

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the concept of protective put using option contracts.

Introduction

Hedging is a strategy implemented by investors to reduce the risk in an existing investment. In financial markets, hedging is an effective tool used by investors to minimize the risk exposure and change the risk profile for any investment in securities. While hedging does not necessarily eliminate the entire risk for any investment, it does limit the potential losses that the investor can incur.

Option contracts are commonly used by market participants (traders, investors, asset managers, etc.) as hedging mechanisms due to their great flexibility (in terms of expiration date, moneyness, liquidity, etc.) and availability. Positions in options are used to offset the risk exposure in the underlying security, another option contract or in any other derivative contract. There are various popular strategies that can be implemented through option contracts to minimize risk and maximize returns, one of which is a protective put.

Buying a protective put

A put option gives the buyer of the option, the right but not the obligation, to sell a security at a predefined date and price.

A protective put also called as a synthetic long option, is a hedging strategy that limits the downside of an investment. In a protective put, the investor buys a put option on the stock he/she holds in its portfolio. The protective put option acts as a price floor since the investor can sell the security at the strike price of the put option if the price of the underlying asset moves below the strike price. Thus, the investor caps its losses in case the underlying asset price moves downwards. The investor has to pay an option premium to buy the put option.

The maximum payoff potential from using this strategy is unlimited and the potential downside/losses is limited to the strike price of the put option.

Market scenario

A put option is generally bought to safeguard the investment when the investor is bullish about the market in the long run but fears a temporary fall in the prices of the asset in the short term.

For example, an investor owns the shares of Apple and is bullish about the stock in the long run. However, the earnings report for Apple is due to be released by the end of the month. The earnings report can have a positive or a negative impact on the prices of the Apple stock. In this situation, the protective put saves the investor from a steep decline in the prices of the Apple stock if the report is unfavorable.

Let us consider a protective position with buying at-the money puts. One of following three scenarios may happen:

Scenario 1: the stock price does not change, and the puts expire at the money.

In this scenario, the market viewpoint of the investor does not hold correct and the loss from the strategy is the premium paid on buying the put options. In this case, the option holder does not exercise its put options, and the investor gets to keep the underlying stocks.

Scenario 2: the stock price rises, and the puts expire in the money.

In this scenario, since the price of the stock was locked in through the put option, the investor enjoys a short-term unrealized profit on the underlying position. However, the put option will not be exercised by the investor and it will expire worthless. The investor will lose the premium paid on buying the puts.

Scenario 3: the stock price falls, and the puts expire out of the money.

In this scenario, since the price of the stock was locked in through the put option, the investor will execute the option and sell the stocks at the strike price. There is protection from the losses since the investor holds the put option.

Risk profile

In a protective put, the total cost of the investment is equal to the price of the underlying asset plus the put price. However, the profit potential for the investment is unlimited and the maximum losses are capped to the put option price. The risk profile of the position is represented in Figure 1.

Figure 1. Profit or Loss (P&L) function of the underlying position and protective put position.

Protective put

Source: computation by the author.

You can download below the Excel file for the computation of the Profit or Loss (P&L) function of the underlying position and protective put position.

The delta of the position is equal to the sum of the delta of the long position in the underlying asset (+1) and the long position in the put option (Δ). The delta of a long put option is negative which implies that a fall in the asset price will result in an increase in the put price and vice versa. However, the delta of a protective put strategy is positive. This implies that in a protective put strategy, the value of the position tends to rise when the underlying asset price increases and falls when the underlying asset prices decreases.

Figure 2 represents the delta of the protective put position as a function of the price of the underlying asset. The delta of the put option is computed with the Black-Scholes-Merton model (BSM model).

Figure 2. Delta of a protective put position.
Delta Protective put
Source: computation by the author (based on the BSM model).

You can download below the Excel file for the computation of the delta of a protective put position.

Example

An investor holds 100 shares of Apple bought at the current price of $144 each. The total initial investment is equal to $14,400. He is skeptical about the effect of the upcoming earnings report of Apple by the end of the current month. In order to avoid losses from a possible downside in the price of the Apple stock, he decides to purchase at-the-money put options on the Apple stock (lot size is 100) with a maturity of one month, using the protective put strategy.

We use the following market data: the current price of Appel stock is $144, the implied volatility of Apple stock is 22.79% and the risk-free interest rate is equal to 1.59%.

Based on the Black-Scholes-Merton model, the price of the put option $3.68.

Let us consider three scenarios at the time of maturity of the put option:

Scenario 1: stability of the price of the underlying asset at $144

The market value of the investment $14,400. The total cost of the initial investment is the cost of acquiring the Apple stocks ($14,400) plus the cost of buying the put options ($368 = $3.68*100), which is equal to $14,768, (i.e. ($14,400 + $368)).

As the stock price is stable at $144, the investor will not execute the put option and the option will expire worthless.

By not executing the put option, the investor incurs a loss which is equal to the price of the put option which is $368.

Scenario 2: an increase in the price of the underlying asset to $155

The market value of the investment $15,500. The total cost of the initial investment is the cost of acquiring the Apple stocks ($14,400) plus the cost of buying the put options ($368 = $3.68*100), which is equal to $14,768, (i.e. ($14,400 + $368)).

As the stock price is at $155, the investor will not execute the put option and hold on the underlying stock.

By not executing the put option, the investor incurs a loss which is equal to the price of the put option which is $368.

