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).

A quick presentation of the M&A field…

May 15, 2022 by Louis DETALLE

A quick presentation of the M&A field…

In this article, Louis DETALLE (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023) explains what does an M&A daily life looks like.

What does M&A consist in?

Mergers & Acquisitions (M&A) is a profession that advises companies wishing to develop their external growth, i.e. growth through the acquisition of a company or through a merger with it. M&A mandates are therefore carried out on the side of the company that wishes to acquire another company, “buy-side”, or on the side of a company that wishes to be acquired, “sell-side”.

What does an analyst work on?

The tasks of an M&A analyst are diverse and include, for example, drawing up a business plan, modelling different scenarios and strategies in Excel, and drafting information memorandums (IMs) on the various deals in progress. All these skills are then widely used for the mergers and acquisitions of companies, in the development of their external strategy, in their financial evaluation or in the analysis of databases. Overall, M&A allows you to move into any sector of finance and this is part of the reason why it is so attractive.

Why does M&A jobs appeal so much to students?

First of all, it is the dynamic working atmosphere that investment banking enjoys that also attracts young graduates. M&A is indeed marked by a culture of high standards and maximum commitment, with highly responsive teams and extremely competent colleagues. Working in a quality team is very stimulating, and often makes it possible to approach the workload with less apprehension and to rapidly increase one’s competence. The remuneration is also much higher than in other professions at the beginning of a professional career for a young graduate and it progresses rapidly. Finally, it is also the exit hypotheses that attract young M&A analysts.

What are the main exits for M&A?

Most professionals who started out in M&A move on to other types of activities where experience in this sector is required. This is particularly the case in private equity. After advising companies on their growth and expansion projects, the young investment banker has all the tools needed to work in investment funds. The skills are indeed transposable to the financial and strategic questions that private equity funds ask themselves in order to obtain a return on investment.

Switching to alternative portfolio management (hedge funds) is also a possibility. Hedge funds can invest in different types of assets such as commodities, currencies, corporate or government bonds, real estate or others. As a former M&A analyst, you have the skills to analyse the market and determine the assets that seem to be the most appropriate and profitable.

Finally, some former M&A bankers switch to corporate M&A, which involves determining which companies or subsidiaries the company should buy or sell. This can be a very interesting area as you have the opportunity to follow the acquisition of a company from start to finish and therefore take a long-term view of the company’s strategy.

Useful resources

Décideurs magazine Rankings for M&A banks in France (league tables)

About the author

The article was written in May 2022 by Louis DETALLE (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023).

Warren Buffet and his basket of eggs

May 12, 2022 by Rayan AKKAWI

Warren Buffet and his basket of eggs

In this article, Rayan AKKAWI (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022) analyzes the two following quotes “Do not put all eggs in one basket” and “Put all your eggs in one basket and watch that basket” often used by Warren Buffet to describe his investment strategy.

“Do not put all eggs in one basket”

I particularly liked this quote first because it is said by the world’s greatest investor and one of the richest people on the planet, Warren Buffet. I aspire this man due to his great investment philosophy which is to invest in great businesses at value for money prices and then by using the “buy and hold strategy” keep the stocks over the long term. He has bought great brands such as Coca Cola, Microsoft, and American Express. Second, I like this quote particularly because it is dedicated to any person who has little or no knowledge in investment, so it is easy to implement.

Analysis

If we analyze the wealthiest people in the world, they are entrepreneurs who have created companies that grew exponentially in value. For example, Bill gates who is the founder of Microsoft (1975), Jeff Bezos who is the founder of Amazon (1994), and Mark Zuckerberg who is the founder of Facebook (2004). And as we continue to analyze these founders, we come to realize that they have made their wealth by putting all their eggs in one basket at least early in their lives. However, not all of us have this entrepreneurial spirit and business success such as these brilliant men. Thus, when Warren Buffet said “do not put all eggs in one basket” he was referring to an average person who has little knowledge in investments. Therefore, he advocates investment into index tracker or passive funds which have the benefit of low charges, better performance, and large diversification than most active managed funds. This involves a buy and hold strategy which keeps share dealing charges low. Thus, it is always recommended to have 80% of investments in passive funds which are low cost, predictable, and conservative funds and 20% of investments in satellite which usually involve higher charges with greater volatility and greater returns.

Another way of looking at it is the following. One might decide to invest a certain number of personal wealth in a new business or in crypto. This would be a risky type of investment because another competitor might release a better and more attractive or even more affordable version of the product or service. Eventually, this might put you out of business if a customer writes a bad review of your product or business or if the bitcoin value drops.

So before you invest more time and money in your business, consider how you can manage your risk. First, you must think about your risk tolerance which depends on your age and current financial obligations. Second, you need to keep sufficient liquidity in your portfolio by setting aside an emergency fund that should be equal to 6 to 8 months’ expenses. For ensuring that there is easy accessibility to emergency funds, you should have low-risk investment options like Liquid Funds and Overnight Funds in your accounts. Then you need to determine an asset allocation strategy that works which refers to investing in more than one asset class for reducing the investment risks and this strategy also provides you with optimal returns. You can invest in a perfect mix of key asset classes like Equity, Debt, Mutual Funds, real estate, etc. One of the asset allocation strategies is to invest in a combination of asset classes that are inversely correlated to each other. After you have found the best mix of asset classes for your portfolio, you can reduce the overall investment risk by diversifying your investment in the same asset class. Think about diversifying by offering more than one product or service. To avoid liquidity risk, it is always better to stay invested in blue chip stock or fund. Investors should check the credit rating of debt securities to avoid default risk.

“Put all your eggs in one basket and watch that basket”

At the same time, Warren Buffet believes that diversification makes little sense if a person doesn’t know exactly what he or she is doing. Diversification is a protection against ignorance and is for people who do not know how to analyze businesses. Sometimes it is enough to invest in two or three companies that are resistant to competition rather than fifty average companies due to less risk. That is why it is as critical for a person to invest in a company where its values and vision are similar to that of the investor and to be able to watch closely the performance of that business and its stocks.

Thus, Warren Buffet believes that it is extremely crucial to be able to “watch your basket” or your stocks closely to better understand the stock market. For example, when the stock market is going down, it is the best way to start buying stocks because businesses will be selling at a discount.

Why should I be interested in this post?

One would be interested to read this post because it introduces the basics of investing in stock markets for an average person who has little knowledge in investments or for a student studying business. As a student, it is crucial and important to be able to have at least a general idea of the basic rules of investments and especially those stated by one of the most famous investors in the world such as Mr. Warren Buffet. Whether you are interested in buying stocks yourself or whether you are not, as a business student, you might be asked about investments and the financial market one time in your life and knowing some useful information about investments will be impressive for you. It will allow you to understand the bigger picture of financial markets, give some recommendations for your family and friends, and help you invest yourself in the safest and most successful way.

Useful resources

Berkshire Hathaway

About the author

The article was written in May 2022 by Rayan AKKAWI (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022).

Big data in the financial sector

May 12, 2022 by Rayan AKKAWI

Big data in the financial sector

In this article, Rayan AKKAWI (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022) explains the role of big data in the financial sector.

Big data is a term used for contemporary technologies and methodologies that are used to collect, process, and analyze complex data. Today, data is being created at an exponential rate. In fact, and according to a 2015 IBM study, 90% of the data in the world has been created in the past two years. As big data gets bigger, it becomes even more important and essential for executives in the financial sector to stay ahead of the curve. Also, it is expected that data creation will continue to grow moving forward in time.

Big Data in The Financial Sector

For decades, financial analysts have relied on data to extract insights. Today, with the rise of data science and machine learning, automated algorithms and complex analytical tools are being used hand in hand to get a head of the curve in diferetn areas of the financial sector.

Fraud prevention

First, data has helped with fraud prevention such as identity theft and credit card schemes. Abnormally high transactions from conservative spenders and out of region purchases often signal credit card fraud. Whenever this happens, the card is automatically blocked, and a notification is sent out to the card owner. This protects users, insurance companies, and banks from huge financial loses in a small period. This also made things even easier and more practical avoiding the hassle of having to call and cancel the card. Data science comes in the form of tool like random forests that can detect a certain suspicion. In addition, and to lower the chance of identity theft, data has helped ease this process through 3D passwords, text messages, and PINT code which have backed up the safety of online transactions.

Anomaly detection

Second, data has helped the financial sector through anomaly detection. Data analysis is not only created to avoid a problem but also to detect it. For example, data today helps with catching illegal insider traders. To do so, data analysts created anomaly detection algorithms that can analyze history in trading patterns and thus detect and catch abnormal transactions of illegal traders.

Customer analytics

Third, data has helped with improving customer analytics. Data analyzes previous behavioral trends of consumers based on historical transactions and then makes future predictions of how consumers are likely to act. With the help of socioeconomic characteristics, we can create clusters of consumers and group customers based on how much money we expect to gain or lose from each client in the future. Following that, we can come up with decisions to focus on a certain type of clients to make profits and cut on other customers to make savings. Thus, financial institutions minimize human errors by utilizing data science. To achieve that, first, by identifying uncertain interactions and then monitor them going forward. Finally, prioritizing the investments most vulnerable at a given time. For example, banks use this approach to create adaptive real risk score time models to identify risky clients and those who are suitable for a mortgage or a loan.

Algorithmic trading

Fourth and most importantly data has created algorithmic trading. Machines make trading based on algorithms multiple times every second with no need for approval by a stand-by analyst. These trades can be in any market and even in multiple markets simultaneously. Thus, algorithmic trading has mitigated opportunity costs. Thus, there are algorithmic rules that can help in identifying if there is a need to trade or not to trade and reinforces business models where errors are highly penalized and then adjust hyper parameters. We can see algorithms that exploit arbitrage opportunities where they can find inconsistencies and make trades which can cause problems. The huge upside is that it is high frequency trading; whenever it will find an opportunity to make a trading, it will. However, the downside is that imprecision could lead to huge losses due to lack of human supervision. That is why sometimes human interventions are needed.

Conclusion

Thus, we can say that data has become the hottest commodity that results in getting an edge over competition. Financial institutions spend a huge amount of money to get exclusive rights to data. By having more information, they can construct better models. The most valuable commodities are not analysts but the data itself. That is how the data science has revolutionized finance.

Characteristics of Big Data

When talking about Big Data, four main characteristics need to be considered to understand the why Big Data plays a transformational role in the financial sector: volume, variety, velocity, and value.

Volume

First, the amount also known as volume of data being produced on daily basis by users has been increasing exponentially by users. This large output of data has helped create Zettabytes (10¹² Gigabyte) and Yottabytes (10¹⁵ Gigabyte) of datasets in which companies can benefit by extracting knowledge and insights out of it. However, this amount of data cannot be processed using regular computers and laptops. Since they would require a lot of processing power.

Variety

Second, as the massive amount of data is being generated by multiple sources, the output of this data is unstructured making it hard to organize the data extract insights. Raw data extracted from the source without being processed does not provide any value to business as it does provide stakeholders with the ability to analyze it.

Velocity

Third, to address the issue of processing technological advancements have brought us to the tipping point where technologies such as cloud computing have enabled companies to process this large amount of data by utilizing the ability to share computational power. Furthermore, cloud platforms have not only helped in the processing part of data but by the emergence or cloud solution such as data lakes and data warehouses. Businesses are able to store this data in its original from to make sure that they can benefit from it.

Value

Finally, this brings us to the most important aspect of Big Data and that in being able to extract insights and value out of the data to understand what it is telling us. This process is tedious and time consuming however with ETL tool (Extract Transform Load) the data in its raw format is transformed so that standardized data sets can be produced. Insights can be extracted through Business Intelligence (BI) tools to create visualization that help business decisions. As well as predictive artificial intelligence models that help business predict when to take a strategic decision. In the case of financial markets, these decisions are when to buy or sell assets, and how much to invest.

