Value at Risk

August 24, 2021 by Jayati WALIA

Value at Risk

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents value at risk.

Introduction

Risk Management is a fundamental pillar of any financial institution to safeguard the investments and hedge against potential losses. The key factor that forms the backbone for any risk management strategy is the measure of those potential losses that an institution is exposed to for any investment. Various risk measures are used for this purpose and Value at Risk (VaR) is the most commonly used risk measure to quantify the level of risk and implement risk management.

VaR is typically defined as the maximum loss which should not be exceeded during a specific time period with a given probability level (or ‘confidence level’). Investments banks, commercial banks and other financial institutions extensively use VaR to determine the level of risk exposure of their investment and calculate the extent of potential losses. Thus, VaR attempts to measure the risk of unexpected changes in prices (or return rates) within a given period.

Mathematically, the VaR corresponds to the quantile of the distribution of returns on the investment.

VaR was not widely used prior to the mid 1990s, although its origin lies further back in time. In the aftermath of events involving the use of derivatives and leverage resulting in disastrous losses in the 1990s (like the failure of Barings bank), financial institutions looked for better comprehensive risk measures that could be implemented. In the last decade, VaR has become the standard measure of risk exposure in financial service firms and has even begun to find acceptance in non-financial service firms.

Computational methods

The three key elements of VaR are the specified level of loss, a fixed period of time over which risk is assessed, and a confidence interval which is essentially the probability of the occurrence of loss-causing event. The VaR can be computed for an individual asset, a portfolio of assets or for the entire financial institution. We detail below the methods used to compute the VaR.

Parametric methods

The most usual parametric method is the variance-covariance method based on the normal distribution.

In this method it is assumed that the price returns for any given asset in the position (and then the position itself) follow a normal distribution. Using the variance-covariance matrix of asset returns and the weights of the assets in the position, we can compute the standard deviation of the position returns denoted as σ. The VaR of the position can then simply computed as a function of the standard deviation and the desired probability level.

VaR Formula

Wherein, p represents the probability used to compute the VaR. For instance, if p is equal to 95%, then the VaR corresponds to the 5% quantile of the distribution of returns. We interpret the VaR as a measure of the loss we observe in 5 out of every 100 trading periods. N^-1(x) is the inverse of the cumulative normal distribution function of the confidence level x.

Figure 1. VaR computed with the normal distribution.

VaR computed with the normal distribution

For a portfolio with several assets, the standard deviation is computed using the variance-covariance matrix. The expected return on a portfolio of assets is the market-weighted average of the expected returns on the individual assets in the portfolio. For instance, if a portfolio P contains assets A and B with weights w_A and w_B respectively, the variance of portfolio P’s returns would be:

Variance of portfolio

In the variance-covariance method, the volatility can be computed as the unconditional standard deviation of returns or can be calculated using more sophisticated models to consider the time-varying properties of volatility (like a simple moving average (SMA) or an exponentially weighted moving average (EWMA)).

The historical distribution

In this method, the historical data of past returns (for say 1,000 daily returns or 4 years of data) are used to build an historical distribution. VaR corresponds to the (1-p) quantile of the historical distribution of returns.
This methodology is based on the approach that the pattern of historical returns is indicative of future returns. VaR is estimated directly from data without estimating any other parameters hence, it is a non-parametric method.

Figure 2. VaR computed with the historical distribution.

VaR computed with the historical distribution

Monte Carlo Simulations

This method involves developing a model for generating future price returns and running multiple hypothetical trials through the model. The Monte Carlo simulation is the algorithm through which trials are generated randomly. The computation of VaR is similar to that in historical simulations. The difference only lies in the generation of future return which in case of the historical method is based on empirical data while it is based on simulated data in case of the Monte Carlo method.

The Monte Carlo simulation method is used for complex positions like derivatives where different risk factors (price, volatility, interest rate, dividends, etc.) must be considered.

Limitations of VaR

VaR doesn’t measure worst-case loss

VaR gives a percentage of loss that can be faced in a given confidence level, but it does not tell us about the amount of loss that can be incurred beyond the confidence level.

VaR is not additive

The combined VaR of two different portfolios may be higher than the sum of their individual VaRs.

VaR is only as good as its assumptions and input parameters

In VaR calculations especially parametric methods, unrealistic or inaccurate inputs can give misleading results for VaR. For instance, using the variance-covariance VaR method by assuming normal distribution of returns for assets and portfolios with non-normal skewness.

Different methods give different results

There are many approaches that have been defined over the years to estimate VaR. However, it essential to be careful in choosing the methodology keeping in mind the situation and characteristics of the portfolio or asset into consideration as different methods may be more accurate for specific scenarios.

Useful Resources

Academic research articles

Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath, (1999) Coherent Measures of Risk, Mathematical Finance, 9, 203-228.

Jorion P. (1997) “Value at Risk: The New Benchmark for Controlling Market Risk,” Chicago: The McGraw-Hill Company.

Longin F. (2000) From VaR to stress testing: the extreme value approach Journal of Banking and Finance, N°24, pp 1097-1130.

Longin F. (2016) Extreme events in finance: a handbook of extreme value theory and its applications Wiley Editions.

Longin F. (2001) Beyond the VaR Journal of Derivatives, 8, 36-48.

About the author

The article was written in September 2021 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Plain Vanilla Options

June 4, 2024August 23, 2021 by Jayati WALIA

Plain Vanilla Options

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents plain vanilla options.

Introduction

An option contract is a financial derivative that gives its holder the right (but not the obligation) to trade an underlying asset at a price and a date set in advance.

In finance, plain vanilla refers to the most basic version of any financial instrument with standard features. Thus, a plain vanilla option simply refers to a contract that provides the option to buy or sell an underlying stock (or any financial asset) at a fixed price (known as the exercise/strike price) at an expiration date in the future. The expiration date (or maturity) of the option is the date when the holder can exercise her option if she wants.

In the US, options were first traded on an exchange on 26th April 1973. The Chicago Board Options Exchange (CBOE) was the first to create standardized, listed options. Today, there are over 50 exchanges worldwide that trade options.

When an option is bought, its holder pays a fixed amount to the option writer as the cost for the flexibility of trading that the option provides. This cost, which is essentially the value of an option (and the margin taken by the issuer), is known as the premium. The premium depends on the characteristics of the option like the strike price and the maturity, and on market data like the price of the underlying asset and especially its volatility. Many different underlying assets can be traded through options including stocks, bonds, commodities, foreign currencies.

Types of options

Vanilla options are of two types: call and put.

Call options

The holder of a call option has the right to buy a particular asset at a strike price K at maturity T. If the asset price at maturity denoted by S_T is higher than K, then it is beneficial for the call option holder to exercise his option at time T as the price set in the call option contract K is lower than the market price S_T. If the asset price at maturity S_T is lower than K, then it is not beneficial for the call option holder to exercise his option at time T as the price set in the call option contract K is higher than the market price S_T; he is then better off to buy the asset on the market at price S_T than at price K.

For example, consider a call option on BNP Paribas stock with a strike price of €50 and a maturity date March 31^st. The holder of this call option thus has the right but not the obligation to buy one BNP Paribas stock for €50 at maturity. He will exercise his option on March 31^st if and only if the stock price is higher than €50.

The equation below gives the pay-off function of a call option that is the value of the call option at maturity T denoted by C_T as a function of the price of the underlying asset S_T.

Payoff formula for a call option

Figure 1 gives a graphical representation of the pay-off function of a call option that is the value of the call option at maturity T as a function of the price of the underlying asset at maturity T, S_T, for a given strike price (equal to €50 in the figure).

Figure 1. Pay-off function of a call option

Payoff for a call option

Put options

Similarly, the holder of a put option has the right to sell a particular asset at a strike price K at maturity T. If the asset price at maturity denoted by S_T is higher than K, then it is beneficial for the put option holder not to exercise his option at time T as the price set in the put option contract K is lower than the market price S_T; he is then better off to sell the asset on the market at price S_T than at price K. If the asset price at maturity S_T is lower than K, then it is beneficial for the put option holder to exercise his option at time T as the price set in the put option contract K is higher than the market price S_T.

For example, consider a put option on BNP Paribas stock with a strike price of €50 and a maturity date March 31^st. The holder of this put option thus has the right but not the obligation to sell one BNP Paribas stock for €50 at maturity. He will exercise his put option on March 31^st if and only if the stock price is lower than €50.

The equation below gives the pay-off function of a put option that is the value of the put option at maturity T denoted by P_T as a function of the price of the underlying asset S_T.

Payoff formula for a put option

Figure 2 gives a graphical representation of the pay-off function of a put option that is the value of the put option at maturity T as a function of the price of the underlying asset S_T for a given strike price (equal to €50 in the figure).

Figure 2. Pay-off function of a put option

Payoff for a put option

Types of exercise

Options can be categorized based on their exercise restrictions.

American options

American options have the most flexible arrangement allowing holders to exercise their options at any time prior to the expiration date. They are widely traded over listed exchanges.

European options

European options provide less flexibility and allow holders to exercise options on only one specific date, which is the expiration date. They thus have a lower value compared to American options and are generally traded OTC.

Bermudan options

There are also Bermudan options that allow exercise of options on a set of specific dates before the expiration and thus provide holders a level of flexibility midway between American and European Options.

Moneyness

Options can also be characterized by their “moneyness” which compares the current price of the underlying asset to the option strike.

In-the-money options

An option with a positive intrinsic value is said to be ‘in the money’. This is the case for a call option if the current market price of the asset is higher than the strike price, and similarly for a put option if the current market price of the asset is lower than the strike price.

Out-of-the-money options

An option with a zero intrinsic value is said to be ‘out of the money’. This is the case for a call option if the current market price of the asset is lower than the strike price, and similarly for a put option if the current market price of the asset is higher than the strike price.

At-the-money options

An option with a strike price close or equal to the current market price is said to be ‘at the money’.