Scenario 3: a decrease in the price of the underlying asset to $140

The market value of the investment $14,000. The total cost of the initial investment is the cost of acquiring the Apple stocks ($14,400) plus the cost of buying the put options ($368 = $3.68*100), which is equal to $14,768, (i.e. ($14,400 + $368)).

As the stock price has decreased to $140, the investor will execute the put option and sell the Apple stocks at $144. By executing the put option, the investor will protect himself from incurring a loss of $400 (i.e.($144-$140)*100) due to a decrease in the Apple stock prices.

▶ All posts about Options

▶ Akshit GUPTA Options

▶ Akshit GUPTA The Black-Scholes-Merton model

▶ Akshit GUPTA Option Greeks – Delta

▶ Akshit GUPTA Covered call

▶ Akshit GUPTA Option Trader – Job description

Useful Resources

Black F. and M. Scholes (1973) “The Pricing of Options and Corporate Liabilities” The Journal of Political Economy, 81, 637-654.

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 10 – Trading strategies involving Options, 276-295.

Merton R.C. (1973) “Theory of Rational Option Pricing” Bell Journal of Economics, 4(1): 141–183.

Wilmott P. (2007) Paul Wilmott Introduces Quantitative Finance, Second Edition, Chapter 8 – The Black Scholes Formula and The Greeks, 182-184.

About the author

Article written in January 2022 by Akshit GUPTA (ESSEC Business School, Grande Ecole Program -Master in Management, 2019-2022).

Straddle and strangle strategy

January 1, 2026January 12, 2022 by Akshit GUPTA

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the strategies of straddle and strangle based on options.

Introduction

In financial markets, hedging is implemented by investors to minimize the risk exposure and maximize the returns for any investment in securities. While hedging does not necessarily eliminate the entire risk for an investment, it does limit or offset any potential losses that the investor can incur.

Option contracts are commonly used by investors / traders as hedging mechanisms due to their great flexibility (in terms of expiration date, moneyness, liquidity, etc.) and availability. Positions in options are used to offset the risk exposure in the underlying security, another option contract or in any other derivative contract. Option strategies can be directional or non-directional.

Directional strategy is when the investor has a specific viewpoint about the movement of an asset price and aims to earn profit if the viewpoint holds true. For instance, if an investor has a bullish viewpoint about an asset and speculates that its price will rise, she/he can buy a call option on the asset, and this can be referred as a directional trade with a bullish bias. Similarly, if an investor has a bearish viewpoint about an asset and speculates that its price will fall, she/he can buy a put option on the asset, and this can be referred as a directional trade with a bearish bias.

On the other hand, non-directional strategies can be used by investors when they anticipate a major market movement and want to gain profit irrespective of whether the asset price rises or falls, i.e., their payoff is independent of the direction of the price movement of the asset but instead depends on the magnitude of the price movement. There are various popular non-directional strategies that can be implemented through a combination of option contracts to minimize risk and maximize returns. In this post, we are interested in straddle and strangle.

Straddle

In a straddle, the investor buys a European call and a European put option, both at the same expiration date and at the same strike price. This strategy works in a similar manner like a strangle (see below). However, the potential losses are a bit higher than incurred in a strangle if the stock price remains near the central value at expiration date.

A long straddle is when the investor buys the call and put options, whereas a short straddle is when the investor sells the call and put options. Thus, whether a straddle is long or short depends on whether the options are long or short.

Market Scenario

When the price of underlying is expected to move up or down sharply, investors chose to go for a long straddle and the expiration date is chosen such that it occurs after the expected price movement. Scenarios when a long straddle might be used can include budget or company earnings declaration, war announcements, election results, policy changes etc.
Conversely, a short straddle can be implemented when investors do not expect a significant movement in the asset prices.

Example

In Figure 1 below, we represent the profit and loss function of a straddle strategy using a long call and a long put option. K₁ is the strike price of the long call i.e., €98 and K₂ is the strike price of the long put position i.e., €98. The premium of the long call is equal to €5.33, and the premium of the long put is equal to €3.26 computed using the Black-Scholes-Merton model. The time to maturity (T) is of 18 days (i.e., 0.071 years). At the time of valuation, the price of the underlying asset (S₀) is €100, the volatility (σ) of the underlying asset is 40% and the risk-free rate (r) is 1% (market data).

Figure 1. Profit and loss (P&L) function of a straddle position.

Source: computation by the author.

You can download below the Excel file for the computation of the straddle value using the Black-Scholes-Merton model.

Strangle

In a strangle, the investor buys a European call and a European put option, both at the same expiration date but different strike prices. To benefit from this strategy, the price of the underlying asset must move further away from the central value in either direction i.e., increase or decrease. If the stock prices stay at a level closer to the central value, the investor will incur losses.

Like a straddle, a long strangle is when the investor buys the call and put options, whereas a short strangle is when the investor sells (issues) the call and put options. The only difference is the strike price, as in a strangle, the call option has a higher strike price than the price of the underlying asset, while the put option has a lower strike price than the price of the underlying asset.

Strangles are generally cheaper than straddles because investors require relatively less price movement in the asset to ‘break even’.

Market Scenario

The long strangle strategy can be used when the trader expects that the underlying asset is likely to experience significant volatility in the near term. It is a limited risk and unlimited profit strategy because the maximum loss is limited to the net option premiums while the profits depend on the underlying price movements.

Similarly, short strangle can be implemented when the investor holds a neutral market view and expects very little volatility in the underlying asset price in the near term. It is a limited profit and unlimited risk strategy since the payoff is limited to the premiums received for the options, while the risk can amount to a great loss if the underlying price moves significantly.