Challenges Solved by Big Data in the Financial Industry

Utilizing Big Data in the finance industry presents a lot of benefits and helps the industry to overcome multiple challenges.

Data Quality

As previously mentioned, the multiple data sources present a huge challenge from a data management standpoint. Making it an ongoing and a tedious effort to maintain the integrity and the reliability of the records collected. Therefore, adding information processing systems and standardizing the data gathering and transformation processes helps improve the accuracy of the decision-making process, especially in financial services companies where real-time data enables fast decision making and elevates the performance of companies.

Data Silos

Since financial data comes from multiple sources (applications, emails, documents, and more), the use of data integration tools help simplifies and consolidate the data of the institution. These technologies facilitate processes and make them faster and more agile, which are important characteristics in the financial markets.

Robo-Advisory

Big Data and analytics have had a huge impact on the financial advisory sector. Where financial advisors are being replaced by machine learning algorithms and AI models to manage portfolio and provide customers with personalized advice and without human intervention.

Why should I be interested in this post?

This article is just an eye opener on the trends and the future state of the financial industry.

Like many other industries, the financial sector is becoming one of the most data driven field. Therefore, as future leaders it is vital to keep track and push towards data driven solutions to excel and succeed within the financial sector.

Useful resources

The Future of Cognitive Computing

Five Ways to Use RPA in Finance

About the author

The article was written in May 2022 by Rayan AKKAWI (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022).

Momentum Trading Strategy

May 8, 2022 by Akshit GUPTA

Momentum Trading Strategy

This article written by Akshit GUPTA (ESSEC Business School, Master in Management, 2019-2022) explains the momentum trading strategy.

Introduction

The momentum trading strategy is a strategy where a trader buys a security when its market price starts to rise and then sells it when its price seems to have reached a top. Similarly, a trader sells (or short sells) a security when its market price starts to fall and then buys it back when its price seems to have reached a bottom. In other words, if we observe a positive price change or return today, we are long tomorrow, and if we observe a negative price change or return today, we are short tomorrow.

This trading strategy is based on the direction of the price trend (up or down) in the market and its relative strength. The rationale behind the momentum trading strategy is that, for an upward trend, if there is enough buying force behind the rise in the price of an asset, it will keep on rising until a strong selling pressure is seen in the market to reverse the trend. Similarly, for a downward trend, if there is enough selling force behind the fall in the price of an asset, it will keep on falling until a strong buying pressure is seen in the market to reverse the trend.

Momentum trading is a trading strategy with a short-term horizon where traders try to capture and profit from the price trend. The period for implementing a momentum strategy can range from a trend forming within a day or over several days. Momentum traders try to identify the strength of an ongoing trend in a particular direction and take a position. The strength can measured by different technical indicators discussed below. Once the strength of the trend begins to fall, the trader exits the position at a profit.

Momentum traders are least concerned about the fundamentals of the company for which the stock is to be traded. They rather use various technical indicators to understand the trend in the stock price, especially its strength.

Implementation

Figure 1 below illustrates the implementation of the momentum trading strategy for Apple stock over the period from April 1, 2020 to March 31, 2021.

Figure 1. Implementation of the momentum trading strategy for Apple stock.

Source: computation by the author (data source: Yahoo Finance).

In Figure 1, an upward trend can be seen forming in the period from November 22, 2020 to November 25, 2020 in the price of Apple stock. The trader following a momentum strategy will go long on the Apple stock till the momentum is in the upward direction. The right time to exit the long position is around December 2, 2020. By following this trend, the trader can capture a price movement of around $10 which is approximately 8%-9%, by going long on the Apple stock.

Momentum trading indicators

Momentum trading indicators help the trader to look for the formation of a trend and the signal of an entry/exit point, and also indicate the strength of that signal. We present below some of the most common indicators used to assess the strength of the trend: relative strength index (RSI), moving-average convergence-divergence (MACD) and Bollinger bands.

Relative Strength Index (RSI)

The RSI indicator is a technical indicator and is plotted on a chart which ranges from 0 to 100. It helps a trader in knowing the relative strength of a trend formation. The indicator is an oscillator which provides overbought or oversold signals based on the positioning of the line in the chart. During the uptrend, if the line crosses the 70 mark, an overbought signal is considered for the given security. Symmetrically, during a downtrend, if the line crosses the 30 mark, an oversold signal is considered. Momentum traders generally take a position in between in the indicator instead of waiting for a price reversal when the line crosses the given thresholds. For example, a trader can use the halfway mark of 50 to get an idea about the formation of a trend. If the RSI line crosses the 50 mark and is moving in an upward direction, it can show the high strength of the upward forming trend and the trader can take a long position in the respective stock.

Figure 2. Relative Strength Index of Apple stock.

Source: computation by the author (data source: Yahoo Finance).

Moving-average convergence-divergence (MACD)

The moving-average convergence-divergence (MACD) is a technical indicator based on the moving averages of prices over a period of time. The indicator helps in understanding the direction and strength of a trend. It also helps in understanding the rate at which the change in trend is happening.

The indicator is shown by two lines namely, the MACD line and the signal line. The MACD line is the difference between two exponential moving-averages, a long-term moving-average like a 26-day moving average and a short-term moving-average like the 12-day moving average. The signal line is made up of the 9-day exponential moving-average of the MACD itself and is placed on the same graph. A bar graphs plotted on the zero-line (X axis) showing the difference by which the MACD line is below/above the signal line. Generally, the indicator is used to understand the degree of the bullish or bearish sentiments in the market. If the MACD line crosses the signal line from below the zero-level moving upwards, it indicates a bullish trend. In such a scenario, a trader practicing momentum strategy would take a long position in the market seeing the trend.

Figure 3. Moving-average convergence-divergence of Apple stock.
MACD of Apple stock
Source: computation by the author (data source: Yahoo Finance).

Bollinger bands

The Bollinger bands is a very popular technical indicator that represents the volatility in the prices of a financial asset. The indicator consist of three lines, namely, a simple moving-average (SMA), and an upper band and a lower band. The simple moving average is usually computed over a rolling period of 20 trading day (about a calendar month for the equity market). The upper and lower bands are usually set by default to two standard deviations away from the simple moving average.

The width between the upper and lower Bollinger bands provides a range for price changes in the market (an indicator of volatility). The bands help to identify the overbought or oversold situations in the market for an asset. They can be used by a trader to identify possible entry or exit prices to implement the momentum trading strategy.

Figure 4 represents the Bollinger bands for Apple stocks. The price of the Apple stock is touching the lower band on November 2, 2020 and reverting just after that. This can be a signal for the momentum trader showing a trend reversal and the trader can take a long position in this stock till the price touches the 20-day SMA line which happens around November 5, 2020, thereby capturing a price movement of $8 approximately.

Figure 4. Bollinger bands of Apple stock.

Source: computation by the author (data source: Yahoo Finance).

Market conditions

Market liquidity and market volatility play a major role in the implementation of a momentum strategy.

A liquid market is generally preferred by traders in order to quickly enter and exit the market.

Stock price volatility is a major factor affecting a momentum trader’s decision to enter/exit a trade. A highly volatile stock can provide a good opportunity for a trader to earn high profits using this strategy as the asset prices can change dramatically in a short period of time. But a high stock volatility can also lead to huge losses if the prices move in an unfavorable direction.

The figure below represents the historical daily volatility (standard deviation of returns over rolling 10-day periods) of Apple stock over the period from April 1, 2020 to March 31, 2021.

Figure 5. Volatility of Apple stock.

Source: computation by the author (data source: Yahoo Finance).

You can download below the Excel file for the computation of the different momentum trading indicators mentioned above.

Risks associated with momentum trading

Although momentum trading is a commonly used strategy, the risks associated with it are quite high. The trader using this strategy should be careful about:

Entering the position too early
Exiting the position too late
Relying on rumors and fake news
Missing the indication of a reversal in the direction of the trend
Not applying a strict stop loss rule

Link with market efficiency

Market efficiency refers to the degree to which all the relevant information about an asset is incorporated in the market prices of that asset. Fama (1970) distinguished three forms of market efficiency: weak, semi-strong, and strong according to the set of information considered (market data, public information, and private information).

In the weak form of the market efficiency hypothesis, the current market price of an asset incorporates all the historical market data (past transaction prices and volumes). The current market price of the asset is then the best predictor of its future price.

In a market efficient in the weak sense, the autocorrelation of asset price changes or returns is close to zero.

A positive autocorrelation coefficient would imply that after a price increase, we should likely observe another price increase, and symmetrically, after a price decrease, we should likely observe another price decrease, leading in both cases to price trends.

The implementation of a momentum strategy assumes that the autocorrelation of price changes is positive, which contradicts the efficient market hypothesis.

In a market which is efficient in the weak sense (implying an autocorrelation close to zero), momentum trading strategies should not exhibit extra profit as traders are not be able to beat the market on the long run.

▶ Jayati WALIA Bollinger bands

▶ Jayati WALIA Moving averages

▶ Akshit GUPTA Growth investment strategy

Useful resources

Academic research

Fama E.F. (1970) Efficient Capital Markets: A Review of Theory and Empirical Work, The Journal of Finance 25(2): 383-417.

Fama E.F. (1991) Efficient Capital Markets II: A Review of Theory and Empirical Work, The Journal of Finance 46(5): 1575-1617.

Business analysis

Fidelity Learning center: Momentum trading strategy

About the author

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

Returns

April 20, 2022 by Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains how returns of financial assets are computed and their interpretation in the world of finance.

Introduction

The main focus of any investment in financial markets is to make maximum profits within a coherent risk level. Returns in finance is a metric that inherently refers to the change in the value of any investment. Positive values of returns are interpreted as gains whereas negative values are interpreted as losses.

Returns are generally computed over standardized frequencies such as daily, monthly, yearly, etc. They can also be computed for specific time periods such as the holding period for ease of comparison and analysis.

Computation of returns

Consider an asset for a time period [t -1, t] with an initial price P_t-1 at time t-1 and final price P_t at time t (one period, two dates). Different forms of defining returns for the asset over period [t -1, t] are discussed below.

Arithmetic (percentage) returns

This is the simplest way for computation of returns.

The return over the period [t -1, t], denoted by R_t, is expressed as:

Arithmetic returns

Logarithmic returns

Logarithmic returns (or log returns) are also used commonly to express investment returns. The log return over the period [t-1, t], denoted by R_t is expressed as:

Logarithmic returns

Log returns provide the property of time-additivity to the returns which essentially means that the log returns over a given period can be simply added together to compute the total return over subperiods. This feature is particularly useful in statistical analysis and reduction of algorithmic complexity.

Logarithmic returns additivity

Log returns are also known as continuously compounded returns because the rate of log returns is equivalent to the continuously compounding interest rate for the asset at price P₀ and time period t.

img_SimTrade_compounded_returns

Link between arithmetic and logarithmic returns

The arithmetic return (R^ari) and the logarithmic return (R^log) are linked by the following formula:

Relation between arithmetic and logarithmic returns

Components of total returns

The total return on an investment is essentially composed of two components: the yield and the capital gain (or loss). The yield refers to the periodic income or cash-flows that may be received on the investment. For example, for an investment in stocks, the yield corresponds to the revenues of dividends while for bonds, it corresponds to interest payments.

On the other hand, capital gain (or loss) refers to the appreciation (or depreciation) in the price of the investment. Thus, the capital gain (or loss) for any asset is essentially the price change in the asset.