Option writers

The above discussion mainly revolves around option purchasers. However, there is also someone who is liable to sell (for a call) or buy (for a put) the underlying security whenever any holder exercises an option. The writer of an option is the person who is obligated to buy/sell the underlying in case of a call/put exercise. As a counterpart, the writer also receives the option premium from the holder.

The best-case scenario for a writer would be that the option is not exercised by its holder as the option remains out of the money (the writer earning the premium without being obliged to pay the cash flow at maturity). However, option writers are exposed to downside risks especially if the options they write are not covered i.e., holding a long or short position already in the underlying security depending on the option written.

Benefits

For traders with strong market views looking to leverage benefits from small to medium-term fluctuations in market price, buying options is an efficient means to offset their risk exposure. The buyer only risks a small amount of investment, and the downside is only limited to the initial premium whereas the upside is a high payoff if the speculation is in her/his favor. The traders can also take up multiple positions in different assets through options and leverage trade opportunities with profitable positions covering more than the hedging costs.

Option Trading

Most vanilla options are traded through exchanges that make it convenient to match buyers with sellers and vice versa. Trading of standardized contracts also promotes liquidity of the instruments in the market. Vanilla options generally come in series of standardized strike prices and expiration dates. For instance, for an option contract on an Apple Inc. stock (AAPL) expiring on 20^th August 2021, the offered strike prices are $115, $120, $125, $130 and so on. Similarly, the expiration dates for listed stock options is generally the third Friday of the month in which the contract expires. If the Friday falls on a holiday, the expiration date becomes Thursday immediately before the third Friday.

Option pricing

The value an option is known at maturity as it is given by the contract. But what is the value of an option at the time of its issuance or at a time before maturity? Many mathematical models have been developed to answer this question. The most famous model is the Black-Scholes-Merton option pricing model. It uses a Brownian motion to model the behavior of stock market prices.

Use of options

Hedging

Options are commonly used in hedging. For instance, you can purchase an option on a stock to limit your losses to say 15% of your position, should the stock decline more than that during the option period.

Speculation

If one has a strong view about the potential market direction of an underlying security, one can make great returns on exploiting options, provided the view was right. This is essentially speculation in option trading. For instance, if you have a bullish opinion regarding a stock, you can purchase a call option on it that will allow you to purchase the stock at the strike price that will be lower than the future price (hopefully!). Thus, if you are right, you could exercise the option and your payoff would be the price difference between the stock price and the strike price. If you are wrong, you lose out on the premium you paid for the option.

Volatility

The volatility of the underlying asset affects positively option prices: stocks with higher volatility have more expensive option contracts that those with low volatility. In fact, the implied volatility (IV) of an option is that value of the volatility of the underlying instrument for which an option pricing model (such as the Black-Scholes-Merton model) will return a theoretical value equal to the current market price of that option. Hence, when the implied volatility increases, the price of options increases as well, assuming all other factors remain constant. When the implied volatility increases after a trade has been placed, it is good news for the option owner and, conversely bad news for the seller. Inversely, when the implied volatility decreases after a trade has been placed, it is bad news for the option owner and, conversely good news for the seller.

Note that the implied volatility tends to depend on the strike price and maturity date of the options for a given underlying asset. Once the implied volatility for the at-the-money contracts is determined in any given expiration month, market makers use pricing models and volatility skews to calculate implied volatility at other strike prices that are less heavily traded. So, every option has an associated volatility and risk profiles can vary drastically among options. Traders may at times balance out the risk of volatility by hedging one option with another.

Thus, it is essential to interpret and analyze risks before venturing into option trading. There are also many strategies that can be applied to vanilla options in order to benefit better and limit risk such as long and short calls/puts, bull and bear spreads, straddles and strangles, butterflies, condors among many.

Useful Resources

Nasdaq Historical data for Apple stock

AVATRADE What are vanilla options

TheStreet Options Trading

About the author

The article was written in August 2021 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Derivatives Market

August 23, 2021 by Jayati WALIA

Derivatives Market

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents an overview of derivatives market.

Introduction

A financial market refers to a marketplace where various kinds of financial securities such as stocks, bonds, commodities, etc. are traded. The term ‘market’ can also refer to exchanges that are legal organizations that facilitate the trade of financial securities between buyers and sellers. In any case, these markets are categorized based of the type of financial securities that are traded through them. One such financial market is the Derivatives Market.

Derivatives market thus refers to the financial marketplace where derivative instruments such as futures, forwards and options contracts are traded between counterparties.

It was during the 1980s and 1990s that the financial markets saw a major growth in the trade of derivatives. A derivative is a financial instrument whose value is derived from the value of an underlying asset such as stocks, bonds, currencies, commodities, interest rates and/or different market indices. These underlying assets have fluctuating prices and returns, and derivatives provides a means to investors to reduce the risk exposure and leverage profits on these assets. Thus, derivatives are an essential class of financial instruments and central to the modern financial markets providing not just economic benefits but also resilience against risks. The most common derivatives include futures, forwards, options and swap contracts.

As per the European Securities and Markets Authority (ESMA), derivatives market has grown impressively (around 24 percent per year in the last decade) into a truly global market with over €680 trillion of notional amount outstanding. The interest rate derivatives (IRDs) accounted for 82% of the total notional amount outstanding followed by currency derivatives at 11%.

Main types of derivative contracts

Derivatives derive their value from an underlying asset, or simply an ‘underlying’. There is a wide range of financial instruments that can be an underlying for a derivative such as equities or equity index, fixed-income instruments, foreign currencies, commodities, and even other securities. And thus, depending on the underlying, derivative contracts can derive their values from corresponding equity prices, interest rates, foreign exchange rates, prices of commodities and probable credit events. The most common types of derivative contracts are elucidated below:

Forwards and Futures

Forward and futures contracts share a similar feature: they are an agreement between two parties to buy or sell a specified quantity of an underlying asset at a specified price (or ‘exercise price’) on a predetermined date in the future (or ‘expiration date’). While forwards are customized contracts i.e., they can be tailor-made according to the asset being traded, expiry date and price, and traded Over-the-Counter (OTC), futures are standardized contracts traded on centralized exchanges. The party that buys the underlying is said to be taking a long position while the party that sells the asset takes a short position and both parties are obligated to fulfil their part of the contract.

Options

An option contract is a financial derivative that gives its holder the right (but not the obligation) to trade an underlying asset at a price set in advance irrespective of the market price at maturity. When an option is bought, its holder pays a fixed amount to the option writer as cost for this flexibility of trading that the option provides, known as the premium. Options can be of the types: call (right to buy) or put (right to sell).

Swaps

Swaps are agreements between two counterparties to exchange a series of cash payments for a stated period of time. The periodic payments charged can be based on fixed or floating interest rates, depending on contract terms decided by the counterparties. The calculation of these payments is based on an agreed-upon amount, called the notional principal amount (or just notional).

Exchange-traded vs Over-the-counter Derivatives Market

Exchange-traded derivatives markets

Exchange-traded derivatives markets are standardized markets for derivatives trading and follows rules set by the exchange. For instance, the exchange sets the expiry date of the derivatives, the lot-size, underlying securities on which derivatives can be created, settlement process etc. The exchange also performs the clearing and settlement of trades and provide credit guarantee by acting as a counterparty for every trade of derivatives. Thus, exchanges provide a transparent and systematic course of action for any derivatives trade.

Over-the-counter markets

Over-the-counter (also known as “OTC”) derivatives markets on the other hand, provide a lesser degree of regulations. They were almost entirely unregulated before the financial crisis of 2007-2008 (also a time when derivatives markets were criticized, and the blame was placed on Credit Default Swaps). OTCs are customized markets and run by dealers who hedge risks by indulging in derivatives trading.

Types of market participants

The participants in the derivative markets can be categorized into different groups namely,

Hedgers

Hedging is a risk-neutralizing strategy when an investor seeks to protect a current or anticipated position in the market by limiting their risk exposure. They can do so by taking up an offset or counter position through derivative contracts. Parties such as individuals or companies who perform hedging are called Hedgers. The hedger thus aims to eliminate volatility against fluctuating prices of underlying securities and protect herself/himself from any downsides.

Speculators

Speculation is a very common technique used by traders and investors in the derivatives market. It is based on when traders have a strong viewpoint regarding the market behavior of any underlying security and though it is risky, if the viewpoint is correct, the speculation may reward with attractive payoffs. Thus, speculators use derivative contracts with a view to make profit from the subsequent price movements. They do not have any risk to hedge, in fact, they operate at a relatively high-risk level in anticipation of profits and provide liquidity in the market.

Arbitrageurs

Arbitrage is a strategy in which the participant (or arbitrageur) aims to make profits from the price differences which arise in the investments made in the financial markets as a result of mispricing. Arbitrageurs aim to earn low risk profits by taking two different positions in the same or different contracts (across different time periods) or on different exchanges to in-cash on price discrepancies or market inefficiencies.

Margin Traders

Margin is essentially the collateral amount deposited by an investor investing in a financial instrument to the counterparty in order to cover for the credit risk associated with the investment. In margin trading, the trader or investor is not required to pay the total value of your position upfront. Instead, they only need pay the margin amount which may vary and are usually fixed by the stock exchanges considering factors like volatility. Thus, margin traders buy and sell securities over a single session and square off their position on the same day, making a quick payoff if their speculations are right.

Criticism of derivatives

While derivatives provide numerous benefits and have significantly impacted modern finance and markets, they pose many risks too. In a 2002 letter to Berkshire Hathaway shareholders, Warren Buffet even described derivatives as “financial weapons of mass destruction”.

Derivatives are more highly leveraged due to relatively relaxed regulations surrounding them, and where one may need to put up half the money or more with buying other securities, derivatives traders can get by with just putting up a few percentage points of the total value of a derivatives contract as a margin. If the price of the underlying asset keeps falling, covering the margin account can lead to enormous losses. Derivatives are thus often criticized as they may allow investors to obtain unsustainable positions that elevates systematic risk so much that it can be equated to legalized gambling. Derivatives are also exposed to counterparty credit risk wherein there is scope of default on the contract by any of the parties involved in the contract. The risk becomes even greater while trading on OTC markets which are less regulated.