Example

In Figure 2 below, we represent the profit and loss function of a strangle strategy using a long call and a long put option. K₁ is the strike price of the long call i.e., €98 and K₂ is the strike price of the long put position i.e., €108. The premium of the long call is equal to €5.33, and the premium of the long put is equal to €9.47 computed using the Black-Scholes-Merton model. The time to maturity (T) is of 18 days (i.e., 0.071 years). At the time of valuation, the price of the underlying asset (S₀) is €100, the volatility (σ) of the underlying asset is 40% and the risk-free rate (r) is 1% (market data).

Figure 2. Profit and loss (P&L) function of a strangle position.

Source: computation by the author..

You can download below the Excel file for the computation of the strangle value using the Black-Scholes-Merton model.

▶ All posts about Options

▶ Akshit GUPTA Options

▶ Akshit GUPTA The Black-Scholes-Merton model

▶ Akshit GUPTA Option Spreads

▶ Akshit GUPTA Option Trader – Job description

Useful resources

Academic research articles

Black F. and M. Scholes (1973) “The Pricing of Options and Corporate Liabilities” The Journal of Political Economy, 81, 637-654.

Merton R.C. (1973) “Theory of Rational Option Pricing” Bell Journal of Economics, 4, 141–183.

Books

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 10 – Trading strategies involving Options, 276-295.

Wilmott P. (2007) Paul Wilmott Introduces Quantitative Finance, Second Edition, Chapter 8 – The Black Scholes Formula and The Greeks, 182-184.

About the author

Article written in January 2022 by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Option Spreads

January 7, 2022 by Akshit GUPTA

Option Spreads

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the different option spreads used to hedge a position in financial markets.

Introduction

In financial markets, hedging is implemented by investors to minimize the risk exposure for any investment in securities. While hedging does not necessarily eliminate the entire risk for an investment, it does limit or offset any potential losses that the investor can incur.

Option contracts are commonly used by traders and investors as hedging mechanisms due to their great flexibility (in terms of expiration date, moneyness, liquidity, etc.) and availability. Positions in options are used to offset the risk exposure in the underlying security, another option contract or in any other derivative contract. Option strategies can be directional or non-directional.

Spreads are hedging strategies used in trading in which traders buy and sell multiple option contracts on the same underlying asset. In a spread strategy, the option type used to create a spread has to be consistent, either call options or put options. These are used frequently by traders to minimize their risk exposure on the positions in the underlying assets.

Bull Spread

In a bull spread, the investor buys a European call option on the underlying asset with strike price K₁ and sells a call option on the same underlying asset with strike price K₂ (with K₂ higher than K₁) with the same expiration date. The investor expects the price of the underlying asset to go up and is bullish about the stock. Bull spread is a directional strategy where the investor is moderately bullish about the underlying asset, she is investing in.

When an investor buys a call option, there is a limited downside risk (the loss of the premium) and an unlimited upside risk (gains). The bull spread reduces the potential downside risk on buying the call option, but also limits the potential profit by capping the upside. It is used as an effective hedge to limit the losses.

Market Scenario

When the price of underlying asset is expected to moderately move up, investors chose to execute a bull spread and the expiration date is chosen such that it occurs after the expected price movement. If the price decreases significantly by the expiration of the call options, the investor loses money by using a bull spread.

Example

In Figure 1 below, we represent the profit and loss function of a bull spread strategy using a long and a short call option. K₁ is the strike price of the long call i.e., €88 and K₂ is the strike price of the short call position i.e., €110. The premium of the long call is equal to €12.62, and the premium of the short call is equal to €1.16 computed using the Black-Scholes-Merton model. The time to maturity (T) is of 18 days (i.e., 0.071 years). At the time of valuation, the price of the underlying asset (S₀) is €100, the volatility (σ) of the underlying asset is 40% and the risk-free rate (r) is 1% (market data).

Figure 1. Profit and loss (P&L) function of a bull spread.

Profit and loss (P&L) function of a bul spread

Source: computation by the author.

You can download below the Excel file for the computation of the bull spread value using the Black-Scholes-Merton model.

Bear Spread

In a bear spread, the investor expects the price of the underlying asset to moderately decline in the near future. In order to hedge against the downside, the investor buys a put option with strike price K₁ and sells another put option with strike price K₂, with K₁ lower than < K₂. Initially, this initial position leads to a cash outflow since the put option bought (with strike price K₁) has a higher premium than put option sold (with strike price K₂) as K₁ is lower than < K₂.

Market Scenario

When the price of underlying asset is expected to moderately move down, investors chose to execute a bear spread and the expiration date is chosen such that it occurs after the expected price movement. Bear spread is a directional strategy where the investor is moderately bearish about the stock he is investing in. If the price increases significantly by the expiration of the put options, the investor loses money by using a bear spread.

Example

In Figure 2 below, we represent the profit and loss function of a bear spread strategy using a long and a short put option. K₁ is equal to the strike price of the short put i.e., €90 and K₂ is equal to the strike price of the long put i.e., €105. The premium of the short put is equal to €0.86, and the premium long put is equal to €7.26 computed using the Black-Scholes-Merton model.

The time to maturity (T) is of 18 days (i.e., 0.071 years). At the time of valuation, the price of the underlying asset (S₀) is €100, the volatility (σ) of stock is 40% and the risk-free rate (r) is 1% (market data).

Figure 2. Profit and loss (P&L) function of a bear spread.