Total returns for a stock over the period [t -1, t], denoted by R_t, can hence be expressed as:

Total returns

Where
   P_t: Stock price at time t
   P_t-1: Stock price at time t-1
   D_t-1,t: Dividend obtained over the period [t -1, t]

Price changes and returns

Consider a stock with an initial price of 100€ at time t=0. Suppose the stock price drops to 50€ at time t=1. Thus, there is a change of -50% (minus sign representing the decrease in price) in the initial stock price.

Now for the stock price to reach back to its initial price (100€ in this case) at time t=2 from its price of 50€ at time t=1, it will require an increase of (100€-50€)/50€ = 100%. With arithmetic returns, the increase (+100%) has to be higher than the decrease (-50%).

Similarly, for a price drop of -25% in the initial stock price of 100€, we would require an increase of 33% in the next time period to reach back the initial stock price. Figure 1 illustrates this asymmetry between positive and negative arithmetic returns.

Figure 1. Evolution of price change as a measure of arithmetic returns.
img_SimTrade_price_change_evolution
Source: computation by the author.

If the return is defined as a logarithmic return, there is a symmetry between positive and negative logarithmic returns as illustrated in Figure 2.

Figure 2. Evolution of price change as a measure of logarithmic returns.
img_SimTrade_price_change_evolution
Source: computation by the author.

You can also download below the Excel file for computation of arithmetic returns and visualise the above price change evolution.

Internal rate of return (IRR)

Internal rate of return (IRR) is the rate at which a project undertaken by the firm break’s even. It is a financial metric used by financial analysts to compute the profitability from an investment and is calculated by equating the initial investment and the discounted value of the future cashflows i.e., making the net present value (NPV) equal to zero. The IRR is the sprecail value of the discount rate which makes the NPV equal to zero.

The IRR for a project can be computed as follows:

IRR formula

Where,
CF_t : Cashflow for time period t

The higher the IRR from a project, the more desirable it is to pursue with the project.

Ex ante and ex post returns

Ex ante and ex post are Latin expressions. Ex ante refers to “before an event” while ex post refers to “after the event”. In context of financial returns, the ex ante return corresponds to prediction or estimation of an asset’s potential future return and can be based on a financial model like the Capital Asset Pricing Model (CAPM). On the other hand, the ex post return corresponds to the actual return generated by an asset historically, and hence are lagging or backward-looking in nature. Ex post returns can be used to forecast ex ante returns for the upcoming period and together, both are used to make sound investment decisions.

About the author

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

Supply and Demand

April 15, 2022 by Diana Carolina SARMIENTO PACHON

Supply and Demand

In this article, Diana Carolina SARMIENTO PACHON (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022) explains the economic concept of supply and demand, which is key to understand the way markets work.

Supply and demand are the fundamental concepts that shape the way we make business and operate in the world. They construct both simple transactions such as the purchase of coffee or more compounded transactions such as the operations in the financial world. For this reason, it’s crucial to understand and uncover them deeper.

The basic concepts

Supply is referred to the amount available of a product that firms offer, whereas demand is the amount desired by consumers or households. When these quantities are equal, an equilibrium is reached and consequently a transaction takes place, leading to the well-known law of supply & demand which shapes the behavior of daily transactions and shifts in the economy. If price increases, then supply also increases; nonetheless, demand decreases as it’s more expensive for consumers to a buy good; on the contrary, if prices decline then supply also decline since producers would make less revenue whereas demand goes up as it is cheaper to buy. This dynamic takes place until the quantities of supply and demand are equal so that the optimum equilibrium is found.

Figure 1. Supply and demand.
img_Simtrade_risk_reduction_stocks
Source: computation by the author.

From another perspective, if demand escalates then price rises due to the high desirability of the good, meanwhile when demand drops it can create a surplus of supply which can drag the price down. Likewise, this scenario can be applied in financial markets e.g., in the case of a bullish sentiment in the market, there can be a positive speculation which creates a higher desirability for certain stock resulting in a decrease in price; nevertheless, when demand is low the price may drop because of a low or negative speculation on a specific stock.

Furthermore, the fundamental law of supply and demand can also explain the price movements seen in the financial markets. To illustrate, for a commodity such as coffee, if the surface of cultivation expands or if the harvest is good, it is very likely that the coffee price will sink as its supply will be abundant. Therefore, it is essential to consider the information about the market regularly as it can have a significant influence on the speculation of investors which will eventually define their demands and so the price of a stock. Consequently, it is very important to be able to determine how an announcement or any kind of information can affect the demand or even the supply of a stock, commodity, or financial instrument since this will define how markets will behave.

Special cases

However, it’s also important to mention that there are industries and situations in which the law of supply and demand does not apply. An instance of this is the luxury industry, in which the higher the product price set by firms, the higher the demand from consumers. This may be due to the value that costumers perceive by purchasing such items. Alternatively, oil is another example to be mentioned as its price has a low-price sensitivity which means that any change in its price won’t result in any significant demand changes, this could be due to the high necessity of oil in all industries which makes it crucial for daily operations.

Useful resources

Krugman, P. & Wells, R. (2012) Economics. 3rd edition. United States: Macmillan Learning.

Mankiw, G. (2016) The Market Forces of Supply and Demand (table of content) Principles of Economics. 8th edition. Boston: Cengage Learning.

Mankiw, G. (2016) The Market Forces of Supply and Demand (slides) Principles of Economics. 8th edition. Boston: Cengage Learning.

Deskara Supply and Demand: Law, Curves, and Examples

International Energy Agency (IEA) Supply and demand for oil

Sabiou M. Inoua and Vernon L. Smith The Classical Theory of Supply and Demand

About the author

The article was written in April 2022 by Diana Carolina SARMIENTO PACHON (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022)

Risk Aversion

April 14, 2022 by Diana Carolina SARMIENTO PACHON

Risk Aversion

In this article, Diana Carolina SARMIENTO PACHON (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022) explains the economic concept of risk aversion, which is key to understand the behavior of participants in financial markets.

Risk Aversion refers to the level of reluctance that an individual possesses towards risk. Specifically, it refers to the attitude of investors towards the risk underlying investments which will directly determine how portfolios are allocated or even how a stock may behave depending on market conditions. To elaborate, when market participants have higher risk aversion due to unfavorable market shocks e.g., natural disasters, bad news or scandals that affect a company or a security, this situation will cause a perception of higher risk leading to many selling, and thus decreasing prices. Therefore, risk aversion should be analyzed carefully.

Risk aversion and investor’s characteristics

It’s important to note that risk aversion can be highly variable over time as this notion changes along with investor profile, in other words with age, income, culture and other key factors, making it even more complex to evaluate than it appears in the traditional economics literature. To illustrate more accurately some of the factors that define an investor profile are:

Age

The older the person is, the more risk averse he or she is. On the contrary, younger individuals tend to be less risk averse which may be due to their high expectations and eagerness to attempt something new as well as the longer timeframe they have, whereas older people prefer safety and stability in their lives.

Income

Individuals with a smaller budget tend to have a higher risk aversion since they have fewer resources, and a loss would make a greater impact on them than a wealthy individual.

Past Losses

When an individual has already experienced some loss, she or he will be more wary of it since it’s now too costly to bear another loss; therefore, risk aversion will be significantly higher. An example of this is the post-crisis, as people have lost so much and this has had a negative impact on their lives, they tend to become more cautious of risk.

Investment Objective

For crucial events such as retirement or education, risk version tends to be higher as the individual cannot bear to risk for such a fundamental matter of his or her life.

Investment Horizon

Investors focused on short-term horizon tend to be more risk averse as they cannot take too much risk due to the short timeline.

Risk aversion and financial investments

Furthermore, risk aversion also takes into account more factors apart from those mentioned above, for this reason most of the time before creating the respective portfolio for an investor, financial advisors shape their client’s risk preferences in order to adjust the portfolio allocation to them. Many times, these can be conducted by questionnaires and tests that will accordingly assign a risk profile concluding with certain risk categories:

A Conservative profile refers to more risk averse individuals, the portfolios assigned for this type are mainly composed by both more secure & less volatile securities such as bonds, meanwhile stocks have a minimal participation.
A Moderate profile is attributed to more risk averse individuals who are willing to take more risk, however he or she does not want to step too much further. These portfolios are usually more diversified as they contain more types of securities in different percentages such as government & corporate bonds, and stocks.
An aggressive profile which is allocated to portfolios mainly composed in the highest percentage by the risky securities. For instance, the main securities could be stocks, specifically growth stocks or even crypto.

Due to all sensitive and private information used by financial institutions, financial regulatory entities are important to ensure the protection and transparency of information, thereby the Mifid (The Markets in Financial Instruments Directive) has been created in the European Union to fulfill such task through the use of rules and general standards.

Measure of risk of financial assets

Additionally, there are other mathematical metrics that can interfere in the risk profile, and depending on these the portfolio may be constructed:

Standard Deviation

It refers to the volatility of historical data, in other words how dispersed the data is over time which illustrates how risky the security may be. The higher the standard deviation, the higher the risk since this is suggesting that the stock is more variable and there is more uncertainty, thus a risk averse individual prefers a lower standard deviation.

Beta

It is linked with the systematic risk that comes with a stock, that is to say it illustrates the volatility compared to the market. Firstly, a beta equal to 1 indicates a volatility and movement equalizing the market, secondly a beta higher than 1 is referred to a security that is more volatile than the market, to illustrate B= 1.50 specifies 50% more volatility than the market. Thirdly, a beta less than 1 stipulates less volatility than the market. Therefore, the lower the beta the less risk exposure is found.

Modern Portfolio Theory & Risk

Introduced by Harry Markowitz in 1950s, the Modern Portfolio Theory illustrates the optimum portfolio allocation that maximizes return given a specific level of risk, in which risk is measured by the standard deviation and the return by the average mean of the portfolio. This explanation also leads to the one- single period mean-variance theory which suggests various portfolio allocations depending on the trade-off between return and risk. However, there are more advance models which explain this scenario in a multiperiod by rebalancing or diversifying further.

Risk aversion and economic conditions

Risk aversion does not only shape the portfolio allocation and its diversification, but it also may have a significant impact on the market as a result of expectations. When there are booming economic times, individuals usually feel more confident and thus less risk averse as a consequence of positive expectations of future cash flows; however, when a recession is coming investors may shift to a more risk averse behavior making them feel afraid of the future which influences them to sell certain stocks and, in this way, making the price plump. Although it may be seen as a simple emotion that defines the fear of risk, it still impacts in a very large extent the financial market as it dictates the roles and strategies behind investing, and thereby it is crucial analyze it carefully.

▶ Youssef LOURAOUI Markowitz Modern Portfolio Theory

▶ Youssef LOURAOUI Implementing Markowitz asset allocation model

▶ Jayati WALIA Capital Asset Pricing Model (CAPM)

Useful resources

Díaz A and Esparcia C (2019) Assessing Risk Aversion From the Investor’s Point of View Frontiers in Psychology, 10:1490

Desjardins Online brokerage The Risk Aversion Coefficient

Coursera course Investment management

Crehana course Trading: How to invest in stocks (Trading: Como invertir en Bolsa)

About the author

The article was written in April 2022 by Diana Carolina SARMIENTO PACHON (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022)

A quick overview of the Bloomberg terminal…

April 11, 2022 by Louis DETALLE

A quick overview of the Bloomberg terminal…

In this article, Louis DETALLE (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023) explains everything there is to know about the Bloomberg terminal which is a must-know in finance.

How to use the main functions of the Bloomberg terminal?

One may notice that the keyboard of the Bloomberg terminal is a little strange. Indeed, this keyboard called Starboard, and contains red, blue, green and yellow keys for specific functions in addition to your regular keys.