Derivatives have been associated with a number of high-profile credit events over the past two decades. For instance, in the early 1990s, Procter and Gamble Corporation lost more than $100 million in transactions in equity swaps. In 1995, Barings collapsed when one of its traders lost $1.4 billion (more than twice its then capital) in trading equity index derivatives.

The amounts involved with derivatives-related corporate financial distresses in the 2000s increased even more. Two such events were the bankruptcy of Enron Corporation in 2001 and the near collapse of AIG in 2008. The point of commonality among these events was the role of OTC derivative trades. Being an AAA-rated company, AIG was being exempted from posting collateral on most of its derivatives trading in 2008. In addition, AIG was unique among CDS market participants and acted almost exclusively as credit protection seller. When the global financial crisis reached its peak in late 2008, AIG’s CDS portfolios recorded substantial mark-to-market losses. Consequently, the company was asked to post $40 billion worth of collateral and the US government had to introduce a $150 billion financial package to prevent AIG, once the world’s largest insurer by market value, from filing for bankruptcy.

Conclusion

Derivatives were essentially created in response to some fundamental changes in the global financial system. If correctly handled, they help improve the resilience of the system, hedge market risks and bring economic benefits to the users. Thus, they are expected to grow further with financial globalization. However, past credit events have exposed many weaknesses in the organization of their trading. The aim is to minimize the risks associated with such trades while enjoying the benefits they bring to the financial system. An important challenge is to design new rules and regulations to mitigate the risks and to promote transparency by improving the quality and quantity of statistics on derivatives markets.

Useful resources

Role of Derivatives in the 2008 Financial Crisis

ESMA Annual Statistical Report 2020

About the author

The article was written in August 2021 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Linear Regression

August 23, 2021 by Jayati WALIA

Linear Regression

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole – Master in Management, 2019-2022) presents linear regression.

Definition

Linear regression is a basic and one of the commonly used type of predictive analysis. It attempts to devise the relationship between two variables by fitting a linear function to observed data. A simple linear regression line has an equation of the form:

wherein Y is considered to be the dependent variable (i.e., variable we want to predict) and X is the explanatory variable (i.e., the variable we use to predict the dependent variable’s value). The slope of the line is β₁, and β₀ is the x-intercept. ε is the residual (or error) in prediction.

Application in finance

For instance, consider Apple stock (AAPL). We can estimate the beta of the stock by creating a linear regression model with the dependent variable being AAPL returns and explanatory variable being the returns of an index (say S&P 500) over the same time period. The slope of the linear regression function is our beta.

Figure 1 represents the return on the S&P 500 index (X axis) and the return on the Apple stock (Y axis), and the regression line given by the estimation of the linear regression above. The slope of the linear regression gives an estimate of the beta of the Apple stock.

Figure 1. Example of beta estimation for an Apple stock.

Beta_AAPL

Source: computation by the author (Data: Apple).

Before attempting to fit a linear model to observed data, it is essential to determine some correlation between the variables of interest. If there appears to be no relation between the proposed independent/explanatory and dependent, then the linear regression model will probably not be of much use in the situation. A numerical measure of this relationship between two variables is known as correlation coefficient, which lies between -1 and 1 (1 indicating positively correlated, -1 indicating negatively correlated, and 0 indicating no correlation). A popularly used method to evaluate correlation among the variables is a scatter plot.

The overall idea of regression is to examine the variables that are significant predictors of the outcome variable, the way they impact the outcome variable and the accuracy of the prediction. Regression estimates are used to explain the relationship between one dependent variable and one or more independent variables and are widely applied to domains in business, finance, strategic analysis and academic study.

Assumptions in the linear regression model

The first step in the process of establishing a linear regression model for a particular data set is to make sure that the in consideration can actually be analyzed using linear regression. To do so, our data set must satisfy some assumptions that are essential for linear regression to give a valid and accurate result. These assumptions are explained below:

Continuity

The variables should be measured at a continuous level. For example, time, scores, prices, sales, etc.

Linearity

The variables in consideration must share a linear relationship. This can be observed using a scatterplot that can help identify a trend in the relationship of variables and evaluate whether it is linear or not.

No outliers in data set

An outlier is a data point whose outcome (or dependent) value is significantly different from the one observed from regression. It can be identified from the scatterplot of the date, wherein it lies far away from the regression line. Presence of outliers is not a good sign for a linear regression model.

Homoscedasticity

The data should satisfy the statistical concept of homoscedasticity according to which, the variances along the best-fit linear-regression line remain equal (or similar) for any value of explanatory variables. Scatterplots can help illustrate and verify this assumption

Normally-distributed residuals

The residuals (or errors) of the regression line are normally distributed with a mean of 0 and variance σ. This assumption can be illustrated through a histogram with a superimposed normal curve.

Ordinary Least Squares (OLS)

Once we have verified the assumptions for the data set and established the relevant variables, the next step is to estimate β₀ and β₁ which is done using the ordinary least squares method. Using OLS, we seek to minimize the sum of the squared residuals. That is, from the given data we calculate the distance from each data point to the regression line, square it, and calculate sum of all of the squared residuals(errors) together.

Thus, the optimization problem for finding β₀ and β₁ is given by:

After computation, the optimal values for β₀ and β₁ are given by:

Using the OLS strategy, we can obtain the regression line from our model which is closest to the data points with minimum residuals. The Gauss-Markov theorem states that, in the class of conditionally unbiased linear estimators, the OLS estimators are considered as the Best Linear Unbiased Estimators (BLUE) of the real values of β₀ and β₁.

R-squared values

R-squared value of a simple linear regression model is the rate of the response variable variation. It is a statistical measure of how well the data set is fitted in the model and is also known as coefficient of determination. R-squared value lies between 0 and 100% and is evaluated as:

The greater is the value for R-squared, the better the model fits the data set and the more accurate is the predicted outcome.

Useful Resources

Linear regression Analysis

Simple Linear Regression

▶ Louraoui Y. Beta

About the author

The article was written in August 2021 by Jayati WALIA (ESSEC Business School, Grande Ecole – Master in Management, 2019-2022).

The Black Scholes Merton Model

February 4, 2024August 19, 2021 by Akshit GUPTA

The Black-Scholes-Merton model

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the Black-Scholes-Merton Model .

Introduction

Options are one of the most popular derivative contracts used by investors to hedge the risks of their portfolios, to optimize the risk profile of their positions and to make profits (or losses) by means of speculation. The value of options is known at maturity date (or expiration date) as it is given by their pay-off functions defined in their contracts. But what is the value of the option at the issuance date or any date between the issuance and the expiration? The Black-Scholes-Merton model allows to answer this question.

The Black-Scholes-Merton model is an continuous-time option pricing model used to determine the fair price or theoretical value for a call or a put option based on variable factors such as the maturity date and the strike price of the option (option characteristics), and the price of underlying asset, the volatility of the price of underlying asset, and the risk-free rate (market data). It is used to determine the price of a European call option, which refers to the option that can only be exercised on the maturity date.

History

The model was first introduced to the world by a paper titled ‘The Pricing of Options and Corporate Liabilities’ by Fischer Black and Myron Scholes and was officially published in spring 1973. Almost around the same time as Black and Scholes, Robert Merton, who was also a colleague of Scholes at MIT Sloan, presented his contributions to the model in another paper named ‘Theory of Rational Option Pricing’, where he coined the name “Black-Scholes model”. Later, Black and Scholes also published empirical tests of the model in their ‘The Valuation of Option Contracts and a Test of Market Efficiency’ paper. For their significant contribution to the world of financial markets, Merton and Black were awarded the prestigious Nobel Prize in Economic Sciences in 1997 (unfortunately Scholes had passed away in 1995 due to which he was ineligible for the Nobel Prize).

In the BSM model, the value of an option depends on the future volatility of the underlying stock rather than on its expected return. The pricing formula is based on the assumption that the price of the underlying asset follows a geometric Brownian motion.

Option pricing with BSM

The BSM model is used to find the theoretical value of a European option. The model assumes that the price of the underlying asset follows a geometric Brownian motion, which implies that the returns on the underlying asset are normally distributed. It is also assumed that there are no arbitrage opportunities, no transaction costs and the risk-free rate remains constant over time.

The BSM formula

The payoffs for a call option and a put option give the value of these options at the maturity date T:

For a call option:

Formula for the payoff of a call option

For a put option:

BSM Formula for the payoff of a put option

The BSM formula gives the price of European put and call options at any date before the maturity date T. The value of European call and put options for a non-dividend paying stock are given by:

For a call option:

BSM formula for the call option

For a put option:

BSM formula for the put option

where,

Formula for the D1 Formula for the D2

The notations used in the above formulae are described as :

S_t: price of the underlying asset at time t
t: current date (or date of calculation of option price)
T: maturity or 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 (0 ≤ N(.) ≤ 1 )

For a call option, N(+d₂) is the probability that the option will be exercised, and Ke^{(-r(T-t) )} N(+d₂) is what is expected to be paid for the underlying stock if the option is exercised, discounted to today (or the calculation date t).

Similarly, SN(+d₁) is what we can expect to receive from selling the underlying stock, if the option is exercised, also discounted to today (or the calculation date t).

For a put option, N(-d₂) is the probability that the option will be exercised, and Ke^{(-r(T-t) )} N(-d₁ ) is what is expected to be paid for the underlying stock if the option is exercised, discounted to today (or the calculation date t).

Similarly, SN(-d₁ ) is what we can expect to receive from selling the underlying stock, if the option is exercised, also discounted to today (or the calculation date t).

Note that the value of the option given by the BSM formula depends on the maturity date and the strike price of the option (option characteristics), and the price of underlying asset, and the risk-free rate (market data) and the volatility of the price of underlying asset. While the option characteristics are known and the market data are observable, the volatility of the price of underlying asset is the only unknown variable in the formula.