Profit and loss (P&L) function of a bear spread

Source: computation by the author.

You can download below the Excel file for the computation of the bear spread value using the Black-Scholes-Merton model.

Butterfly Spread

In a butterfly spread, the investor expects the price of the underlying asset to remain close to its current market price in the near future. Just as a bull and bear spread, a butterfly spread can be created using call options. In order to profit from the expected market scenario, the investor buys a call option with strike price K₁ and buys another call option with strike price K₃, where K₁ < K₃, and sells two call options at price K₂, where K₁ < K₂ < K₃. Initially, this initial position leads to a net cash outflow.

Market Scenario

When the price of underlying asset is expected to stay stable, investors chose to execute a butterfly spread and the expiration date is chosen such that the expected price movement occurs before the expiration date. Butterfly spread is a non-directional strategy where the investor expects the price to remain stable and close to the current market price. If the price movement is significant (either downward or upward) by the expiration of the call options, the investor loses money by using a butterfly spread.

Example

In Figure 3 below, we represent the profit and loss function of a butterfly spread strategy using call options. K₁ is equal to the strike price of the long call position i.e., €85 and K₂ is equal the strike price of the two short call positions i.e., €98 and K₃ is equal to the strike price of another long call position i.e., €111. The premium of the long call K₁ is equal to €15.332, the premium of the long call K₃ is equal to €0.993 and the premium of the short call K₂ is equal to €5.334 computed using the Black-Scholes-Merton model. The premium of the butterfly spread is then equal to €5.657 (= 15.332 + 0.993 -2*5.334), which corresponds to an outflow for the investor.

The time to maturity (T) is of 18 days (i.e., 0.071 years). At the time of valuation, the price of the (S₀) is €100, the volatility (σ) of stock is 40% and the risk-free rate (r) is 1% (market data).

Figure 3. Profit and loss (P&L) function of a butterfly spread.

Profit and loss (P&L) function of a butterfly spread

Source: computation by the author.

You can download below the Excel file for the computation of the butterfly spread value using the Black-Scholes-Merton model.

Note that bull, bear, and butterfly spreads can also be created from put options or a combination of call and put options.

▶ All posts about options

▶ Gupta A. Options

▶ Gupta A. The Black-Scholes-Merton model

▶ Gupta A. Option Greeks – Delta

▶ Gupta A. Hedging Strategies – Equities

Useful resources

Hull J.C. (2018) Options, Futures, and Other Derivatives, Tenth Edition, Chapter 12 – Trading strategies involving Options, 282-301.

About the author

Article written in January 2022 by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Understanding Options and Options Trading Strategies

December 13, 2021 by Luis RAMIREZ

Understanding Options and Options Trading Strategies

In this article, Luis RAMIREZ (ESSEC Business School, Global BBA, 2019-2023) discusses the fundamentals behind options trading.

Financial derivatives

In order to understand and grasp the concept of options, knowledge of what is a derivative should be established. A financial instrument derivative is ultimately an instrument whose value derives from the value of an underlying asset (or multiple underlying assets). These underlying assets can of course be bonds, stocks, commodities, currencies, etc. Derivatives are widely common and used around the world; investment banks, commercial banks, and corporations (mainly multinational corporations) are all consistent users of derivatives. The purpose, or goal, behind derivatives is to manage risk, whether that be alleviating risk by hedging investments, or by taking on risk through speculative investments. To carry out this process, the investor must undertake one of the four types of derivatives. The four types are the following: options, forwards, futures, and swaps. In this article the focus will be solely placed on options.

What are options?

An option contract provides an investor the chance to either buy (for a call option) or sell (for a put option) the underlying asset, depending on what type of option they possess. Every option contract has an expiry date in which the investor can effectively exercise the option. A very important thing about options is that they provide investors the right, but not the obligation, to either buy or sell an asset (i.e., stock shares) at a price and at a date that have been agreed at the issuing of the cotnract.

Put options vs call options

Firstly, the main two different options are call and put options. Call options give investors (that bought the call option) the right to buy a stock at a certain price and at a certain date, and put options give investors (that bought the put option) the right to sell a stock at a certain price and at a certain date. The first step into acquiring options, either type, is paying a premium. This premium which is spent at the beginning of the process is the only loss that investors will face if the options are not exercised. However, the other side of the coin, options writers (sellers) are more exposed to risk as they are exposed to lose more than only the premium.

Sell-side vs buy-side

In an option contract, the price at which the asset is sold or bought is known as the strike price, or exercise price. When a call option has been bought, and the price of the share has
had a bullish trend and rises above the strike price, the investor can simply exercise his right to buy the share at the strike price, and then immediately sell it at the spot price, resulting in immediate profit. However, if the price of the share had a bearish trend and dropped below the strike price, the investor can decide not to exercise his right and will only lose the amount of premium paid in this case.

Figure 1. Profit and loss (P&L) of a long position in a call option
as a function of the price of the underlying asset at maturity
Profit and loss (P&L) as a function of the price of the underlying asset at maturity

Profit and loss (P&L) as a function of the price of the underlying asset at maturity

Source: production by the author.

On the other hand, selling options differs. Selling options is commonly known as writing options. The way this works is that a writer receives the premium from a buyer, this is the maximum profit a writer can receive by selling call options. Normally, a call option writer is bearish, therefore he believes that the price of the stack will fall below that of the strike price. If indeed the share price falls below the strike price, the writer would profit the premium paid by the buyer, since the buyer would not exercise the option. However, if the share price surpassed the strike price, the writer would have to sell shares at the low strike price. The writer would then experience a loss, the size of the loss depends on how many shares and price the writer would have to use to cover the entire option contract. Clearly, the risk for call writers is much higher than the risk exposure call buyers when acquiring an option. To summarize, the call buyer can only lose the premium paid, and the call writer can face infinite risk because the price of a share can keep increasing.