Functions are unique Bloomberg applications that provide analysis and information on securities,
sectors, regions and more. Each function is accessed by typing in its unique mnemonic (a short, memorable name) and then pressing the key located in the lowest-right sided area of the keyboard.

Let’s review together the different functions of the buttons:

The HELP button is perhaps the most useful button for those just starting out. If you have questions about anything on the terminal, simply press the button once and a Bloomberg specialist will be there to start a live chat with you to resolve your questions.

In order to benefit from the latest news, users can simply type NEWS and press enter to get the latest information on market trends, movements and other relevant news.

Those in the finance industry chat via Bloomberg Messaging, which is essentially equivalent to Facebook Messenger but on Bloomberg. It enables you to send a message to anyone on the device. This means that anyone in the industry can technically contact each other instantly. No need to ask for someone else’s number or find out the best way to get in touch.

Main users of the Bloomberg terminal

Traders, brokers, analysts, portfolio managers, investors and executives are the Terminal’s primary consumer base as they need to access the data provided by Bloomberg easily in order to do their job.

A subscription to the Bloomberg Terminal costs approximately $20,000 a year, but that does not stop its customers from renewing their subscriptions because of its usefulness.

Training webinars

First and foremost, the Bloomberg beginner should work on the document available on Bloomberg website, Getting started on the Bloomberg Terminal, which will give you the main information on the keys and their function.

The best next step to get used to the Bloomberg Terminal is to complete the certification
course: Bloomberg Market Concepts (BMC). BMC is an 8-hour e-learning course that will
provide a visual introduction to the financial markets and covers nearly 70 Terminal functions which is enough for whoever wants to start using Bloomberg.

Useful resources

Bloomberg’s website

Capital Markets (BMC) Certification’s website

About the author

The article was written in April 2022 by Louis DETALLE (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023).

Specific risk

April 1, 2022 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) explains the specific risk of financial assets, a key concept in asset pricing models and asset management in practice.

This article is structured as follows: we start with a reminder of portfolio theory and the central concept of risk in financial markets. We then introduce the concept of specific risk of an individual asset and especially its sources. We then detail the mathematical foundation of risk. We finish with an insight of the relationship between diversification and risk reduction with a practical example to test this concept.

Portfolio Theory and Risk

Markowitz (1952) and Sharpe (1964) created a framework for risk analysis based on their seminal contributions to portfolio theory and capital market theory. All rational profit-maximizing investors attempt to accumulate a diversified portfolio of risky assets and borrow or lend to achieve a risk level consistent with their risk preferences given a set of assumptions. They established that the key risk indicator for an individual asset in these circumstances is its correlation with the market portfolio (the beta).

The variance of returns of an individual asset can be decomposed as the sum of systematic risk and specific risk. Systematic risk refers to the proportion of the asset return variance that can be attributed to the variability of the whole market. Specific risk refers to the proportion of the asset return variance that is unconnected to the market and reflects the unique nature of the asset. Specific risk is often regarded as insignificant or irrelevant because it can be eliminated in a well-diversified portfolio.

Sources of specific risk

Specific risk can find its origin in business risk (in the assets side of the balance sheet) and financial risk (in the liabilities side of the balance sheet):

Business risk

Internal or external issues might jeopardize a business. Internal risk is directly proportional to a business’s operational efficiency. An internal risk would include management neglecting to patent a new product, so eroding the company’s competitive advantage.

Financial risk

This pertains to the capital structure of a business. To continue growing and meeting financial obligations, a business must maintain an ideal debt-to-equity ratio.

Mathematical foundations

Following the Capital Asset Pricing Model (CAPM), the return on asset i, denoted by R_i can be decomposed as

img_SimTrade_return_decomposition

Where:

R_i the return of asset i
E(R_i) the risk premium of asset i
β_i the measure of the risk of asset i
R_M the return of the market
E(R_M) the risk premium of the market
R_M – E(R_M) the market factor
ε_i represent the specific part of the return of asset i

The three components of the decomposition are the expected return, the market factor and an idiosyncratic component related to asset only. As the expected return is known over the period, there are only two sources of risk: systematic risk (related to the market factor) and specific risk (related to the idiosyncratic component).

The beta of the asset with the market is computed as:

Beta

Where:

σ_i,m : the covariance of the asset return with the market return
σ_m² : the variance of market return

The total risk of the asset measured by the variance of asset returns can be computed as:

Decomposition of total risk

Where:

β_i² * σ_m² = systematic risk
σ_εi² = specific risk

In this decomposition of the total variance, the first component corresponds to the systematic risk and the second component to the specific risk.

Decomposition of returns

We analyze the decomposition of returns on Apple stocks. Figure 1 gives for every month of 2021 the decomposition of Apple stock returns into three parts: expected return, market factor (systematic return) and an idiosyncratic component (specific return). We used historical price downloaded from the Bloomberg terminal for the period 1999-2022.

Figure 1. Decomposition of Apple stock returns:
expected return, systematic return and specific return.
Decomposition of asset returns Computation by the author (data: Bloomberg).

You can download below the Excel file which illustrates the decomposition of returns on Apple stocks.

Why should I be interested in this post?

Investors will be less influenced by single incidents if they possess a range of firm stocks across several industries, as well as other types of assets in a number of asset classes, such as bonds and stocks.

An investor who only bought telecommunication equities, for example, would be exposed to a high amount of unsystematic risk (also known as idiosyncratic risk). A concentrated portfolio can have an impact on its performance. This investor would spread out telecommunication-specific risks by adding uncorrelated positions to their portfolio, such as firms outside of the telecommunication market.

Useful resources

Academic research

Evans, J.L., Archer, S.H. 1968. Diversification and the Reduction of Dispersion: An Empirical Analysis. The Journal of Finance, 23(5): 761–767.

Markowitz, H. 1952. Portfolio Selection. The Journal of Finance, 7(1): 77-91.

Mossin, J. 1966. Equilibrium in a Capital Asset Market. Econometrica, 34(4): 768-783.

Sharpe, W.F. 1963. A Simplified Model for Portfolio Analysis. Management Science, 9(2): 277-293.

Sharpe, W.F. 1964. Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance, 19(3): 425-442.

Tole T.M. 1982. You can’t diversify without diversifying. The Journal of Portfolio Management. Jan 1982, 8 (2) 5-11.

About the author

The article was written in April 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Systematic risk

April 1, 2022 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the systematic risk of financial assets, a key concept in asset pricing models and investment management theories more generally.

This article is structured as follows: we introduce the concept of systematic risk. We then explain the mathematical foundation of this concept. We present an economic understanding of market risk on recent events.

Portfolio Theory and Risk

Markowitz (1952) and Sharpe (1964) developed a framework on risk based on their significant work in portfolio theory and capital market theory. All rational profit-maximizing investors seek to possess a diversified portfolio of risky assets, and they borrow or lend to get to a risk level that is compatible with their risk preferences under a set of assumptions. They demonstrated that the key risk measure for an individual asset is its covariance with the market portfolio under these circumstances (the beta).

The fraction of an individual asset’s total variance attributable to the variability of the total market portfolio is referred to as systematic risk, which is assessed by the asset’s covariance with the market portfolio. Systematic risk can be decomposed into the following categories:

Interest rate risk

We are aware that central banks, such as the Federal Reserve, periodically adjust their policy rates in order to boost or decrease the rate of money in circulation in the economy. This has an effect on the interest rates in the economy. When the central bank reduces interest rates, the money supply expands, allowing companies to borrow more and expand, and when the policy rate is raised, the reverse occurs. Because this is cyclical in nature, it cannot be diversified.

Inflation risk

When inflation surpasses a predetermined level, the purchasing power of a particular quantity of money reduces. As a result of the fall in spending and consumption, overall market returns are reduced, resulting in a decline in investment.

Exchange Rate Risk

As the value of a currency reduces in comparison to other currencies, the value of the currency’s returns reduces as well. In such circumstances, all companies that conduct transactions in that currency lose money, and as a result, investors lose money as well.

Geopolitical Risks

When a country has significant geopolitical issues, the country’s companies are impacted. This can be mitigated by investing in multiple countries; but, if a country prohibits foreign investment and the domestic economy is threatened, the entire market of investable securities suffers losses.

Natural disasters

All companies in countries such as Japan that are prone to earthquakes and volcanic eruptions are at risk of such catastrophic calamities.

Following the Capital Asset Pricing Model (CAPM), the return on asset i, denoted by R_i can be decomposed as

img_SimTrade_return_decomposition

Where:

R_i the return of asset i
E(R_i) the risk premium of asset i
β_i the measure of the risk of asset i
R_M the return of the market
E(R_M) the risk premium of the market
R_M – E(R_M) the market factor
ε_i represent the specific part of the return of asset i

The beta of the asset with the market is computed as:

Beta

Where:

σ_i,m : the covariance of the asset return with the market return
σ_m² : the variance of market return

The total risk of the asset measured by the variance of asset returns can be computed as:

Decomposition of total risk

Where:

β_i² * σ_m² = systematic risk
σ_εi² = specific risk

In this decomposition of the total variance, the first component corresponds to the systematic risk and the second component to the specific risk.

Systematic risk analysis in recent times

The volatility chart depicts the evolution of implied volatility for the S&P 500 and US Treasury bonds – the VIX and MOVE indexes, respectively. Implied volatility is the price of future volatility in the option market. Historically, the two markets have been correlated during times of systemic risk, like as in 2008 (Figure 1).

Figure 1. Volatility trough time (VIX and MOVE index).

Sources: BlackRock Risk and Quantitative Analysis and BlackRock Investment Institute, with data from Bloomberg and Bank of America Merrill Lynch, October 2021 (BlackRock, 2021).

The VIX index has declined following a spike in September amid the equity market sell-off. It has begun to gradually revert to pre-Covid levels. The periodic, albeit brief, surges throughout the year underscore the underlying fear about what lies beyond the economic recovery and the possibility of a wide variety of outcomes. The MOVE index — a gauge of bond market volatility – has remained relatively stable in recent weeks, despite the rise in US Treasury yields to combat the important monetary policy to combat the effect of the pandemic. This could be a reflection of how central banks’ purchases of government bonds are assisting in containing interest rate volatility and so supporting risk assets (BlackRock, 2021).

The regime map depicts the market risk environment in two dimensions by plotting market risk sentiment and the strength of asset correlations (Figure 2).

Figure 2. Regime map for market risk environment.

Source: BlackRock Risk and Quantitative Analysis and BlackRock Investment Institute, October 2021 (BlackRock, 2021).

Positive risk sentiment means that riskier assets, such as equities, are outperforming less risky ones. Negative risk sentiment means that higher-risk assets underperform lower-risk assets.

Due to the risk of fast changes in short-term asset correlations, investors may find it challenging to guarantee their portfolios are correctly positioned for the near future. When asset correlation is higher (as indicated by the right side of the regime map), diversification becomes more difficult and risk increases. When asset prices are less correlated (on the left side of the map), investors have greater diversification choices.

When both series – risk sentiment and asset correlation – are steady on the map, projecting risk and return becomes easier. However, when market conditions are unpredictable, forecasting risk and return becomes substantially more difficult. The map indicates that we are still in a low-correlation environment with a high-risk sentiment, which means that investors are rewarded for taking a risk (BlackRock, 2021). In essence, investors should use diversification to reduce the specific risk of their holding coupled with macroeconomic fundamental analysis to capture the global dynamics of the market and better understand the sources of risk.

Why should I be interested in this post?