Beyond the formula itself for the option prices, the BSM model also gives a method to manage the option over time (delta hedging) as an option is equivalent (under the assumption of no arbitrage) to a portfolio composed of the underlying asset and risk-free bond.

Example – Call and Put option pricing using Black-Scholes-Merton model

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 40€ with a maturity of 0.50 years. The price of the underlying asset is 50€ at time t and volatility is 40%. The risk-free rate is assumed to be 1%.

Figure 1. Call option Pricing using BSM formula Covered call
Source: computation by the author (based on the BSM model).

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 40€ with a maturity of 0.50 years. The price of the underlying asset is 50€ at time t and volatility is 40%. The risk-free rate is assumed to be 1%.

Figure 2. Put option Pricing using BSM formula Covered call
Source: computation by the author (based on the BSM model).

You can download below the Excel file used for the computation of the Call and Put option prices using the BSM Model.

Conclusion

The option-pricing model developed by Black, Scholes and Merton in 1973 provides a way of computing the prices of option contracts and has been widely used by traders since its publication. Following the seminal works by Black, Scholes and Merton, there haven been many extensions of their model, which have broadened its applicability to other instruments such as more complex options and insurance contracts.

Limitations of the BSM model

However, the model is sometimes criticized due to its weaknesses emerging from unrealistic sets of assumptions, which cause errors in estimation and model’s predictions. For instance, the BSM model assumes a constant value for volatility of the price of the underlying asset and also neglects any dividend payments from stocks which is certainly not the case in real life. Also, the model is only applicable to European options and would not be able to accurately determine the value of an American option which can be exercised at any time until the expiry date. Researchers have worked on amending the model to incorporate more realistic assumptions and have concluded that despite the model’s weaknesses, its application is still extremely useful in analyzing option prices.

Useful resources

Academic research

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

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

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

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

About the author

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

Call – Put Parity

August 19, 2021 by Akshit GUPTA

Call-Put Parity

This article written by Akshit GUPTA (ESSEC Business School, Master in Management, 2019-2022) presents the subject of call-put parity.

Introduction

The call-put parity (also written the put-call parity) is a concept introduced in the 1960s by the economist Hans R. Stoll in a paper named “The Relationship Between Put and Call Option Prices”. The call-put parity shows the relationship between the prices of a put option, a call option, and the underlying asset. The call option and the put option are written on the same underlying asset and have the same expiration date and strike price. The call-put parity is applicable only on European options with a fixed time to expiration (it is not applicable to American options).

Call-put parity relation

The call-put parity relation is given by the equality:

Formula for the call put parity

Where t is the evaluation date (any date between the issuance date and the maturity date of the option), C_t the price of the call option, P_t the price of the put option, S_t the price of the underlying asset, K the strike price of the two options (same strike price for the call and put options), T the maturity date of the two options (same maturity date for the call and put options) and r the risk-free rate.

The call-put parity relation is sometimes written in different ways:

Formula for the call put parity styles

Demonstration

Let us try to find the call-put parity relation for a put option and a call option, which are European options with the same strike price K and the same maturity date T.

Let us consider a portfolio composed a long position in the underlying asset, a long position in the put option, a short position in the call option and a short position of a zero-coupon bond maturing at time T and of final value K.

Let us compute the value of this position at time T. The underlying asset is worth S_T. The zero-coupon bond is worth K. Regarding the call and put options, we can distinguish two cases: S_T > K and S_T < K.

In the first case, the put option finishes out of the money and the call finishes in the money and is worth S_T – K. The value of the position is then equal to: S_T + 0 – (S_T – K) – K, which is equal to zero.

In the second case, the call option finishes out of the money and the put finishes in the money and is worth K – S_T. The value of the position is then equal to: S_T + (K – S_T) – 0 – K, which is equal to zero.

If the value of the position at time T is also equal to 0, then the value of the position at time t is also equal to 0. If there is no arbitrage, then the value of the position by detailing its components satisfies:

Formula for the call put parity without arbitrage

which leads to the formula given above.

Application

The call-put parity formula helps the investors to calculate the price of a put option from the price of a call option, or inversely, to calculate the price of a call option from the price of a put option (the call option and the put option are written on the same underlying asset and have the same expiration date T and strike price K).

Implication

If the put-call parity does not hold true, there exists an arbitrage opportunity for investors. An arbitrage opportunity helps the investors earn profits without taking any risks. But the chances of finding an arbitrage opportunity is low given the high liquidity in the markets.

Example of application of the call-put parity

Assuming the stock of APPLE is trading at $25 in the market, the strike price of a 3-month European call option on Apple stock is $24 and the premium is $5. The risk-free rate is 8%.

Now, using the call-put parity,

Formula for the call put parity styles

we can calculate the price of the 3-month European put option on Apple stock with the same strike price, which is as follows:

The price of the call option (C) is $5, the price of the underlying asset (S) is $25, the present value of the strike price (K) is $23.52, and the risk-free rate (r) is 8% (market data).

As per the formula: P = $5 – $25 + $23.52, the price of the put option (P) is approximately equal to $3.52.

Useful resources

Academic research

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 11 – Properties of Stock Options, 256-275.

Stoll H.R. (1969) “The Relationship Between Put and Call Option Prices,” The Journal of Finance, 24(5): 801-824.

About the author

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

Option Greeks – Theta

February 4, 2024August 19, 2021 by Akshit GUPTA

Option Greeks – Theta

This article written by Akshit GUPTA (ESSEC Business School, Master in Management, 2019-2022) presents the technical subject of theta, an option Greek used in option pricing and hedging to deal with he passing of time.

Introduction

Theta is a type of option Greek which is used to compute the sensitivity or rate of change of the value of an option contract with respect to its time to maturity. The theta is denoted using the symbol (θ). Essentially, the theta is the first partial derivative of the price of the option contract with respect to the time to maturity of the option contract.

It is shown as:

Formula for the theta

Where V is the value of the option contract and T the time to maturity for the option contract.

Theoretically, as the option contract approaches maturity, the theta of on option contract increases and moves towards zero as the time value or the time value of the option decreases. This is referred to as “theta decay”.

For example, an option contract is trading at a premium of $10 and has a theta of -0.8. Thus, with theta decay, the option price will decrease to $9.2 after one day and further to $6 after five days.

The figure below represent the theta of a call option as a function of the time to maturity:

Figure 1. Theta of a call option as a function of time to maturity.

Source: computation by the author (Model: Black-Scholes-Merton).

Intrinsic and time value of an option contract

Essentially, the price of an option contract consists of two values namely, the intrinsic value and the time value (sometimes called extrinsic value). The intrinsic value in the price of an option contract is the real value or the fundamental value of an option based on the price of the underlying asset at a given point in time.

For example, a call option contract has a strike price of $10 and the underlying asset has a market price of $17. Theoretically, the buyer of a call option can execute the contract and buy the asset at $10 and sell it in the market for $17. He/she can make an immediate profit of $7 if they decide to exercise the option. Thus, the intrinsic value of the option contract is $7.

If the current call option price/premium is $9 in the market and the intrinsic value is $7, then the time value can be calculated as:

Time Value for the theta

Thus, the time value is $9-$7 is equal to $2. The $2 is the time value of an option contract which is determined by the factors other than the price of the underlying asset. As the option approaches maturity, the time value of the option contract declines and tends to zero. The price of an option contract which is at the money or out the money, it consists entirely of the time value as there is no intrinsic value involved.

For example, a call option contract with a strike price of $20, the underlying asset price of $15, and option premium of $3, has a time value equal to the option premium, $3, since the option is out of money.

Calculating Theta for call and put options

The theta for a non-dividend paying stock in a European call and put option is calculated using the following formula from the Black-Scholes Merton model:

Formula for the theta of a call and a put option

Where N’(d₁) represents the first order derivative of the cumulative distribution function of the normal distribution given by:

First_ derivative_Normal_distribution_d1

d₁ is given by:

Formula for d1

d₂ is given by:

Formula for d2

And N(-d₂) is given by:

Formula for -d2

Where S is the price of the underlying asset (at the time of valuation of the option), σ the volatility in the price of the underlying asset, T time to option’s maturity, K the strike price of the option contract and r the risk-free rate of return.

Excel pricer to calculate the theta of an option

You can download below an Excel file for an option pricer (based on the Black-Scholes-Merton or BSM model) which allows you to calculate the theta of a European-style call option.

Example for calculating theta

Let us consider a call option contract with the following characteristics: the underlying asset is an Apple stock, the option strike price (K) is equal to $300 and the time to maturity (T) is of one month (i.e., 0.084 years).

At the time of valuation, the price of the Apple stock (S) is $300, the volatility (σ) of Apple stock is 30% and the risk-free rate (r) is 3% (market data).

The theta of a call option is approximately equal to -0.2636 per trading day.

Using the above example, we can say that after one trading day, the price of the option will decrease by $0.2636 (approximately) due to time decay.

Useful resources

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 19 – The Greek Letters, 424–431.

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

About the author

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

Option Greeks – Vega

January 30, 2024August 19, 2021 by Akshit GUPTA

Option Greeks – Vega

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains the technical subject of vega, the option Greek used in option pricing and hedging to take into account the volatility of the underlying asset.

Introduction

Vega is a type of option Greek which is used to compute the sensitivity or rate of change of the value of an option contract with respect to the volatility of the underlying asset. The Vega is denoted using the Greek letter (ν). Essentially, the vega is the first partial derivative of the value of the option contract with respect to the volatility of the underlying asset.

The vega formula for an option is given by

Formula for the gamma

Where V is the value of the option contract and σ is the volatility of the underlying asset.

If the Vega is a very high positive or a negative number, this means that the option price is highly sensitive to the volatility of the underlying asset. The Vega is maximum when the option price is at the money. For example, the strike of an option contract is €100, and the price of the underlying asset is €100. The option is at the money (ATM) and has an intrinsic value of zero. So, the option premium entirely consists of the time value of the option. Thus, the Vega is the highest for at the money option contract since the option value are mostly dependent on the time value (sometimes called the extrinsic value). An increase/decrease in volatility can change the option value significantly for at-the-money options.