Figure 2. Profit and loss (P&L) of a short position in a call option
as a function of the price of the underlying asset at maturity
Profit and loss (P&L) as a function of the price of the underlying asset at maturity

Source: production by the author.

As for put options, put buyers usually believe the share price will decrease under the strike price. If this does eventually happen, the investor can simply exercise the put and sell at strike price, instead of a lower spot price. If the investor wants to go long, he can substitute the shares used in the option contract and buy them for a cheaper spot price after the put has been exercised. However, if the spot price is above the strike price, and the investor choses to not exercise the put, the loss will once again only be the cost of the premium.

Figure 3. Profit and loss (P&L) of a long position in a put option
as a function of the price of the underlying asset at maturity
Profit and loss (P&L) as a function of the price of the underlying asset at maturity

Source: production by the author.

On the other hand, put writers think the share price will have a bullish trend throughout the duration of the option lifecycle. If the share price rises above strike price, the contract will expire, and the seller’s profit is the premium he received. If the share price decreases, and falls under the strike price, then the writer is obliged to buy shares at a strike price which higher than the spot price. This is when the risk is at the highest for a put writer, if the share price falls. Just like call writing, the loss can be hefty. Only that in the case of put writing, it happens if the share price tumbles down.

Figure 4. Profit and loss (P&L) of a short position in a put option
as a function of the price of the underlying asset at maturity
Profit and loss (P&L) as a function of the price of the underlying asset at maturity

Source: production by the author.

This can be shrunk down to knowing that call buyers can benefit from buying securities or assets at a lower price if the share price rises during the length of the option contract. Put buyers can benefit from selling assets at a higher strike price if the share price falls during the length of the option contract. As per writers, they receive a premium fee when writing options. However, it is not all positive points, option buyers need to pay the premium fee and discount this from their potential profit, and writers face an indefinite risk subject to the share price and quantity.

Figure 5. Market scenarios for buying and selling call and put options

Source: production by the author.

Option Trading Strategies

Four trading strategies have already been mentioned, selling or buying either puts or calls. However, there are several different option trading strategies and new ones are being produced frequently, anyhow the article will focus on five trading strategies that most, if not all, investors are familiar with.

Covered Call

This trading strategy consists in the writer selling call options against the stock that he already owns. It is ‘covered’ because it covers the writer when the buyer of the option exercises his right to buy the shares, due to the writer already owning them, meaning that the writer can deliver the shares. This strategy is often used as an income stream from premiums. This is an employable strategy for those who believe that the asset they own will only experience a small change in price. The covered call is considered a low-risk strategy, and if used appropriately with a reliable stock, it can be a source of income.

Figure 6. Profit and loss (P&L) of a covered call
as a function of the price of the underlying asset at maturity

Source: production by the author.

Married put

Like a covered call, in a married put the investor buys an asset and then buys a put option with the strike price being equal to the spot price. This is done to be protected against a decrease in the asset price. Of course, when buying an option, a premium must be paid, which is a downside for a married put strategy. However, the married put limits the loss an investor could incur in case of a price decrease. On the other hand, if the price increases, profit is unlimited. This strategy is often used for volatile stocks.

Figure 7. Profit and loss (P&L) of a married put
as a function of the price of the underlying asset at maturity

Source: production by the author.

Protective Collar

This strategy is done when an investor buys a put option where the strike price is lower than the spot price, as well as instantly writing a call option where the strike price is higher than the spot price, this must be done by the investor owning said asset. This strategy protects the investor from a decrease in price. If the share price increases, large profits will be capped, however large losses will be also capped. When performing a protective collar, the best possibility for an investor is that the share price rises to the call strike price.

Figure 8. Profit and loss (P&L) of a protective collar
as a function of the price of the underlying asset at maturity

Source: production by the author.

Bull Call Spread

In order to execute this strategy, an investor buys calls at the same time that he sells the equivalent order of calls, which have a higher strike price. Of course, both calls must be tied to the same asset. As seen on the name of this strategy, it is a strategy that an investor employs when he predicts a bullish trend. Just like the protective collar, it limits both, gains and losses.

Figure 9. Profit and loss (P&L) of a bull call spread
as a function of the price of the underlying asset at maturity

Source: production by the author.

Bear Put Spread

This strategy is like the Bull Call Spread, only that it is in terms of a put option. The investor buys put options while he sells put options at a lower strike price. This can be done when the investor foresees a bearish trend, just like its call counterpart, the Bear Put Spread limits losses and gains.

Figure 10. Profit and loss (P&L) of a bear put spread
as a function of the price of the underlying asset at maturity

Source: production by the author.

Importance of options on financial markets

As seen on the variety of option trading strategies, and the different factors that play into each strategy mentioned, and dozens of other out there to explore, this instrument is a very utilized tool for investors, and financial institutions. The ‘options within options’ are of a huge variety and so much could be done. Many people have strong feelings towards this derivative, whether it is a negative, or positive stance, it all depends on the profits it brings. There is a lot of work behind options, and just like any other investment, due diligence is a key aspect of the procedure.

Useful Resources

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition.