Market risks fluctuate throughout time, sometimes gradually, but also in some circumstances dramatically. These adjustments typically have a significant impact on the right positioning of a variety of different types of investment portfolios. Investors must walk a fine line between taking enough risks to achieve their objectives and having the proper instruments in place to manage sharp reversals in risk sentiment.

Useful resources

Academic research

Markowitz, H. 1952. Portfolio Selection. The Journal of Finance, 7(1): 77-91.

Mossin, J. 1966. Equilibrium in a Capital Asset Market. Econometrica, 34(4): 768-783.

Sharpe, W.F. 1963. A Simplified Model for Portfolio Analysis. Management Science, 9(2): 277-293.

Sharpe, W.F. 1964. Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance, 19(3): 425-442.

Business analysis

BlackRock, 2021. Market risk monitor

About the author

The article was written in April 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

The Monte Carlo simulation method for VaR calculation

January 30, 2024March 24, 2022 by Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole – Master in Management, 2019-2022) explains the Monte Carlo simulation method for VaR calculation.

Introduction

Monte Carlo simulations are a broad class of computational algorithms that rely majorly on repeated random sampling to obtain numerical results. The underlying concept is to model the multiple possible outcomes of an uncertain event. It is a technique used to understand the impact of risk and uncertainty in prediction and forecasting models.

The Monte Carlo simulation method was invented by John von Neumann (Hungarian-American mathematician and computer scientist) and Stanislaw Ulam (Polish mathematician) during World War II to improve decision making under uncertain conditions. It is named after the popular gambling destination Monte Carlo, located in Monaco and home to many famous casinos. This is because the random outcomes in the Monte Carlo modeling technique can be compared to games like roulette, dice and slot machines. In his autobiography, ‘Adventures of a Mathematician’, Ulam mentions that the method was named in honor of his uncle, who was a gambler.

Calculating VaR using Monte Carlo simulations

The basic concept behind the Monte Carlo approach is to repeatedly run a large number of simulations of a random process for a variable of interest (such as asset returns in finance) covering a wide range of possible scenarios. These variables are drawn from pre-specified probability distributions that are assumed to be known, including the analytical function and its parameters. Thus, Monte Carlo simulations inherently try to recreate the distribution of the return of a position, from which VaR can be computed.

Consider the CAC40 index as our asset of interest for which we will compute the VaR using Monte Carlo simulations.

The first step in the simulation is choosing a stochastic model for the behavior of our random variable (the return on the CAC 40 index in our case).
A common model is the normal distribution; however, in this case, we can easily compute the VaR from the normal distribution itself. The Monte Carlo simulation approach is more relevant when the stochastic model is more complex or when the asset is more complex, leading to difficulties to compute the VaR. For example, if we assume that returns follow a GARCH process, the (unconditional) VaR has to be computed with the Monte Carlo simulation methods. Similarly, if we consider complex financial products like options, the VaR has to be computed with the Monte Carlo simulation methods.

In this post, we compare the Monte Carlo simulation method with the historical method and the variance-covariance method. Thus, we simulate returns for the CAC40 index using the GARCH (1,1) model.
Figure 1 and 2 illustrate the GARCH simulated daily returns and volatility for the CAC40 index.

Figure 1. Simulated GARCH daily returns for the CAC40 index.
img_SimTrade_CAC40_GARCH_ret
Source: computation by the author.

Figure 2. Simulated GARCH daily volatility for the CAC40 index.
img_SimTrade_CAC40_GARCH_vol
Source: computation by the author.

Next, we sort the distribution of simulated returns in ascending order (basically in order of worst to best returns observed over the period). We can now interpret the VaR for the CAC40 index in one-day time horizon based on a selected confidence level (probability).

For instance, if we select a confidence level of 99%, then our VaR estimate corresponds to the 1st percentile of the probability distribution of daily returns (the bottom 1% of returns). In other words, there are 99% chances that we will not obtain a loss greater than our VaR estimate (for the 99% confidence level). Similarly, VaR for a 95% confidence level corresponds to bottom 5% of the returns.

Figure 3 below represents the unconditional probability distribution of returns for the CAC40 index assuming a GARCH process for the returns.

Figure 3. Probability distribution of returns for the CAC40 index.
img_SimTrade_CAC40_MonteCarloVaR
Source: computation by the author.

From the above graph, we can interpret VaR for 99% confidence level as -3% i.e., there is a 99% probability that daily returns we obtain in future are greater than -3%. Similarly, VaR for 95% confidence level as -1.72% i.e., there is a 95% probability that daily returns we obtain in future are greater than -1.72%.

You can download below the Excel file for computation of VaR for CAC40 stock using Monte Carlo method involving GARCH(1,1) model for simulation of returns.

Advantages and limitations of Monte Carlo method for VaR

The Monte Carlo method is a very powerful approach to VAR due its flexibility. It can potentially account for a wide range of scenarios. The simulations also account for nonlinear exposures and complex pricing patterns. In principle, the simulations can be extended to longer time horizons, which is essential for risk measurement and to model more complex models of expected returns.

This approach, however, involves investments in intellectual and systems development. It also requires more computing power than simpler methods since the more is the number of simulations generated, the wider is the range of potential scenarios or outcomes modelled and hence, greater would be the potential accuracy of VaR estimate. In practical applications, VaR measures using Monte Carlo simulation often takes hours to run. Time requirements, however, are being reduced significantly by advances in computer software and faster valuation methods.

Useful resources

Jorion P. (2007) Value at Risk, Third Edition, Chapter 12 – Monte Carlo Methods, 321-326.

About the author

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

Implementing Black-Litterman asset allocation model

March 21, 2022 by Youssef LOURAOUI

In this article, Youssef Louraoui (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents an implementation of the Black-Litterman model, used to determine the expected return of a portfolio by integrating investor’s views regarding the performance of the underlying assets selected in the investment portfolio.

This article follows the following structure: first, we introduce the Black-Litterman model. We then present the mathematical foundations of this model. We conclude with an explanation of the methodology to build the spreadsheet with the results obtained. You will find in this post an Excel spreadsheet which implement the model.

Introduction

The Black-Litterman asset allocation model, established for the first time in the early 1990’s by Fischer Black and Robert Litterman, is a sophisticated strategy for dealing with unintuitive, highly concentrated, and input-sensitive portfolios. The most likely reason that more portfolio managers do not use the Markowitz model, which maximises return for a given degree of risk, is input sensitivity, a well-documented issue with mean-variance optimization.

The Black-Litterman Model employs a Bayesian technique to integrate an investor’s subjective views of expected returns on one or more assets with the market equilibrium vector (prior distribution) of expected returns to obtain a new, mixed estimate of expected returns. The new vector of returns (the posterior distribution) is a weighted complex average of the investor’s views and market equilibrium.

Mathematical foundation

The idea of the Black Litterman estimates is not to find the optimum portfolio weights as in the Markowitz optimization framework, but instead to find the expected return that would be used as an input to compute the optimum portfolio weights. This approach is referred as reversion portfolio optimization technique. The idea behind is that optimum weights are already observed in the market and captured in the market portfolio. We can approach the reasoning by maximizing the following utility function adjusted to the risk:

img_SimTrade_mathematical_foundation_Black_Litterman_6

w^T = transposed of portfolio weights
Π = Implied equilibrium excess return vector
A = price of risk or risk aversion factor
Σ = variance-covariance matrix

We take the partial derivative of U in terms of weights (w) and we derive the following expression:

img_SimTrade_mathematical_foundation_Black_Litterman_5

By setting the partial derivative equal to zero, we can maximize the utility function in term of weights. The proposed approach in the Black Litterman approach is that instead of seeking the optimal weights, which are incorporated in the market portfolio and thus computable via the market capitalization of the equities in the portfolio, we’ll isolate the Π (implied equilibrium excess return) to obtain the optimal expected returns for the portfolio:

We can deconstruct the Black-Litterman model as

img_SimTrade_mathematical_foundation_Black_Litterman_3

τ= scalar
P = Linking matrix
∑ = Variance-covariance matrix
Π= implied equilibrium excess return
A = Price of risk
w = weight vector
Ω = uncertainty of views

The first term of the formula is introduced in order to respect the constraint that the portfolio weights should be equal to one:

The second term of the formula is to compute a weighted average of the implied equilibrium excess return adjusted to the uncertainty of the returns by the view vector weighted with the uncertainty of views:

The final output E(R) is a vector of return n x 1 that represent the equilibrium returns of the market adjusted to investors views.

Implementation of the Black-Litterman asset allocation model in practice

To model a Black-Litterman portfolio allocation, we obtained a large time series to obtain useful results by downloading the equivalent of 23 years of market data from a data provider (in this example, Bloomberg). We generate the variance-covariance matrix after obtaining all necessary statistical data, which includes the expected return and volatility indicated by the standard deviation of the returns for each stock during the provided period.

The data is derived from the Bloomberg terminal. The first step is to calculate the logarithmic returns and excess returns on the selected assets (returns minus the risk-free rate). After calculating the logarithmic returns on each asset, we can estimate the capital asset pricing model’s returns (CAPM) expected returns. This information will be used to calculate the Black-Litterman expected returns on a comparative basis.

1. The first input for the model is the price of risk A, which represents the risk aversion of investor and is obtained by subtracting the expected return of the market the risk-free rate and divided by the variance of the market:

img_SimTrade_Black_Litterman_formulas_for_spreadsheet_1

E(r_m)= expected market returns
r_f = risk-free rate
σ²_m = variance of market

2. We extract the respective market capitalization of each security to obtain their market weights in the portfolio. Given that our investable universe is made of five stocks, we can retrieve their respective market capitalization and compute the weights of each stock in relation to the sum of total market-capitalization in the portfolio.

img_SimTrade_Black_Litterman_formulas_for_spreadsheet_2

Table 1 depicts the optimal weights obtained from their respective market capitalisation, coupled with the respective expected return and volatility.

Table 1. Asset characteristics of Apple, Amazon, Microsoft, Goldman Sachs, and Pfizer.

img_SimTrade_Black_Litterman_spreadsheet_2

Source: computation by the author.

3. We compute the variance-covariance matrix of logarithmic returns using the data analysis tool pack available in Excel (Table 2).

Table 2. Variance-covariance matrix of asset returns

img_SimTrade_Black_Litterman_spreadsheet_5

Source: computation by the author.

4. We compute the implied equilibrium excess return (Π) as the matrix calculation of the price of risk (A) times the matrix multiplication of the weights computed in step 4 times the variance-covariance matrix computed in step 3.

img_SimTrade_Black_Litterman_formulas_for_spreadsheet_3

Π= implied equilibrium excess return
A = Price of risk
w = weight vector

5. The views are incorporated into the model. To achieve this, we provide three views to include into the model. If there are no views, the values will correspond to the market portfolio. The investment manager’s views for the expected return on certain of the portfolio’s assets regularly diverge from the Implied Equilibrium Return Vector (), which serves as the market-neutral starting point for the Black-Litterman model that quantifies the uncertainty associated with each view. The Black-Litterman Model can be used to depict such views in absolute or relative terms. As an illustration, let us suppose that the real and simulated portfolio will have the same views:

View 1: Apple will outperform Microsoft by .05 percent
View 2: Amazon will outperform Microsoft by .1 percent
View 3: Apple will outperform Amazon by .05 percent

To incorporate the vector Q of views, we create a link matrix P where the rows sum to zero. Figure 3 depicts the workings done in the spreadsheet.

Table 3. Views vector and Link Matrix (P)

img_SimTrade_Black_Litterman_spreadsheet_1

Source: computation by the author.