Figure 1 below represents the vega of a call option as a function of the price of the underlying asset. The parameters of the call option are a maturity of 3 months and a strike of €100. The market data are a price of the underlying asset between €50 and €150, a volatility of the underlying asset of 40%, a risk-free interest rate of 3% and a dividend yield of 0%.

Figure 1. Vega of a call option as a function of the price of the underlying asset.

Source: computation by the author (Model: Black-Scholes-Merton).

Calculating the vega for call and put options

The vega for a European call or put option is calculated using the following formula:

Formula for the gamma

where

N’(d₁) represents the first order derivative of the cumulative distribution function of the normal distribution given by:

First_ derivative_Normal_distribution_d1

and d₁ is given by:

Formula for d1

where S is the price of the underlying asset (at the time of valuation of the option), σ the volatility in the price of the underlying asset, T time to option’s maturity, K the strike price of the option contract and r the risk-free rate of return.

Example for calculating vega

At the time of valuation, the price of the Apple stock (S) is $300, the volatility (σ) of Apple stock is 30% and the risk-free rate (r) is 3% (market data).

The vega of the call option is approximately equal to 0.3447963.

Using the above value, we can say that due to a 1% change in the volatility of the underlying asset, the price of the option will change approximately by $0.3447.

Excel pricer to calculate the vega of an option

You can download below an Excel pricer (based on the Black-Scholes-Merton or BSM model) to calculate the vega of an option (call or put).

Useful resources

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 19 – The Greek Letters, 424–431.

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

About the author

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

Cash flow statement

August 18, 2021 by Bijal GANDHI

Cash flow statement

In this article, Bijal GANDHI (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains the meaning of cash flow statement.

This read will help you understand in detail the meaning, structure, components of cash flow statement along with relevant examples.

Cash Flow statement

The cash flow statement is one of the three most important financial statements which acts as a bridge between the balance sheet and the income statement. It is a summary of all the cash and cash equivalents that have entered or left the company in the previous years. It helps to understand how well a company manages its cash position. In many countries, it is a mandatory part of the financial statements for large firms.

Structure of Cash Flow statement

The cash flow statement is divided into three of the following major activity categories: operating activities, investing activities and financing activities.

Cash from operating activities

The operating activities includes all the sources and uses of cash related to the production, sale and delivery of the company’s products and services. Few examples of the operating activities include,

• Sale of goods & services
• Payments to suppliers
• Advertisements and marketing expenses
• Rent and salary expenses
• Interest payments
• Tax payments

Cash from investing activities

As the name suggests, investing activities includes all those sources and use of cash from a company’s investments, assets, and equipment. A few examples of investing activities include,

Purchase and sale of an asset
Loans to suppliers
Loans received from customers
Expenses related to mergers and acquisitions

Cash from financing activities

Financing activities are those that include all the sources and use of cash from investors. All the inflow and outflow of cash such as,

Capital raised through sale of stock
Dividends paid
Interest paid to bondholders
Net borrowings
Repurchase of company’s stock

LVMH Example: Cash Flow Statement

Here, we again take the example of LVMH. The French multinational company LVMH Moët Hennessy Louis Vuitton was founded in 1987. The company headquartered in Paris specializes in luxury goods and stands at a valuation of $329 billion (market capitalization in June 2021). It is a consortium of 75 brands controlled under around 60 subsidies. Here, you can find a snapshot of LVMH Cash flow statement for three years: 2018, 2019 and 2020.

Credit Rating

Importance and use of cash flow statement

The cash flow statement is a very important indicator of the financial health of a company. This is because a company might make enough profits but might run out of cash to be able to operate. Also, it indicates the company’s abilities to meet its interest obligations and dividend payments if any. Basically, it provides a true picture of a company’s liquidity and financial flexibility. Therefore, a cash flow statement used in conjunction with the income statement and the balance sheet helps provide a holistic view of a company’s strength and weaknesses. The cash flow statement is therefore of great use to the following stakeholders:

Potential and current debtholders (creditors and bondholders)
Potential and current shareholders
Management team and company’s directors

About the author

Article written in July 2021 by Bijal GANDHI (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Operating vs Non-Operating Revenue

August 18, 2021 by Bijal GANDHI

Operating vs Non-Operating Revenue

In this article, Bijal GANDHI (ESSEC Business School, Master in Management, 2019-2022) explains the difference between operating and non-operating revenue.

This read will help you understand in detail various terminologies related to revenue and income statement.

What is operating revenue?

The revenue generated from the primary or core activities of a company is referred to as operating revenue. It is important to differentiate between operating and non-operating revenue to gain insights into the efficiency of a firm’s core operations.

For example, the revenue generated from the total sale of iPhones worldwide is an operating revenue for Apple, whereas the revenue generated from sale of old office furniture would be a non-operating revenue.

What is non-operating revenue?

Non-Operating revenue refers to the revenue generated from operations that are not part of a company’s core business. The items in this section are generally unique in nature and therefore they do not show a true picture of the efficiency of a company’s core business. It is rather attributable to a company’s managerial and financial decisions.

For example, research grants obtained by universities are non-operating revenues as they are not generated from the core business (tuition fees).

How are revenue recorded in the income statement?

We know from the income statement that the COGS is deducted from revenue to derive the gross profit. The operating expenses are further deducted from the gross profit to attain the operating profit. The non-operating revenues and expenses are then combined and deducted from the operating profit to derive the net profit.

LVMH example

Let us once take the example of Moët Hennessy Louis Vuitton (LVMH). The French multinational company LVMH was founded in 1987. The company headquartered in Paris specializes in luxury goods and stands at a valuation (market capitalization in June 2021) of $329 billion. It is a consortium of 75 brands controlled under around 60 subsidiaries. Here, you can find a snapshot of LVMH Income statement for three years: 2018, 2019 and 2020.

Here, you can see that the highlighted part; “other financial income and expenses” are combined to derive the net profit before taxes

About the author

Article written in August 2021 by Bijal GANDHI (ESSEC Business School, Master in Management, 2019-2022).

Gains vs Revenue & Losses vs Expenses

August 18, 2021 by Bijal GANDHI

Gains vs Revenue & Losses vs Expenses

In this article, Bijal GANDHI (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains the difference between gains and revenue, and losses and expenses.

This read is for the students who wish to have a clear and theoretical understanding of the basic terms used in accounting and finance.

Revenue

We know from revenue, that it is referred to the money brought into a company from the sale of either goods, services, or both. Revenue is synonymous to sales and top line. This is because it first line on the income statement and it is a good indicator of a business’s performance. Revenue consists of two components, the price and the number of products/services sold. It is then calculated in the following manner:

Gains

Gains refers to the income generated through non-primary operations of the company. Any positive monetary value (profit) generated from secondary sources is a capital gain. For example, profit from the sale of real estate is to be treated as capital gain. Other such examples include the following,
• Profit from sale of equity holdings in any company
• Profit on investment in mutual fund
• Profit from winning a lawsuit.
• Profit from disposing an asset.

Gains can be from short-term holdings or long-term holdings. Short term could be defined as one to two years depending on accounting standards and type of financial instrument. It is important to take this in consideration while investing as both have different taxation guidelines.

Expenses

Expenses refers to the cost of operations incurred by a company. The basic goal of any company is to keep the expenses in check to ensure maximum profits. Expenses are broadly defined under the following two categories,
• Operating Expenses: The costs related to the main activities of the company such as cost of goods sold, salary, rent, legal, advertisement, etc.
• Non-Operating Expenses: These are the expenses that are not directly related to the core operations of a business. For example, profit from the sale of real estate would be a non-operating expense for a company who does not regularly deal in real estate. Similarly, the expenses such as interest payments on debt is also a non-operating expense since it does not arise from the company’s core business.

Losses

A loss in accounting terms refers to the money lost through non-primary operations of the company. Any negative monetary value (loss) incurred due to secondary sources is recorded as a capital loss. For example, the loss on an investment in equity shares of another company is a capital loss.
Like gains, it is important to identify whether a loss is from a short-term holding or a long-term holding. This is because in taxation, gains can be offset against corresponding losses.

About the author

Article written in July 2021 by Bijal GANDHI (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Quantitative Finance: Introduction and Scope

August 9, 2021 by Jayati WALIA

Quantitative Finance: Introduction and Scope

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole – Master in Management, 2019-2022) presents an overview of Quantitative Finance.

Quantitative Finance: Introduction and Scope

Quantitative finance has become an integral part of modern finance with the advent of innovative technologies, trading platforms, mathematical models, and sophisticated algorithms. In lay man terms, it is essentially the application of high-level mathematics and statistics to finance problems. Quantitative finance majorly focuses on most frequently traded securities. The very basis of it involves observation and quantitative analysis of market prices (stock prices, exchange rates, interest rates, etc.) over time, along with applying them to stochastic models and deducing results to make security pricing, trading, risk assessment, hedging and many other investment decisions. Hence, the heavy involvement of mathematics and especially stochastic calculus. However, it is not limited to that. In fact, theories and concepts from many other disciplines including physics, computer science, etc. have contributed to put together what we know as quantitative finance today.

Brief History

It was in the 20th century that the foundations of Quantitative Finance were laid starting off with the ‘Theory of Speculation’ PhD thesis by the French mathematician Louis de Bachelier. Bachelier applied the concept of Brownian motion to asset price behavior for the first time. Later the Japanese mathematician Kiyoshi Îto wrote a paper on stochastic differential equations and founded the stochastic calculus theory that is also named after him (Îto calculus) and is widely used in option pricing. The major breakthrough however, came in the 1970s when Robert Merton’s ‘On the pricing of corporate debt: the risk structure of interest rates’ and Fischer Black and Myron Scholes’ ‘The pricing of options and corporate liabilities’ research papers were published which inherently presented a call and put option pricing model and after that there was no looking back. The Black-Sholes-Merton model known as “BSM” model is widely used and is creditable for the boom of the options market. Today many more stochastic models have been devised to extend the BSM model, setting the benchmarks of quantitative analysis higher and benefitting the global economy.