Prof. Longin’s website Pricer d’options standards sur actions – Calls et puts (in French)

About the author

Article written in December 2021 by Luis RAMIREZ (ESSEC Business School, Global BBA, 2019-2023).

Covered call

December 11, 2021 by Akshit GUPTA

Covered Call

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the concept of covered call used in equities option contracts.

Introduction

Covered call

The covered call strategy is a two-part strategy that essentially involves an investor writing a call option on an underlying security while simultaneously holding a long position in the same underlying. This action of buying an asset and writing calls on it at the same time is commonly referred as ‘buy write’. By writing a call option, the investor locks in the price of the underlying asset, thereby enjoying a short-term gain from the premium received.

Market scenario

The covered call is generally ideal if the investor has a neutral or slightly bullish outlook of the market wherein the potential future upside of the underlying asset owned by the investor is limited. This strategy is used by investors when they would prefer booking short-term profits on the assets than to keep holding it.

For instance, consider a ‘buy write’ situation where an investor buys shares of a stock (i.e., holds a long position in the stock) and simultaneously writes call options on them. The investor has a neutral view on the stock and doesn’t expect the price to rise much.
To book a short-term profit and also hedge any minor downsides in the stock price, the investor is writing call options on the stock at a strike price greater than or equal to the current price of the stock (i.e. out-of-the-money or at-the-money call options). The buyer of those call options would pay the investor a premium on those calls, whether or not the option is exercised. This is the covered call strategy in a nutshell.

Let us consider a covered call position with writing at-the money calls. One of following three scenarios may happen:

Scenario 1: the stock price does not change, and calls expire at the money

In this scenario, the market viewpoint of the investor holds correct and the profit from the strategy is the premium earned on the call options. In this case, the option holder does not exercise its call options, and the investor gets to keep the underlying stocks too.

Scenario 2: the stock price rises, and calls expire in the money

In this scenario, since the price of the stock was already locked in through the call, the investor enjoys a short-term profit along with the premium. However, this also poses a risk in case the price of the stock rises substantially because the investor misses out on the opportunity.

Scenario 3: the stock price falls and calls expire out of the money

This is a negative scenario for the investor. There is limited protection from the downside through the premium earned on the call options. However, if the stock price falls below a certain break-even point, the losses for the investor can be considerable since there will be a fall in its underlying position.

Risk profile

In a covered call, the total cost of the investment is equal to the price of the underlying asset minus the premium earned by writing the call. However, the profit potential for the investment is limited and the maximum loss can be significantly high. The risk profile of the position is represented in Figure 1.

Figure 1. Risk profile of covered call position.

Source: computation by the author (based on the BSM model).

You can download below the Excel file for the computation of the Profit or Loss (P&L) function of the underlying position and covered call position.

The delta of the position is equal to the sum of the delta of the long position in the underlying asset (+1) and the short position in the call option (-Δ).

Figure 2 represents the delta of the covered call position as a function of the price of the underlying asset. The delta of the call option is computed with the Black-Scholes-Merton model (BSM model).

Figure 2. Delta of a covered call position.

Source: computation by the author (based on the BSM model).

You can download below the Excel file for the computation of the delta of a protective put position.

Example

An investor holds 100 shares of Apple bought at the current price of $144 each. The total investment is then equal to $14,400. She is neutral about the short-term prospects of the market. In order to gain from her market scenario, she decides to write an at-the-money call option at $144 on the Apple stock (lot size is 100) with a maturity of one month, using the covered call strategy.

We use the following market data: the current price of Appel stock is $144, the implied volatility of Apple stock is 22.79%, and the risk-free interest rate is equal to 1.59%.

Based on the Black-Scholes-Merton model, the price of the call option is $3.87.

Let us consider three scenarios at the time of maturity of the call option:

Scenario 1: stability of the price of the underlying asset at $144

The total cost of the initial investment is the cost of acquiring the Apple stocks ($14,400) minus the premium received on writing the calls ($387 = $3.87*100), which is equal to $14,013, i.e. $14,400 – $387.

As the stock price ($144) is equal to the strike price of the call options ($144), the value of the call options is equal to zero, and the investor earns a profit which is equal to the initial price of the call options (the premium), which is equal to $387.

Scenario 2: an increase in the price of the underlying asset to $155

The total cost of the initial investment is the cost of acquiring the Apple stocks ($14,400) minus the premium on writing the calls ($387 = $3.87*100), which is equal to $14,013, i.e. $14,400 – $387.

As the stock price has risen to $155, the call options are exercised by the option buyer, and the investor will have to sell the Apple stocks at the strike price of $144.

By executing the covered call strategy, the investor earns $387 (i.e. ($144-$144)*100 +$387) but misses the opportunity of earning higher profits by selling the stock at the current market price of $155.

Scenario 3: a decrease in the price of the underlying asset to $142

As the stock price is at $142, the call options are not exercised by the option buyer and the options expire worthless (out of the money).

As the buyer does not exercise the call options, the investor earns a profit which is equal to the price of the call options which is equal to $387. But his net profit decreases by the amount of the decrease in his position in the APPLE stocks which is equal to -$200 (i.e. ($142-$144)*100).

▶ All posts about Options

▶ Akshit GUPTA Options

▶ Akshit GUPTA Option Trader – Job description

▶ Akshit GUPTA The Black-Scholes-Merton model

▶ Akshit GUPTA Protective Put

▶ Akshit GUPTA Option Greeks – Delta

Useful Resources

Research articles

Black F. and M. Scholes (1973) “The Pricing of Options and Corporate Liabilities” The Journal of Political Economy, 81, 637-654.