6. We compute omega to determine the degree of uncertainty associated with the views. While Black-Litterman paper used a value of tau equal to 0.25, an important number of academics went for calculating the tau equal to one. For the sake of simplifying the model, consider tau to be equal to one. This input is obtained by multiplying the Linking matrix by the variance-covariance matrix and transposing the Linking matrix (P).

img_SimTrade_Black_Litterman_formulas_for_spreadsheet_4

τ= scalar
P = Linking matrix
∑ = Variance-covariance matrix

7. We integrate all the values computed previously in the Black-Litterman model. Table 4 depicts the results obtained via the Black-Litterman allocation model.

Table 4. Results of the Black-Litterman allocation

img_SimTrade_Black_Litterman_spreadsheet_4

Source: computation by the author.

We can see that the results converge slightly to those from CAPM. Additionally, we can see that the views are reflected in the Black-Litterman expected returns. As a result, we can determine whether or not each view is satisfied. Indeed, Apple outperforms Amazon and Microsoft, while Amazon outperforms Microsoft.

You can download an Excel file to help you construct a portfolio via the Black-Litterman allocation model.

Why should I be interested in this post?

Modern Portfolio Theory is at the heart of modern finance, shaping the modern investing landscape. MPT has become the cornerstone of current financial theory and practice. MPT’s thesis is that winning the market is difficult and requires diversification and taking higher-than-average risks.

MPT has been around for nearly sixty years and shows no signs of slowing down. His theoretical contributions paved the way for more portfolio theory study. But Markowitz’s portfolio theory is sensitive to and depends on further ‘probabilistic’ expansion. This paper expanded on Markowitz’s previous work by incorporating investor views into the asset allocation process.

Useful resources

Academic research

Black, F. and Litterman, R. 1990. Asset Allocation: Combining Investors Views with Market Equilibrium. Goldman Sachs Fixed Income Research working paper

Black, F. and Litterman, R. 1991. Global Asset Allocation with Equities, Bonds, and Currencies. Goldman Sachs Fixed Income Research working paper

Black, F. and Litterman, R. 1992. Global Portfolio Optimization.Financial Analysts Journal, 28-43.

Idzorek, T.M. 2002. A step-by-step guide to Black-Litterman model. Incorporating user-specified confidence levels. Working Paper, 2-11.

Markowitz, H., 1952. Portfolio Selection. The Journal of Finance, 7(1): 77-91.

About the author

The article was written in Mars 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Implementing Markowitz asset allocation model

March 20, 2022 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) explains how to implement the Markowitz asset allocation model. This model is used to determine optimal asset portfolios based on the risk-return trade-off.

This article follows the following structure: first, we introduce the Markowitz model. We then present the mathematical foundations of this model. We conclude with an explanation of the methodology to build the spreadsheet with the results obtained. You will find in this post an Excel spreadsheet which implements the Markowitz asset allocation model.

Introduction

Markowitz’s work is widely regarded as a pioneer work in financial economics and corporate finance due to its theoretical foundations and applicability in the financial sector. Harry Markowitz received the Nobel Prize in 1990 for his contributions to these disciplines, which he outlined in his 1952 article “Portfolio Selection” published in The Journal of Finance. His major work established the foundation for what is now commonly referred to as “Modern Portfolio Theory” (MPT).

To find the portfolio’s minimal variance, the Markowitz model uses a constrained optimization strategy. The goal of the Markowitz model is to take into account the expected return and volatility of the assets in the investable universe to provide an optimal weight vector that indicates the best allocation for a given level of expected return or the best allocation for a given level of volatility. The expected return, volatility (standard deviation of expected return), and the variance-covariance matrix to reflect the co-movement of each asset in the overall portfolio design are the major inputs for this portfolio allocation model for an n-asset portfolio. We’ll go over how to use this complex method to find the best portfolio weights in the next sections.

Mathematical foundations

The investment universe is composed of N assets characterized by their expected returns μ and variance-covariance matrix V. For a given level of expected return μ_P, the Markowitz model gives the composition of the optimal portfolio. The vector of weights of the optimal portfolio is given by the following formula:

With the following notations:

w_P = vector of asset weights of the portfolio
μ_P = desired level of expected return
e = identity vector
μ = vector of expected returns
V = variance-covariance matrix of returns
V^-1 = inverse of the variance-covariance matrix
t = transpose operation for vectors and matrices

A, B and C are intermediate parameters computed below:

img_SimTrade_implementing_Markowitz_2

The variance of the optimal portfolio is computed as follows

img_SimTrade_implementing_Markowitz_3

To calculate the standard deviation of the optimal portfolio, we take the square root of the variance.

Implementation of the Markowitz asset allocation model in practice

We generated a large time series to obtain useful results by downloading the equivalent of 23 years of market data from a data provider (in this example, Bloomberg). We generate the variance-covariance matrix after obtaining all necessary statistical data, which includes the expected return and volatility indicated by the standard deviation of the returns for each stock during the provided period. Table 1 depicts the expected return and volatility for each stock retained in this analysis.

Table 1. Asset characteristics of Apple, Amazon, Microsoft, Goldman Sachs, and Pfizer.
img_SimTrade_implementing_Markowitz_spreadsheet_1
Source: computation by the author.

We use the data analysis tool pack supplied in Excel to run a variance-covariance matrix for ease of computation (Table 2).

Table 2. Variance-covariance matrix of asset returns.
img_SimTrade_implementing_Markowitz_spreadsheet_4
Source: computation by the author.

We can start the optimization task by setting a desirable expected return after computing the expected return, volatility, and the variance-covariance matrix of expected return. With the data that is fed into the appropriate cells, the model will complete the optimization task. For a 10% desired expected return, we get the following results (Table 3).

Table 3. Asset weights for an optimal portfolio.
img_SimTrade_implementing_Markowitz_spreadsheet_2
Source: computation by the author.

To demonstrate the effect of diversification in the reduction of volatility, we can form a Markowitz efficient frontier by tilting the desired anticipated return with their relative volatility in a graph. The Markowitz efficient frontier is depicted in Figure 1 for various levels of expected return (Figure 1).

Figure 1. Markowitz efficient portfolio frontier.
img_SimTrade_implementing_Markowitz_spreadsheet_3
Source: computation by the author.

You can download the Excel file below to use the Markowitz portfolio allocation model.

Why should I be interested in this post?

Modern Portfolio Theory (MPT) is at the heart of modern finance, shaping the modern investing landscape. MPT has become the cornerstone of current financial theory and practice. MPT has been around for nearly sixty years and shows no signs of slowing down. His theoretical contributions paved the way for more portfolio theories. This post helps you to grasp the theoretical model and its implementation.

Useful resources

Academic research

Petters, A. O., and Dong, X. 2016. An Introduction to Mathematical Finance and Applications. Springer Undergraduate Texts in Mathematics and Technology.

Markowitz, H., 1952. Portfolio Selection. The Journal of Finance, 7(1): 77-91.

About the author

The article was written in Mars 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Monte Carlo simulation method

March 17, 2022 by Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole – Master in Management, 2019-2022) explains the Monte Carlo simulation method and its applications in finance.

Introduction

The Monte Carlo method was invented by John von Neumann (Hungarian-American mathematician and computer scientist) and Stanislaw Ulam (Polish mathematician) during World War II to improve decision making under uncertain conditions. It is named after the popular gambling destination Monte Carlo, located in Monaco and home to many famous casinos. This is because the random outcomes in the Monte Carlo modeling technique can be compared to games like roulette, dice and slot machines. In his autobiography, ‘Adventures of a Mathematician’, Ulam mentions that the method was named in honor of his uncle, who was a gambler.

How Monte Carlo simulation works

The main idea is to repeatedly run a large number of simulations of a random process for a variable of interest (such as an asset price in finance) covering a wide range of possible situations. The outcomes of this variables are drawn from a pre-specified probability distribution that is assumed to be known, including the analytical function and its parameters. Thus, Monte Carlo simulations inherently try to recreate the entire distribution of asset prices.

Example: Apple stock

Consider the Apple stock as our asset of interest for which we will generate stock prices according to the Monte Carlo simulation method.

The first step in the simulation is choosing a stochastic model for the behavior of our random variable (the Apple stock price in our case). A commonly used model is the geometric Brownian motion (GBM) model. The model assumes that future asset price changes are uncorrelated over time and the probability distribution function of the future price is a log-normal distribution. The movements in price in GBM process can be expressed as:

img_SimTrade_GBM_process

with dS being the change in asset price in continuous time dt. dW is the Wiener process (W_t+1 – W_t is a random variable from the normal distribution N(0, 1)). σ represents the price volatility considering the unexpected changes that can result from external effects (σ is assumed to be constant over time). μdt together represents the deterministic return within the time interval with μ representing the growth rate of the asset price or the ‘drift’.

Integrating dS/S over a finite interval, we get :

img_SimTrade_simulated_asset_price

Where ε is a random number generated from a normal distribution N(0,1).

This equation thus gives us the evolution of the asset price from a simulated model from day t-1 to day t.

We can now generate a simulation path for 100 days using the above formula.

The figure below shows five simulations for the price of the Apple stock over 100 days with Δt = 1 day. The initial price for Apple stock (i.e, price at t=0) is $146.52.

Figure 1. Simulated Apple stock prices according to the Monte Carlo simulation method.
img_SimTrade_Apple_MonteCarloSim
Source: computation by author.

Thus, we can observe that the prices obtained by just these five simulations range from $100 to over $220.

You can download below the Excel file for generating Monte Carlo Simulations for Apple stock.

Applications in finance

The Monte Carlo simulation method is widely used in finance for valuation and risk analysis purposes.

One popular application is option pricing. For option contracts with complicated features (such as Asian options) or those with a combination of assets as their underlying, Monte Carlo simulations help generate multiple potential payoff scenarios for the option which are averaged out to determine the option price at the issuance date.

The Monte Carlo method is also used to assess potential risks by generating simulations of market variables affecting portfolios such as asset returns, interest rates, macroeconomic factors, etc. over different time periods. These simulations are then assessed as required for risk modelling and to compute risk metrics such as the value at Risk (VaR) of a position.

Other applications include personal finance planning and corporate project finance where simulations are generated to construct stochastic financial models for sensitivity analysis and net present value (NPV) projections.

Useful resources

Hull, J.(2008) Risk Management and Financial Institutions, Fifth Edition, Chapter 7 – Valuation and Scenario Analysis.

About the author

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

Excel functions for mortgage

February 5, 2024March 14, 2022 by Liangyao TANG

Excel functions for mortgage

In this article, Liangyao TANG (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022) explains the functions in Excel that are useful to study a mortgage. Mastery of Excel is an essential skill nowadays in financial analysis and modelling tasks. Proficiency in using Excel formulas can help analysts quickly process the data and build the models more concisely.

Mortgage

A mortgage is the type of loan used in real estate, vehicles, and other types of property purchasing activities. There are two parties in the mortgage contract: the borrower and the lender. The contract sets the terms and conditions about the principal amount, interest rate, interest type, payment period, maturity, and collaterals. The borrower is contracted to pay back the lender in a series of payments that contains part of the principal as well as the interests before the maturity date.

The mortgage is also subject to different terms according to the bank’s offers and macroeconomic cycle. There are two types of interest rates: the fixed-rate loan and the floating (variable) rate loan, in which the interest rate is a pre-determined rate (at the beginning of the period) and post-determined rate (at the end of the period).

Example of repayment schedule.

In this post, I will use the following example: a mortgage of $300,000 for property purchasing. The mortgage specifies a 5% fixed annual interest rate for 30 years, and the borrower should pay back the loan on a monthly basis. We can use Excel functions to calculate the periodic (monthly) payment and its two components, the principal repaid and the interests paid for a given period. The calculations are shown in the sample Excel file that you can download below.

PMT

The “PMT” (Payment) Excel function calculates the periodic mortgage payment.