Market participants

Quantitative Finance is used by many market participants: banks, financial institutions, investors and businesses who want better and automated control over their finances given the fluctuating behavior of the assets they trade. Initially, quantitative finance was majorly used in modelling market finance problems like pricing and managing derivative products for trading, managing risk of the investments in contracts, etc. basically in the sell-side of the firms such as Investment Banking. However, with continuous advancements, we see increased usage in buy-side as well among areas like Hedge Funds and Asset Management through development of quantitative models to analyze asset behavior and predict market movements in order to leverage potential trading opportunities.

Thus, any firm or investor that deals in financial derivatives (futures and options), portfolios of stocks and/or bonds, etc. need to use Quantitative Finance. These participants have specialized analysts to work on the quantitative finance and they are generally known as Quantitative Analysts or ‘quants’. Once referred to as ‘the rocket scientists of Wall Street’, quants have sound understanding of finance, mathematics and statistics combined with the acumen of programming/coding. With the dramatic changes in industry witnessed over the past years, quants with a stellar combination of the mentioned disciplines are greatly in demand.

Types of Quants

Quants create and apply financial models for derivative pricing, market prediction and risk mitigation. There are however many variations in quant roles, some of which are explained below:

Front Office Quant: Work in proximity with traders and salespersons on the trade floor. Implement pricing models used by traders to spot out new opportunities and provide guidance on risk strategies.
Quant Researcher: Essentially the Back Office quants, they research and design high frequency algorithms, pricing models and strategies for traders and brokerage firms.
Quant Developer: They are essentially software developers in a financial firm. They translate business requirements provided by researchers into code applications.
Risk Management Quant: They build models for keeping in check credit and regulatory operations and assessing credit risk, market risk, ALM (Asset and Liability Management) risk etc. They are the Middle Office quants and perform risk analysis of markets and assets and stress testing of the models too.

The Future of Quantitative Finance

Quants and Quantitative finance are here to stay! With firms becoming larger than life and the tremendous data and money involved, the scope and demand for quantitative finance is escalating like never before. Quantitative Finance is no more just about complex mathematics and stochastic models. With finance becoming more technical, data science, machine and deep learning and artificial intelligence are taking over the domain’s informative decision-making strategies. Thus, quantitative finance is being driven to new heights by the power of high processing computer algorithms that enable us to analyze enormous data and run model simulations within nanoseconds. To quote Rob Arnott, American entrepreneur and founder of Research Affiliates: “To a man with a hammer, everything looks like a nail. To a quant, anything that can’t be quantified is ignored. And historical data is our compass, even though we know that past performance is no guarantee of future results.”

Useful resources

Quantitative Finance
What is Quantitative Finance?
2020 Quants predict next decade in global finance

About the author

The article was written in July 2021 by Jayati WALIA (ESSEC Business School, Grande Ecole – Master in Management, 2019-2022).

Option Greeks – Gamma

February 4, 2024August 8, 2021 by Akshit GUPTA

Option Greeks – Gamma

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the technical subject of gamma, an option Greek used in option hedging.

Introduction

Gamma is a type of option Greek which is used to compute the sensitivity or rate of change of delta (Δ) of an option contract with respect to a change in the price of the underlying in the option contract (S). The gamma of an option is expressed in percentage terms. Denoted by the Greek letter (Γ), the gamma is defined by

Formula for the gamma of an option

Where (Δ) is the delta of the option and S the price of the underlying asset.

Essentially, the gamma is the second partial derivative of the value of the option contract (V) with respect to the price of the underlying asset (S). It measures the convexity of the value of the option contract with respect to the price of the underlying asset. The gamma then corresponds to

Formula for the gamma of an option

Where V is the value of the option and S the price of the underlying asset.

The gamma of an option contract is at its maximum when the price of the underlying asset is equal to the strike price of the option (an at-the-money option). If the price of the underlying moves deeper in the money or out of the money, the value of the gamma approaches zero.

The gamma as a function of the price of the underlying asset for a call option is given below.

Figure 1. Gamma of a call option.

Source: computation by the author (Model: Black-Scholes-Merton).

Also, if the gamma of the option contract is small, it means that the delta of the option moves slowly with the price of the underlying asset.

Calculating gamma for call and put options

The gamma for European call or put options on a non-dividend paying stock is calculated using the following formula from the Black-Scholes-Merton model is:

Formula for the gamma of a call/put option

Where,N’d₁ represents the first order derivative of the cumulative distribution function of the normal distribution given by:

First_ derivative_Normal_distribution_d1

and d₁ is given by:

Formula for d1.png

Excel pricer to calculate the gamma of an option

You can download below an Excel file for an option pricer (based on the Black-Scholes-Merton or BSM model) which allows you to calculate the gamma of a European-style call option.

Delta-gamma hedging

A trader holding a portfolio of option contracts uses gamma hedging to offset the risks associated with the price movement in the underlying asset by buying and selling the option contracts to maintain a constant delta. Generally, the delta is maintained near or at the zero level to attain delta neutrality. The neutrality in the gamma for the option is required to protect the portfolio’s value against sharp price movements in the price of the underlying asset.

Formula for the gamma hedging of a call option

Limitations of gamma hedging

The limitation of gamma hedging includes the following:

Transaction cost – Gamma hedging requires constantly monitoring the markets and buying or selling the option contracts. Due to this practice of buying and selling frequently, the transaction costs are quite high to execute a gamma hedge. Thus, gamma hedging is an expensive strategy to practice.
Loosing delta neutrality – Whenever a trader executes a gamma hedge and trades in option contracts, it is often accompanied with a move in the portfolio’s delta. Thus, to achieve delta neutrality again, the trader must buy or sell additional quantities of the underlying asset, which is time consuming and comes with a transaction cost.

Useful resources

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 19 – The Greek Letters, 424–431.

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

About the author

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

Option Greeks – Delta

February 4, 2024August 5, 2021 by Akshit GUPTA

Option Greeks – Delta

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the technical subject of delta, an option Greek used in option pricing and hedging.

Introduction

Option Greeks are sophisticated financial metric used by trader to calculate the sensitivity of option contracts to different factors related to the underlying asset including the price of the underlying, its volatility, and time value. The Greeks are used as an effective tool to practice different hedging strategies and eliminate risks in a position. They also help to optimize the options positions at any point in time.

Delta is a type of option Greek which is used to compute the sensitivity or rate of change in price of the option contract with respect to the change in price of the underlying asset. It is denoted by the Greek letter (Δ). The formula for calculating the delta of an option contract is:

Formula for the delta of an option

Where V is the value of the option and S the price of the underlying asset.

For example, if an option on Apple stock has a delta of 0.3, it essentially means that a $1 change in the price of the underlying asset i.e., Apple stock, will lead to a change of $0.3 in the price of the option contract.

When a trader takes a position based on the delta sensitivity of any option contract, it is called delta hedging. The goal is to achieve a delta-neutral portfolio and eliminate the risks associated with movement in the prices of the underlying. Due to the complexity of the tool, delta hedging is generally practiced by professional traders in large financial institutions. In options, the delta of any call option is always positive whereas the delta of a put option is always negative.

Delta formula

Call option

According the Black-Scholes-Merton model, the formula for calculating the delta for a European-style call option on a non-dividend paying stock is given by:

Formula for the delta of a call option

Where N represents the cumulative distribution function of the normal distribution and d₁ is given by:

Formula for d1

Where S is the price of the underlying asset (at the time of valuation of the option), σ the volatility in the price of the underlying asset, T time to maturity of the option, K the strike price of the option, and r the risk-free rate of return.

Put option

According the Black-Scholes-Merton model, the formula for calculating the delta for a European-style put option on a non-dividend paying stock is given by:

Formula for the delta of a put option

Delta as a function of the price of the underlying asset

Call option

The delta as a function of the price of the underlying asset for a European-style call option is represented in Figure 1.

Figure 1. Delta of a call option.

Source: computation by the author (Model: Black-Scholes-Merton).

For a call option, the delta increases from 0 (out-of-the-money option) to 1 (in-the-money option).

Put option

The delta as a function of the price of the underlying asset for a European-style put option is represented in Figure 2.

Figure 2. Delta of a put option.

Source: computation by the author (Model: Black-Scholes-Merton).

For a put option, the delta increases from -1 (in-the-money option) to 0 (out-of-the-money option).

Excel pricer to calculate the delta of an option

You can download below an Excel file for an option pricer (based on the Black-Scholes-Merton or BSM model) which allows you to calculate the delta of a European-style call option.

Delta Hedging

A trader holding an option contract uses delta hedging to offset the risks associated with the price movement in the underlying asset by continuously buying and selling the underlying asset to achieve delta neutrality. This is used by option traders in financial institutions to manage their option book (the delta is computed at the option level and aggregated at the book level) and generate the margin the bank of the option writing activity.

The delta of an option contract keeps on changing as the prices of the underlying and the option contract changes. So, to maintain the delta neutrality the trader must constantly monitor the markets and execute trades to achieve neutrality. The process of continuously buying or selling the underlying asset is called dynamic hedging in options.

At the first order, the change of the value of a delta-hedged call option over the period from t to t+ δt would be equal to the risk-free rate (r) over the period:

Formula for the delta hedging of a call option

Limitations of delta hedging

Although delta hedging is a useful tool to offset the risks associated to the movement in the price of an underlying, it comes with some limitations which are:

Transaction cost

Since delta hedging requires constantly buying or selling the underlying asset, it comes with a high transaction cost. This makes delta hedging an expensive tool to optimize the portfolio against price risk. In practice, traders would adjust their option position from time top time.

Illiquid Markets

When the market for an asset is illiquid, it is difficult to practice delta hedging as the trader will not be able to constantly buy or sell the underlying asset to neutralize the price impact.