Merton R.C. (1973) “Theory of Rational Option Pricing” Bell Journal of Economics, 4(1): 141–183.

Books

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 10 – Trading strategies involving Options, 276-295.

Wilmott P. (2007) Paul Wilmott Introduces Quantitative Finance, Second Edition, Chapter 8 – The Black Scholes Formula and The Greeks, 182-184.

About the author

Article written in December 2021 by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Types of exercise for option contracts

February 13, 2026September 19, 2021 by Akshit GUPTA

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the different types of exercise for option contracts.

Introduction

Exercising a call option contract means the purchase of the underlying asset by the call buyer at the price set in the option contract (strike price). Similarly, exercising a put option contract means the sale of the underlying asset by the put buyer at the price set in the option contract.

The different option contracts can be settled in cash or with a physical delivery of the underlying asset. Normally, the equity, fixed interest security and commodity option contracts are settled using physical delivery and index options are settled in cash.

Majority of options are not exercised before the maturity date because it is not optimal for the option holder to do so. Note that for options with physical delivery, it may be better to close the position before the expiration date). If an option expires unexercised, the option holder loses any of the rights granted in the contract (indeed, in-the-money options are automatically exercised at maturity). Exercising options is a sophisticated and at times a complicated process and option holder need to take several factors into consideration while making the decision about exercise such as opinion about future market behavior of underlying asset in option, tax implications of exercise, net profit that will be acquired after deducting exercise commissions, option type, vested shares, etc.

Different types of exercise for option contracts

The option style does not deal with the geographical location of where they are traded! The contracts differ in terms of their expiration time when they can be exercised. The option contracts can be categorized as per different styles they come in. Some of the most common styles of option contracts are:

American options

American-style options give the option buyer the right to exercise his/her option anytime prior or up to the expiration date of the contract. These options provide greater flexibility to the option buyer but also come at a higher price as compared to the European-style options.

European options

European-style options can only be exercised on the expiration or maturity date of the contract. Thus, they offer less flexibility to the option buyer. However, the European options are cheaper as compared to the American options.

Bermuda options

Bermuda options are a mix of both American and European style options. These options can only be exercised on specific predetermined dates or periods up to the expiration date. They are considered to be exotic option contracts and provide limited flexibility to the option buyer.

Early Exercise

Early exercise is a strategy of exercising options before the expiration date and is possible with American options only. The question is: when the holder of an American option should exercise his/her option? Before the expiration date or at the expiration date? Quantitative models say that it could be optimal to exercise American options before the date of a dividend payout (options are not protected against the payement of dividends by firms) and sometimes for deep in-the-money put options.

There are many strategies that investors follow while exercising option contracts in order to maximize their gains and hedge risks. A few of them are discussed below:

Exercise-and-Hold

Investors can purchase their option shares with cash and hold onto them. This allows them to benefit from ownership in company stock, providing potential gains from any increase in stock value and dividend payments if any. Investors are also liable to pay brokerage commissions fees and taxes.

Exercise-and-Sell

This is a cashless strategy wherein investors purchase the option shares and then immediately sell them. Brokerages generally allow this kind of transaction without use of cash, with the money from the stock sale covering the purchase price, as well as the commissions and taxes associated with the transaction. This choice provides investors with available cash in pocket to invest elsewhere too.

Exercise-and-Sell-to-Cover

In this strategy too, investors exercise the option and then immediately sell enough shares to cover the purchase price, commissions fees and taxes. The remaining shares remain with the investor.

Useful Resources

Academic research

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 10 – Mechanics of options markets, 235-240.

Business analysis

Fidelity Exercising Stock Options

About the author

Article written in August 2021 by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

VIX index

September 12, 2021 by Youssef LOURAOUI

VIX index

In this article, Youssef LOURAOUI (ESSEC Business School, Global Bachelor of Business Administration, 2017-2021) presents the VIX index, which is a financial index that measures the uncertainty in the US equity market.

This article is structured as follows: we begin by defining the grounding notions of the VIX index. We then explain the behavior of this index and its statistical characteristics. We finish by presenting its practical usage in financial markets.

Definition

The CBOE Volatility Index, abbreviated “VIX”, is a measure of the expected S&P 500 index movement calculated by the Chicago Board Options Exchange (CBOE) from the current trading prices of options written on the S&P 500 index.

Known as Wall Street’s “fear index”, the VIX is closely monitored by a broad range of market players, and its level and pattern have become ingrained in market discussion.

Figure 1 illustrates the evolution of the VIX index for the period from 2003 to 2021.
Figure 1 Historical levels of the VIX index from 2003-2021.

Source: computation by the author (Data source: Thomson Reuters).

VIX values greater than 20 are regarded to be high by market participants. If the VIX is between 12 and 20, it is considered normal; if it is less than 12, it is considered low. As it is the case with other indices, the VIX is computed using the price of a basket of tradable components (in this case, options expiring within the next month or so). The profit or loss that option buyers and sellers realize during the option’s life will depend, among other things, on how significantly the S&P 500’s actual volatility will differ from the implied volatility given by the VIX at the start of the period (S&P Global Research, 2017).

Behavior of the VIX index

Statistical distribution of the S&P500 index returns and VIX level

Figure 2 displays the statistical distribution of the price variations in the S&P500 index for different levels of the VIX index The higher the VIX index (by convention, greater than 20), the more severe the distribution tends to be, with negative skewness and high kurtosis indicating heightened volatility in the US market, therefore exacerbating both positive and negative swings. An opposite finding may be made for the VIX level at lower levels (often less than 12), when market swings are less evident due to less skewness and lower kurtosis (S&P Global Research, 2017).