The periodic repayment for a fixed-rate mortgage includes a portion of repayment to the principal and an interest payment. Since the mortgage has a given maturity date, the payment is calculated on a regular basis, for example, every month. All repayments are of equal amount throughout the loan period.

The mathematical formula for the periodic mortgage payment is:

Formula for the periodic mortgage payment

With the following notations:

PMT: the payment
P: the principal value
r: the interest rate
N: the total number of periods

The repayment schedule is a table which gives the periodic payment, and the principal repaid and the interests paid for a given period. It can be a large table. For example, the repayment schedule of a loan with 30 year maturity and monthly payment has 180 lines. In formal terms, the payment schedule would be:

Repayment schedule of a mortgage

The repayment schedule shows the payment amount of each period, and the remaining principal balance after each payment. The ‘P’ represents the principal amount at the beginning of the mortgage, and the remaining principal is subjective to an (1+r) times interests at each period. The remaining principal is the principal balance from last period minus the current payment. Therefore for period 1, the remaining balance is equal to P(1+r), which is the principal with one year of interest, minus the PMT value, which is the payment of the current period.

The syntax for the Excel function to calculate the periodic payment is: PMT(rate, nper, pv, [fv], [type]).

With the following notations:

PMT: the periodic payment of the loan
Nper: the total number of periods of the loan
PV : the principal (present value) of the loan
[fv]: the future value of the loan (optional parameter). Default equal to 0
[type]: when payments are due (optional parameter). 0 = end of period, 1 = beginning of period. Default is 0

The function is used explicitly in the case of a fixed interest rate to compute the (constant) periodic payment.

The PMT function will calculate the loan’s payment at a given level of interest rate, the number of periods, and the total value of the loan for principals at the beginning of the period (principal + interest).

When using the function, it is essential to always align the time unit of the interest rate and the unit of Nper. If the mortgage is compounding on a monthly basis, the number of periods should be the total number of months in the amortization, and the rate should be the monthly interest rate, which equals the annual rate divided by 12. . In the above example, the interest should be paid in a monthly basis, therefore the number of period (Nper) is equal to 12 month x 30 year = 360 periods. Since the annual interest rate is 5%, the monthly interest rate would equal to 5% divide by 12, which is 0.42% per month.

IPMT and PPMT

To supplement on the information about the monthly payment, we can also use the function IPMT and PPMT to calculate the principal repaid and the interest rate paid for a given period.

IPMT

IPMT is the Excel function that calculates the interest portion in each of the periodic payment.

The syntax of the Excel function to calculate the interest portion of the periodic payment is: IPMT(rate, per, nper, pv, [fv], [type]).

With the following notations:

IPMT: interest payment
rate: interest rate
per: current period number
nper: total number of periods
pv: present value
[fv]: the future value of the loan (optional parameter). Default equal to 0
[type]: when payments are due (optional parameter). 0 = end of period, 1 = beginning of period. Default is 0

The rate refers to the periodic interest rate, while the “nper” refers to the total number of payment periods, and the “per” refers to the period for which we want to calculate the interest.

PPMT

PPMT is the Excel function that calculates the principal portion of a periodic payment.

The syntax of the Excel function to calculate the principal portion of a periodic payment is: PPMT(rate, per, nper, pv, [fv], [type]).

With the following notations:

PPMT: principal payment
rate: interest rate
per: current period number
nper: total number of periods
pv: present value
[fv]: the future value of the loan (optional parameter). Default equal to 0
[type]: when payments are due (optional parameter). 0 = end of period, 1 = beginning of period. Default is 0

Those of the results should be consistent with the amortization schedule shown above. The principal repayment should equal to PMT per period minus the interest rate paid (IPMT).

RATE

Contrarily, if the user is given the periodic payment amount information and wants to find out about the interest rate used for the calculation, he/she can use the RATE function in Excel.

The syntax of the Excel function to calculate the rate is: RATE(nper, pmt, pv, [fv], [type], [guess]).

With the following notations:

RATE: the interest rate
nper: the total number of payment periods
pmt: the constant periodic payment
pv: the principal amount
[fv]: the future value of the loan (optional parameter). Default equal to 0
[type]: when payments are due (optional parameter). 0 = end of period, 1 = beginning of period. Default is 0
[guess]: your guess on the rate (optional parameter). Default is 10%

The RATE Excel function will automatically calculate the interest rate per period. The time unit of the interest rate is aligned with the compounding period; for example, if the mortgage is compounding on a monthly basis, the RATE function also returns a monthly interest rate.

Example with an Excel file

The use of the Excel functions PMT, IPMT, PPMT and RATE is illustrated in the Excel file that you can download below.

Useful resources

Forbes What is a mortgage

Rocket mortgage Types of mortgage

Ramsey How Do Student Loans Work?

Prof. Longin’s website Echéancier d’un crédit (mortgage calculator in French)

About the author

The article was written in March 2022 by Liangyao TANG (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022).

Eurobonds

March 7, 2022 by Akshit GUPTA

Eurobonds

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains Eurobonds traded in financial markets.

Introduction

In financial markets, bonds are debt securities used by issuers to raise capital from investors. In return investors get an interest payment on the principal invested over the life of the bond. The bonds can be issued by governments, municipalities, financial institutions, and companies. The duration of the bonds can cover different time periods.

Eurobonds are a special kind of bonds issued by companies or governments to raise capital from financial markets. These bonds are denominated in a currency different from the currency of the country where they originated. The Eurobonds help issuers to raise capital in a foreign currency and at a lower cost. Let’s take the example of an American company which would like to issue debt in euros to finance its operations in Europe. If it borrows in European markets, it will get a higher interest rate as it is less well known in the foreign markets that in the domestic market. With Eurobonds, the company can benefit from the same level of interest rates as for its domestic bonds, thereby lowering its cost of capital.

These instruments have a medium to long term maturity and are highly liquid in the market. They are traded over the counter (OTC) and the market for Eurobonds is made up of several financial institutions, issuers, investors, government bodies, and brokers. Many brokerages across the world provide trading platforms facilities to investors and borrowers for trading in different kinds of Eurobonds.

Characteristics of Eurobonds

Eurobonds are unsecured instruments and investors demand high yields on these instruments based on the credit ratings of the issuer. The issuer can issue Eurobonds in a foreign currency and a foreign land based on their capital needs. The name of a Eurobond carries the name of the currency in which they are dominated. For example, a French company willing to do business in the United States, can issue a Eurobond in the UK financial market denominated in US dollars which will be called as euro-dollar bond.

A Eurobond should not be confused with a foreign bond issued by an issuer in the foreign market denominated in the local currency of the investor. A Eurobond can be issued in a foreign country and can be denominated in a currency different from the local currency of the issuer. For example, a French company willing to invest in Japan can issue a Euro-yen bond in the US markets denominated in the local currency of Japan.

These bonds are traded electronically on different platforms and can have maturities ranging from 5 years to 30 years. The bonds can have fixed or floating interest rates with semi-annual or annual payments. These bonds have a relatively small face value making it attractive even to small investors.

Benefits of Eurobonds

Eurobonds can serve different benefits to issuers and investors.

Major advantage of Eurobonds for the issuers

Access to capital at lower rates – Companies can choose countries with lower interest rates to issue Eurobonds, thereby avoiding interest rate risks
Access to different bond maturities – As Eurobonds can have maturities ranging from 5 years to 30 years, companies can have a wide range of maturities to choose from depending on their requirements
Access to international markets – By issuing Eurobonds denominated in a different currency, companies can access different markets with more ease with a wide investor base.

Major advantage of Eurobonds for the investors

Access to international markets – By buying Eurobonds, investors can gain easy access to international markets thereby diversifying their fixed income portfolios.
Access to different bond maturities – As Eurobonds can have maturities ranging from 5 years to 30 years, borrowers can have a wide range of maturities to choose from depending on their investment profile.
High liquidity – As the market size for Eurobonds is very large, investors can enjoy higher liquidity and can exit their positions as per their needs.

Example

The figure below gives an example of Eurobonds issued by the Federal Republic of Nigeria.

Characteristics of the Eurobonds issuance.

Example of Eurobond issuance

Source: FMDQ.

▶ Akshit GUPTA Green bonds

▶ Jayati WALIA Fixed-income products

▶ Jayati WALIA Credit Risk

Useful resources

International Capital Market Association (ICMA) History of Eurobonds

About the author

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

Green bonds

March 7, 2022 by Akshit GUPTA

Green bonds

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains Green bonds traded in financial markets.

Introduction

A green bond is a fixed-income product that works like a conventional bond, except that the money invested in them is used exclusively to finance green projects that support environment preservation, sustainability and reduction of climate change (low-carbon economy). Green projects can include renewable energy such as solar and wind power, energy-efficient infrastructure, clean transportation and waste management and recycling.

In 2007, the European Investment Bank (EIB) issued the world’s first ever green bond under the name Climate Awareness Bond (CAB), which focused on renewable energy and energy efficiency projects. This was followed by the World Bank issuing its own green bonds, until 2012 when the first corporate green bond was issued. Since then the market for green bonds has grown tremendously creating all-time highs with every passing year. The greatest issuer of green bonds in 2020 was the French government with a combined issue size of nearly 13 billion USD.

Types of green bonds

Green bonds can be classified as the following: green “use of proceeds” bonds, green “use of proceeds revenue” bonds, green project bonds, and securitized green bonds.

Green “use of proceeds” bonds

The funds raised by these green bonds are invested in green projects but they are backed/secured by issuer’s assets. Hence, their ratings are the same as other debt instruments by the issuer. For instance, the Climate Awareness Bond issued by EIB is one such green bond.

Green “use of proceeds revenue” bonds

The funds raised are assigned to eligible green projects. However, bondholders have recourse to a specified revenue stream from the issuers which may or may not be related to the eligible green projects.

Green project bonds

Proceeds from green project bonds are used for specific projects, investors having a direct exposure to the green project itself.

Securitized green bonds

These bonds are backed by a large group of green projects or assets.

Benefits of investing in green bonds for issuers

Lower cost of capital

Green bonds help environment focused companies to raise large amount of initial and working capital at lower costs to fund their ESG activities which require heavy initial investments. For example, companies can raise capital to fund a project focused towards generating renewable energy.

Brand value

Companies issuing green bonds enjoy an increase in the brand value and favourable reputation amongst the investors, as they are becoming more inclined towards sustainability.

Benefits of investing in green bonds for investors

Diversification

Over the years, the financial markets have seen an increased demand for green bonds amongst investors. Various factors have contributed to this increase including portfolio diversification, focus on socially responsible investments opportunities, fulfilment of ESG mandates of the financial institutions, etc.

Tax benefits

Investors can enjoy tax incentives on the investments made in green bonds. The interest incomes generated on these bonds are generally tax exempt or provide tax reductions to the investors. Thus, the issuers also benefit from lower interest rates due to the tax benefits.

Increase in liquidity

As the market size for green bonds is increasing, investors can enjoy higher liquidity and can exit their positions as per their needs.

Examples

The image below shows the listing of green bonds on Euronext.

Listing of green bonds on Euronext.

Listing of green bonds

Source: Euronext.

Useful resources

Corporate Finance Institute Eurobonds

ICMA History of Eurobonds

Euronext Listing of green bonds

About the author

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

The regulation of cryptocurrencies: what are we talking about?

March 5, 2022 by Hugo MEYER

The regulation of cryptocurrencies: what are we talking about?

In this article, Hugo MEYER (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2020-2021) presents the regulation of cryptocurrencies.

Introduction

The first cryptocurrency – Bitcoin – launched in 2008 by Satoshi Nakamoto had for ambition to “break the rules and change the world”.