Example for calculating delta

At the time of valuation, the price of the Apple stock (S) is $300, the volatility (σ) of Apple stock is 30% and the risk-free rate (r) is 3% (market data).

The delta of a call option is approximately equal to 0.50238.

Using the above value, we can say that due to a $1 change in the price of the underlying asset, the price of the option will change by $0.50238.

Useful resources

Research articles

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

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

Books

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 19 – The Greek Letters, 424 – 431.

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

About the author

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

The return of inflation

August 2, 2021 by Alexandre VERLET

The return of inflation

In this article, Alexandre VERLET (ESSEC Business School, Master in Management, 2017-2021) explains how inflation could become an issue again for the first time in 40 years.

Inflation is not something we usually worry about. In fact, few understand what inflation is about beyond the fact that it is characterized by a rise in prices. But since inflation has been around for 40 years without causing any problem, it seems to be absolutely not dangerous and perfectly controlled by central banks. Problem is, the Covid-19 crisis and the economics policies launched by governments and central banks in response are unprecedented. Moreover, an excess of inflation can be a major problem for developed economies: the UK in the 1970’s was Europe’s sick man and had to revolutionize its economy the hard way in order to get out of its stagflation spiral.

So why are we talking about a 40 year old subject? Because for several weeks now, markets have been worried about a sustained return of inflation. Fantasy for some, harsh reality for others: the scenario of a sustainable return of inflation is far from unanimous among economists. None of them, however, disputes the appearance of signals favorable to an at least temporary rise in prices, even if the extent of the phenomenon is debated. Indeed, the latest figures from the United States speak for themselves: in April, prices there rose by 4.2% over one year. This is the first time since September 2008 that the markets have been particularly nervous in recent days. In the euro zone, inflation, although more moderate (+1.6% year-on-year), also seems to be accelerating as economies are recovering from the crisis.

What is inflation and what is causing it to return?

To put it simply, inflation is the sustained rise of general prices over a period of time. It is calculated using a basket of products in which their weight in the GDP is taken into account so the basket represents the economy as a whole. The causes of inflation can be derived from a simple phenomenon: the imbalance between supply and demand of good. In our case, all the ingredients were in place for a rise in prices. Initially, the end of the Covid-19 epidemic in China and the roll-out of the vaccination campaign, particularly in the United States, contributed to the sudden rebound in global demand. But the supply side was not able to keep up with the movement and meet all the needs, since supply chains and production processes are still disorganized. Adding to that, some countries remain closed, and global supply chains cannot be restarted overnight after more than a year of pause. As a result, bottlenecks have developed in some sectors and manufacturers are now facing shortages of raw materials. Companies must also adapt their production processes under the Covid-19 regulation, and all this has a cost.This automatically leads to higher production costs, which companies pass on in their prices.

Beyond the tensions on the goods and services market, other signals are worrying the markets across the Atlantic. Starting with Joe Biden’s three stimulus plans, which will involve almost 30% of US GDP. These massive plans, which are flourishing both in the United States and in Europe, are encouraged by the central banks’ accommodating policy and their unlimited power of money creation which, through asset purchases, allow governments to go into debt at lower cost. But by injecting so much money to stimulate demand, the Fed and the White House are taking the risk of putting the US economy in a state of overheating which could lead to a surge in prices in the US and, by contagion, in Europe. This is the principle of the quantitative theory of money developed by the economist Milton Friedman in 1970 when he stated that “inflation is always and everywhere a monetary phenomenon in the sense that it is and can be generated only by an increase in the quantity of money faster than the increase in output. The other phenomenon fueling fears of a sustained acceleration in prices is the tightness in the US labor market. Some sectors are facing a shortage of labor, including low-skilled workers, which could restart the “wage-price loop”. Several companies, including McDonald’s and Amazon, have already announced a significant increase in their minimum wage and attractive hiring bonuses to attract new candidates to the United States.

How would the return of the inflation impact us?

If it does not exceed a certain level, inflation is not necessarily harmful to the economy and can even be good for some. Keep in mind that the European Central Bank is aiming for an inflation rate close to but below 2% per year. The markets fear the return of inflation, but everyone is waiting for this inflation. Since 2008, the world entered a phase of low inflation but also of risk of deflation. While rising prices cause consumers to lose purchasing power in the short term, they often result in higher wages in the medium term. Not least because the French minimum wage is indexed to inflation, as are a number of social benefits. And an increase in the minimum wage most often results in an increase in the lowest wages, as explained by INSEE in a study on wages in France. In addition, employee representatives usually use inflation as a reason to obtain wage increases during annual negotiations in the company. If the employer accepts an increase at least equal to that of prices, then the purchasing power of employees remains stable. But one of the main winners from an acceleration of inflation is the state. When prices rise across the board, tax revenues increase. Another positive consequence is that inflation increases the capacity to repay public debt, since it increases nominal GDP and thus reduces the debt/GDP ratio. The same mechanism applies to all borrowers. At least if wages keep pace with inflation over time. Let us take the case of an employee earning 2000 euros per month. This person has taken out a fixed-rate loan with a monthly payment of 500 euros. Let us also assume an inflation rate of 2% for three consecutive years. Assuming that wages increase at the same rate, the employee will receive 2122 euros per month three years later but will still have to continue to repay 800 euros. His debt ratio would then fall from 32% to 30%. It would then be easier for him to repay his loan. The opposite is true for savers. When inflation is higher than the rate of return on savings, which is the case for the Livret A, the real return becomes negative. This means that the capital invested loses value. Finally, civil servants or pensioners can also be the big losers of a return of inflation if their income is not revalued in line with inflation, as has been the case in recent years. Provided that it is not excessive, inflation is not always a bad thing and is even often synonymous with growth. The question is therefore to know how much inflation will be and whether it will be sustainable.

In the current context, the prospect of uncontrolled inflation cannot be ruled out. The pre-existing equilibrium was not one of non-existent inflation, but one of well-anchored inflation expectations. The extremely accommodating fiscal and monetary policies are now threatening that balance.

If private agents start to doubt the willingness and ability of their central bank to defend price stability, then expectations may be derailed and a return to normal inflation would require huge sacrifices. To prevent expectations from deteriorating further, the central bank would be forced to absorb liquidity by a reverse quantitative easing, which would cause a rise in long-term rates and a contraction in economic activity. As a consequence, the ability of States to take on debt would become severely limited, which would threaten the sustainability of post-covid recovery plans.

Should we worry about the future because of inflation?

The inflation threat should be definitely be treated seriously by central banks. Nevertheless, the scenario of an uncontrolled inflation remains unlikely, especially in Europe where the stimulus package were far from the size of Biden’s plan. Firstly, the rise in prices in the United States is largely temporary. The shortage of raw materials and labor will eventually fade, so the resulting inflation should do the same. Secondly, the inflation figures observed in April should be put into perspective as they reflect a catch-up phenomenon. Indeed, demand had fallen at the same time last year due to the confinement, which had also pushed prices down. It should also be noted that the increase in prices in the US is highly sectorised: one third of the monthly inflation in April was linked to the evolution of second-hand car prices. And if we exclude volatile prices such as energy and food, US inflation reached 3% over one year. For their part, central banks such as the US Fed point out that a number of deflationary elements have not disappeared, starting with unemployment, which puts the risk of wage inflation into perspective. If inflation anticipations are still strong enough to offset those two trends, central banks will have to raise key rates to cool the economy in order to limit price increases. It would then be the end of the years of “free money”, and that is something that will impact all of us as potential borrowers. So keep an eye on economic indicators over the next few months!

About the author

Article written in August 2021 by Alexandre VERLET (ESSEC Business School, Master in Management, 2017-2021).

Understanding financial derivatives: options

August 2, 2021 by Alexandre VERLET

In this article, Alexandre VERLET (ESSEC Business School, Master in Management, 2017-2021) explains why financial markets invented options and how they function.

A historical perspective on options

The history of options is surrounded by legends.. This story is linked to human’s desire to control the unpredictable, sometimes to protect himself from it, often to profit from it. This story is also that of a flower: the tulip. At the beginning of the seventeenth century, in the Netherlands, the tulip was at the origin of the first known speculative bubble. Furthermore, this was historically the first time that options contracts were used on such a large scale. The possibility of profiting from the rise in the price of tulips by paying only a small part of the price aroused great interest on the part of speculators, thus increasing the price of the precious flower tenfold. Soon the price of the tulip reached levels completely unrelated to its market value. Then, suddenly, demand dried up, causing the price to fall even faster than the previous rise. The crisis that followed had serious consequences and confirmed Amsterdam’s loss of world leadership in finance to the benefit of London, which had already taken over the Dutch capital as the world’s center for international trade. Educated by the Dutch experience, the British became increasingly sceptical about options, so much so that they eventually banned them for over a century. The ban was finally lifted towards the end of the 19th century. It was also at this time that options were introduced in the United States.

The American options market entered a new dimension at the end of the 20th century. Indeed, 1973 was a pivotal year in the history of options in more ways than one. In March 1973, a floating exchange rate regime was adopted as the standard for converting international currencies, creating unprecedented instability in the currency market. This was also the year of the “first oil shock”. Also in 1973, the Chicago Board Options Exchange (CBOE), the first exchange entirely dedicated to options, opened its doors. The same year saw the birth of the Options Clearing Corporation (OCC), the first clearing house dedicated to options. Finally, 1973 saw the publication of the work of Fischer Black and Myron Scholes. This work was completed by Robert Merton, leading to the Black-Scholes-Merton model. This model is of capital importance for the evaluation of the price of options.

What’s an option?