Figure 2. The distribution of 30-day return in the S&P500 index for different VIX index levels.

Source: S&P Global Research (2017).

If the VIX is low, market players may benefit by purchasing options; conversely, if the VIX is high, market participants may profit from selling options. The specific utility of anticipated VIX is that it gives us with a more accurate assessment of whether VIX is high, low, or normal at any point in time (S&P Global Research, 2017). Thus, VIX may be regarded of as a crowd-sourced estimate of the S&P 500’s expected volatility. As with interest rates and dividends, one cannot invest directly in them, even though one can guess on their future worth, one cannot invest directly in VIX, and the significance of a specific VIX level is commonly misinterpreted (S&P Global Research, 2017).

Recent volatility in the S&P500 index and VIX level

Figure 3 demonstrates that the VIX index is strongly correlated with recent market volatility. However, there is considerable variance; for example, a recent volatility level of about 20% has been associated with a VIX level of 34 (point B, when VIX was very “high”) and with a VIX level of 12 (point C, when VIX was relatively “low”). Volatility (realized or implied) has a strong propensity to return to its mean. This insight is not especially original, despite its illustrious past. There is an enormous body of data demonstrating that volatility tends to mean revert across markets, and the pioneers of this field were given the Nobel Prize in part for incorporating their results into volatility forecasts and simulations (S&P Global Research, 2017).

Figure 3. Relation between VIX and recent volatility.

Source: S&P Global Research (2017).

Realized volatility in the S&P500 index and VIX level

Figure 4 represents the relationship between Realized volatility in the S&P500 index over a period and the VIX level at the begining of the period.

Figure 4. VIX versus next realized volatility.

Source: S&P Global Research (2017).

Mean reversion

Figure 5 shows how VIX index converge to a certain llong-term level as time passes. This finding is not due to 15% being exceptional in any manner; this figure for M was calculated using historical volatility levels for the S&P 500 and their evolution. It is not implausible that M (else referred to as long-term average volatility in the US equities market) may change over time; changes in the S&P 500’s sector weightings, trade All of these factors have the ability to influence both the pace and the volume and the point at which mean reversion occurs.

Figure 5. Mean-reversion dynamic in recent volatility.

Source: S&P Global Research (2017).

Use of the VIX index in financial markets

There are two methods for determining an asset’s volatility. Either through a statistical calculation of an asset’s realized volatility, also known as historical volatility, which serves as a pointer to the asset’s volatility behavior. This is a limited method that is based on the premise that past volatility tends to replicate itself in the future, without including a forward-looking study of volatility. The second technique is to extract an asset’s volatility from option prices referred to as “implied volatility”.

Why should I be interested in this post?

When investors make investment decisions, they utilize the VIX to gauge the degree of risk, worry, or stress in the market. Additionally, traders can trade the VIX using a range of options and exchange-traded products, or price derivatives using VIX values.

Useful resources

Business analysis

CBOE , 2021. VIX

Nasdaq, 2021. Realized Volatility

Nasdaq, 2021. Vix Index Volatility

S&P Global Research, 2017. Reading VIX: Does VIX Predict Future Volatility?

S&P Global Research, 2017. A Practitioner’s Guide to Reading VIX

About the author

The article was written in September 2021 by Youssef LOURAOUI (ESSEC Business School, Global Bachelor of Business Administration, 2017-2021).

Introduction

Types of derivatives markets

Market structure: over-the-counter derivatives and exchange-traded derivatives

Types of derivatives contracts

Types of underlying asset classes

Quantitative measures of derivatives market activity and size

Notional amount

Gross market value

Open Interest

Trading Volume

Key sources of statistics on global derivatives markets

Bank for International Settlements (BIS)

International Swaps and Derivatives Association (ISDA)

Futures Industry Association (FIA)

How to get the data

Derivatives market business statistics

Global derivatives market

OTC derivatives market

Exchange-traded derivatives market

Underlying asset classes

Role of the Black–Scholes–Merton (BSM) model

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Related posts on the SimTrade blog

Useful resources

Academic research on option pricing

Academic research on the role of models

Data

About the author

Introduction

Option pricing

Implied volatility

Moneyness

Empirical observation: implied volatility depends on moneyness

Smiles and smirks

Stylized facts about the implied volatility curve across markets

An Empirical Analysis of S&P 500 Implied Volatility

Asynchronous trading and measurement error

Example: options on the S&P 500 index

Economic Insights

Demand for crash protection:

Leverage and volatility feedback effects:

Market frictions and limits to arbitrage:

Conclusion

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Related posts on the SimTrade blog

Option price modelling

Volatility

Useful resources

Academic research on Option pricing

Academic research on Stylized facts

About the author

Introduction

The Black-Scholes-Merton model

Key Parameters

Option payoff

Call option value

Option and volatility

Volatility: the unobservable parameter of the model

Implied Volatility

Calculating Implied volatility

Newton-Raphson Method

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Related posts on the SimTrade blog

Useful resources

Academic research

About the BSM model

Other

About the author

Une immersion complète dans la finance de marché

Des insights concrets pour réussir

Cas pratiques et actualité

Trois concepts financiers à découvrir dans la BD

Les produits dérivés

Les stratégies de couverture

Les Greeks en finance

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Expériences professionnelles

Techniques financières

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