Thirteen years later, cryptocurrency represents a 2$ trillion market, with an increasing institutional presence, from crypto hedge funds to large banks. Behind this bewildering evolution, public authorities lagged behind, slowly empowering with feverish regulation actions.

The lack of regulation in this burgeoning area has created an opening for boundless fraud and money laundering, forcing some countries to get to grips with the cryptocurrency’s pitfalls.

What are cryptocurrencies?

Cryptocurrencies are at the edge of revolutionizing the way we’ve been trading since thousands of years. By definition, a cryptocurrency is an encrypted, digital, and decentralized medium of exchange that allows two parties that could be located everywhere on the globe to transfer funds directly, without relying on any trusted third party.

Instead of being secured by public institutions and/or companies, these transfers are carried out on the Blockchain which is a “digital database or ledger containing information that can be simultaneously used and shared within a large decentralized, publicly accessible network”.

Example: To make it simple, let’s say that A wants to send money to B. This transaction is included in a ‘block’. This block is broadcasted to every member in the network, and then validated or not by them. Once validated, the block is added to the chain, triggering the money to move from A to B.

Figure 1. Process of transaction with the blockchain
Schema of the blockchain
Source: Institut des actuaires.

This distributed network provides an indelible and transparent record of transactions as the chain cannot be counterfeited. If someone tried to change any information contained in one block, the different parties of the network would not approve the transaction as they could check the whole history of the blockchain and compare it to the new one.

Thus, many cryptocurrencies such as Bitcoin, Ethereum and Monero rely on public blockchains to allow transactions in complete security and transparency.

“I do think Bitcoin is the first money that has the potential to do something like change the world” – Peter THIEL.

What is the regulation about?

By definition, regulation tally with the act of controlling something, or enacting an official rule. What does it imply for cryptocurrencies?

A cryptocurrency is entirely defined by its creator, that must foremost determine its characteristics. This creation process is divided into three steps:

Pick or create its blockchain platform
Choose a consensus algorithm
Design the blockchain architecture.

His creator defines the rules around it, while the ecosystem built accordingly to these rules regulate it and make it functional. Once the crypto is launched, it is impossible to modify its architecture and the rules. In this way, a cryptocurrency cannot be regulated, even by his founder. Thus, authorities do not have any grip with cryptocurrencies in themselves. They are auto-regulated by their initial algorithms, and nothing else.

Thereby, what are we talking about when dealing with the regulation of cryptocurrency?

Cryptocurrencies are mainly exchanged through platforms called “exchanges” such as Coinbase, Binance or eToroX. The first existing regulation framework is the accessibility to these platforms. For most of them, requirements like providing its identity are requested, following the Know Your Customer (KYC) compliance.

Secondly, the regulatory framework for these platforms depends on where they are based. Each country has a different approach of cryptocurrency, meaning that the regulation can be different in any of them.

For example, cryptocurrency exchanges are legal in the United States and fall under the regulatory scope of the Bank Secrecy Act (BSA). Therefore, exchanges service providers must register with FinCEN, implement an anti-money laundering (AML) and combating the financing of terrorism (CFT) program, maintain appropriate records, and submit reports to the authorities. It does not mean their trading activities are regulated.

These requirements permit exchanges to operate as licensed Money Service Businesses (MSBs), leading regulators to focus on anti-money laundering (AML) and due diligence measures, but not trading (and all the aspects of market manipulation).

Given the lack of significant regulatory oversight of actual trading activity, it is not surprising to see many cryptocurrency exchanges carry out questionable activities, such as offering leverage to their clients and wash trading, during times of market instability. But these are not the only problems raised by the lack of regulation.

Why should exchanges be more regulated?

The blockchain is a recent technology, understood by a few. As regulation always comes after innovation, the crypto market has been sidelined by public authorities for many years. The question of regulating it has recently appeared in response to the many downsides incurred to cryptos.

Customer protection

When investing in cryptocurrencies, the customer is lacking protection. An investor could be facing fake websites, hacking, and platform bankrupts without any legal recourse to recover his money. These situations could never happen in a traditional investment as it is institutionally regulated. To become more secure, exchanges must follow the example of itBit, an US-based exchange oversighted by the New York Department of Financial Services (DFS) and registered as a bank.

Illegal Financial flows & crime

Cryptocurrency can be used for illicit transactions and for laundering criminal proceeds that may or may not have started as cryptocurrency. These illicit transactions occur on the dark web, including the purchase/sale of illicit drugs and debit and credit card information. According to a study published in 2019 by Oxford Academics, 76$ billion of illegal activity per year involve bitcoin, which represents half of total Bitcoin transactions.

Cryptos can also be used for ransomware attacks, like the one that shut down the Colonial Pipeline in May 2021. This attack was one of many others high-profile instances of hackers seeking Bitcoin ransoms, that should tend to multiply in the upcoming years.

Price stability

Blockchain technology has increasingly become a speculative tool for investing and achieving high returns in the short term, leading to market volatility. These fast and unpredictable price changes also have a direct impact on the velocity, where more and more people hold their cryptocurrencies instead of selling or using them.

Plus, the volatility of cryptos prices may let the market suffer from illiquidity. The notion of liquidity for a financial asset refers to the ease with which an asset can be bought or sold (without a strong price impact, e.g., limit implicit transaction costs).

Tax evasion

One of the first problem that arise from tax evasion is taxation. Many countries have their own regulatory framework, either taxing cryptos as an asset (Israel), a financial asset (Bulgaria), or even a foreign currency (Switzerland). Once the taxation rule found out, authorities will tackle another problem: The investors resistance to report their gains.

Taking the example of USA, authorities ask filers on their income tax forms – like any form of income – whether they received or made any transactions with cryptocurrency. However, third-party reporting in the sector is scarce; making it even more difficult to attribute gains to one natural person.

Thus, how can regulation allow the crypto market to take over these pitfalls?

Worldwide market regulation

“Bitcoin is not unregulated. It is regulated by algorithm instead of being regulated by government bureaucracies” – Andreas Antonopoulos

Despite being a global phenomenon, every country does not hang up with the same type of regulation.

First, some countries have expanded their laws on money laundering, counterterrorism, and organized crimes to include cryptocurrency markets, and require banks and other financial institutions that facilitate such markets to conduct all the due diligence requirements imposed under such laws. For instance, Australia and Canada recently enacted laws to bring cryptocurrency transactions and institutions that facilitate them under the ambit of money laundering and counter-terrorist financing laws.

Some jurisdictions have gone even further and imposed restrictions on investments in cryptocurrencies. Some countries – Algeria, Bolivia, Morocco, Vietnam – explicitly ban any and all activities involving cryptocurrencies. Qatar and Bahrain consider that their citizens are forbidden from engaging in any kind of activities involving cryptocurrencies locally but allow them to do so outside their borders.

There are also countries that, while not banning their citizens from investing in cryptocurrencies, impose indirect restrictions by hindering transactions involving cryptocurrencies, such as China, Iran, or Thailand.

A limited number of countries regulate initial coin offerings (ICOs), which use cryptocurrencies as a mechanism to raise funds. Of the jurisdictions that address ICOs, some (mainly China, Macau, and Pakistan) ban them altogether, while most tend to focus on regulating them.

When it comes to taxation, the challenge appears to be how to categorize cryptocurrencies and the specific activities involving them. This matters primarily because whether gains are categorized as income or capital gains invariably determines the applicable tax bracket. For instance, in Israel, cryptos gains are taxed as assets, while there are subject to income tax in Spain and Argentina.

Advocates of digital currencies say that accepting cryptocurrencies is much more relevant than rejecting it. For instance, El Salvador became the 7th of September 2021 the first country in the world to make Bitcoin a legal tender. One day after, the “Regulation of the Bitcoin Law” entered into force, that establishes standards of conducts supervised by the Superintendency of the Financial System (SSF), the equivalent of the Securities Exchange Commission (SEC) in the United States or the Autorité des Marchés Financiers (AMF) in France. This regulation will bring much more protection to Bitcoin users, while setting up numerous programs in cybersecurity, anti-money laundering, and tax evasion.

Conclusion

As Bitcoin – and other cryptocurrencies – become more and more popular, regulation will have to step up altogether, despite asking extensive questions on its bounding by International Authorities.

Economic threat, exacerbated risks and investigation complications are all issues that can be counteracted by regulation laws on the crypto market. Central banks will play a major role in this governance, going along with their traditional missions such as ensuring price stability and a proper operating financial system.

Nevertheless, regulation may lead to underestimated consequences. As it goes on, crypto investment will progressively become “mainstream” and dismiss the first and most powerful investors. This trend might also push innovators to take a step back from it, thus decreasing the number of cryptocurrencies created and newly innovative blockchains.

Useful resources

Academic research

Sean, F. Jonathan, R K. Talis, P. 2019. Sex, Drugs, and Bitcoin: How much illegal activity is financed through cryptocurrencies?” The Review of Financial Studies. Vol. 32, p. 1798-1853.

Business Analysis

L, S. 2016. Who is Satoshi Nakamoto, The Economist explains.

Thiemann, A. 2021. Cryptocurrencies: An empirical View from a Tax Perspective, JRC Working Papers on Taxation and Structural Reforms. No 12/2021, European Commission, Joint Research Centre, Seville, JRC126109.

Global Legal Research Directorate. 2018. Regulation of Cryptocurrency Around the World. LL File No. 2018-016036 LRA-D-PUB-002438.

Ryan, H. 2021. U.S. Officials send mixed messages on crypto regulation. Here’s what it all means for investors. NextAdvisor.

American Overseas, 2021. Washington Monthly: Catching Bitcoin tax evaders.

Alexis, G. 2021. Crypto doesn’t have to enable tax cheats Bloomberg Opinion.

Douma, S. 2016. Bitcoin: The pros and cons of regulation. s1453297.

About the author

The article was written in March 2022 by Hugo MEYER (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2020-2021).

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).

Introduction

Computation methods for implied volatility

Example

Volatility smile

Volatility surface

CBOE Volatility Index

Related posts on the SimTrade blog

Useful resources

Academic articles

Business

About the author

A quick presentation of the M&A field…

What does M&A consist in?

What does an analyst work on?

Why does M&A jobs appeal so much to students?

What are the main exits for M&A?

Related posts on the SimTrade blog

Useful resources

About the author

Warren Buffet and his basket of eggs

“Do not put all eggs in one basket”

Analysis

“Put all your eggs in one basket and watch that basket”

Why should I be interested in this post?

Related posts on the SimTrade blog

Useful resources

About the author

Big data in the financial sector

Big Data in The Financial Sector

Fraud prevention

Anomaly detection

Customer analytics

Algorithmic trading

Conclusion

Characteristics of Big Data

Volume

Variety

Velocity

Value

Challenges Solved by Big Data in the Financial Industry

Data Quality

Data Silos

Robo-Advisory

Why should I be interested in this post?

Related posts on the SimTrade blog

Useful resources

About the author

Momentum Trading Strategy

Introduction

Implementation

Momentum trading indicators

Relative Strength Index (RSI)

Moving-average convergence-divergence (MACD)

Bollinger bands

Market conditions

Risks associated with momentum trading

Link with market efficiency

Related posts

Useful resources

Academic research

Business analysis

About the author

Introduction

Computation of returns

Arithmetic (percentage) returns

Logarithmic returns

Link between arithmetic and logarithmic returns

Components of total returns

Price changes and returns

Internal rate of return (IRR)

Ex ante and ex post returns

Related posts on the SimTrade blog

About the author

Supply and Demand

The basic concepts

Special cases

Useful resources

About the author

Risk Aversion

Risk aversion and investor’s characteristics