There are two types of option contracts: calls and puts. Since these contracts can be both bought and sold, there are four basic transactions. Thus, in options trading, it is possible to either go long (buy a call contract, buy a put contract), or to be short (sell a call contract, sell a put contract). An option contract can therefore be defined as a contract that gives the counterparty buying the contract (the long) the right, but not the obligation, to buy or sell an asset (the underlying) at a predetermined price (the strike price), date (the maturity date) and amount (the nominal value). It is useful to note that the counterparty selling the contracts (the short) is in a completely different situation. This counterparty must sell or buy the underlying asset if the transaction is unfavorable to it. However, if the transaction is favorable, this counterparty will not receive any capital gain, because the counterparty buying the contract (the long) will not have exercised its call option. To compensate for the asymmetry of this transaction, the counterparty selling the option contracts (the short) will receive a premium at the time the contract is initiated. The selling counterparty therefore has a role similar to that of an insurance company, as it is certain to receive the premium, but has no control over the time of payment or the amount to be paid. This is why it is important to assess the amount of the premium.

The characteristic of an option contract

Options contracts can have as underlying assets financial assets (interest rates, currencies, stocks, etc.), physical assets (agricultural products, metals, energy sources, etc.), stock or weather indices, and even other derivatives (futures or forwards). The other important feature of an option contract is its expiration date. Options contracts generally have standardized expiry dates. Expiry dates can be monthly, quarterly or semi-annually. In most cases, the expiration date coincides with the third Friday of the expiration month. In addition, options whose only possible exercise date is the maturity date are called European options. However, when the option can be exercised at any time between signing and expiration, it is called an American option. Ultimately, what will drive the holder of an option contract to exercise his right is the difference between the underlying price and the strike price. The strike price is the purchase or sale price of the underlying asset. This price is chosen at the time the option contract is signed. The strike price will remain the same until the end of the option contract, unlike the price of the underlying asset, which will vary according to supply and demand. In organised markets, brokers usually offer the possibility to choose between several strike prices. The strike price can be identical to the price of the underlying asset. The option is then said to be “at-the-money” (or “at par”).

In the case of a call, if the proposed strike price is higher than the price of the underlying, the call is said to be “out of the money”.

Are you “in the money”?

Let’s take an example: a share is quoted at 10 euros. You are offered a call with a price of 11 euros. If we disregard the premium, we can see that a resale of the call, immediately after buying it, will result in a loss of one euro. For this reason, the call is said to be “out of the money”. On the other hand, when the strike price offered for a call is lower than the price of the underlying asset, the call is said to be “in the money”. Another example: the stock is still trading at 10 euros. This time you are offered a call with a strike price of 9 euros. If you disregard the premium, you can see that you earn one euro if you sell the call immediately after buying it. This is why this call is called “in the money”. Note that our potential gain of one euro is also called the “intrinsic value” of the call. Of course, the intrinsic value is only valid for “in the money” options. For puts, it is the opposite. A put is said to be “out of the money” if its strike price is lower than the price of the underlying asset.

Finally, a put is said to be “in the money” if its strike price is higher than the price of the underlying asset. If you are one of those people who think that you can make money with options by simply buying and selling calls or puts “in the money”, I have bad news for you! In reality, the premiums of the different contracts are calculated in such a way as to cancel out the advantage that “in the money” contracts offer over other contracts.

Useful resources

ISDA

About the author

Article written in July 2021 by Alexandre VERLET (ESSEC Business School, Master in Management, 2017-2021).

The NFTs, a new gold rush?

July 31, 2021 by Alexandre VERLET

The NFTs, a new gold rush?

In this article, Alexandre VERLET (ESSEC Business School, Master in Management, 2017-2021) explores the latest tech trend, which could revolutionize the art market and so much more.

These three letters are on everyone’s lips right now: NFT (for non-fungible token). But have you figured out what it’s really about? Let’s get into that special world of art, blockchain and people rich enough to buy a single tweet or jpg image. Jack Dorsey, CEO of Twitter, has put his very first tweet up for sale; the current auction is at $2.5 million (about €2 million). Sound like a lot? Canadian artist Grimes (and companion of the whimsical Elon Musk) has put up for sale an entire collection of digital works for nearly 6 million, while the most expensive single work sold to date is an animation showing Donald Trump, naked, being mocked by a blue bird. Its price? $6.6 million. But let’s get back to the basics and technique by detailing what an NFT, or non-fungible token, actually is.

Fungible vs non-fungible

First of all, let’s explain what a fungible element is and how it differs from a non-fungible element. The dictionary gives the following definition of the word “fungible”: things that are consumed by use and can be replaced by things of the same kind, quality, and quantity (e.g., commodities, cash).This means that it is something that has a value, but can be replaced by an equivalent of the same nature. For example, a coin that has no traceability, no serial number and will have the same value as a similar coin. Conversely, a non-fungible item cannot be replaced or substituted. For example, imagine a plane ticket: it is an object that can be consumed (in the sense that it can be bought), but its number, the fact that it is linked to a name and a particular seat on a given flight prevents it from being substituted for any other plane ticket.

What is a NFT (non-fungible token)?

An NFT applies this principle by adding a cryptographic layer based on an ERC (Ethereum Request for Comment) blockchain. This means that an NFT can be registered and exchanged just like an Ethereum (the second largest cryptocurrency after Bitcoin). This unique virtual token can then be used as a certificate for anything and everything, whether it is a real or digital good. Only its holder will be able to justify its possession, while it is possible to check the path of this token throughout its life. An NFT allows you to justify a purchase and prove its authenticity, whatever you have bought. Even a simple tweet, which may one day go down in history and be worth billions of dollars. The very principle of the blockchain ensures the encryption of information and its security, making each NFT unfalsifiable with today’s technical possibilities.

What can you buy with an NFT?

Technically, an NFT can be used as a certificate for anything. A famous painting, an official pair of sneakers… but where NFTs really come into their own is for digital assets. It’s easy to prove you own a painting or a pair of shoes, it’s harder to prove you bought a tweet from Jack Dorsey. But above all, it is a real revolution in the art world since any digital creation can now be identified and recognized as a unique work, thus immediately taking on value, like anything else that is unique. Some things, such as memes, can thus be considered as unique works.

Why buy a virtual image when you can copy it?

A question that often comes up is that of copying. What is the point of buying a 6.6 million dollar video or a single image when they are available everywhere on the net and can be downloaded and admired without any problem? Simply for the art and the joy of owning something unique. In a few seconds, you can find a reproduction of the Mona Lisa on Google Images and there is nothing to stop you from printing it and displaying it in your living room. However good your printer is, you will never own THE Mona Lisa by Leonardo da Vinci. Speculation and the principle of supply and demand do the rest and allow some works to be exchanged for several millions. And this is only the beginning.

The limits of NFTs

In front of this picture of the future that is being painted in real time before our eyes, there are a few fences linked to technical, ethical and legal limits. The biggest one being the cost of the blockchain. The Ethereum blockchain is currently particularly energy-intensive, which makes it expensive to use. From an ecological, ethical and economic point of view, relying on an ERC chain today is a miscalculation. “Today. Cryptocurrencies and blockchain in general are still in their infancy and the arrival of Ethereum 2.0 (a version that completely changes the principle of this blockchain in order to simplify and fluidify its operation expected in the next few years) could well solve these problems. Whatever you think about NFTs being a good investment or not, you will probably hear about them a lot in the coming years.

About the author

Article written in July 2021 by Alexandre VERLET (ESSEC Business School, Master in Management, 2017-2021).

Cryptocurrencies

July 31, 2021 by Alexandre VERLET

Cryptocurrencies

In this article, Alexandre VERLET (ESSEC Business School, Master in Management, 2017-2021) explores explores the latest and most fashionable investment trend.

They are everywhere on the news, in (young) people’s daily conversations, and probably in a corner of your head if you have already invested a bit of money in them. Cryptocurrencies are a daily drama, as it allows people to make or lose big money in record time. Everyone’s heard of it, but few people actually understand where cryptos come from and how they work. You may not necessarily need that to invest in them in the short term, as simply following Elon Musk on twitter might be a quicker and more efficient way to predict its evolution. But in the long run, and to understand the impact it will have on society, you need to know what’s going on. For some, it might become an actual currency in the coming years and will compete with the national currencies. For others, regulation will eventually tame cryptos and people will therefore lose interest in them. What’s for sure is that a public debate will arise at some point, and you might as well have the keys to understand cryptos so you can forge your own opinion. So here we go.

What is a cryptocurrency?

In a nutshell, it’s a virtual currency. What makes it a completely different and original currency is that it is not centrally managed; in other terms, it is the user who has full control over the cryptocurrency in their possession (peer-to-peer). This process is done through the implementation of Blockchain technology: the latter is a distributed and decentralized data storage and transmission technology at its core. The most frequently used analogy is that of a ledger that is accessible to all, indestructible and unpublishable once the data is embedded in the system. Like cryptocurrency, the Blockchain also relies on peer to peer to operate in a decentralized manner. Note that Blockchain can be used for much more than cryptocurrency; being a database, this technology represents a potentially huge evolution in the way we (businesses) deal with data. However, it was with the advent of Bitcoin, the first of many cryptocurrencies, that the distributed blockchain was seen as a potential successor to existing storage technology. The main cryptocurrencies are Bitcoin- the world’s most widely used and legitimate cryptocurrency-, Ethereum – founded in 2015 and known for its enhanced architecture using “smart contracts”-, Litecoin – released in 2011, similar to Bitcoin but with a higher programmed supply limit (84 million units vs 21 million).

Where do cryptos come from?

Before cryptos as we know them were invented, some early cryptocurrency proponents already shared the goal of applying cutting-edge mathematical and computer science principles to solve what they perceived as practical and political shortcomings of “traditional” currencies. It goes back to the 1980s when an American cryptographer named David Chaum invented a “blinding” algorithm that allowed for secure, unalterable information exchanges between parties, laying the groundwork for future electronic currency transfers. Then, the late 1990s and early 2000s saw the rise of more conventional digital finance intermediaries, such as Elon Musk’s Paypal. But no true cryptocurrency emerged until the late 2000s when Bitcoin came onto the scene. Bitcoin is widely regarded as the first modern cryptocurrency, because it combined decentralized control, user anonymity, record-keeping via a blockchain, and built-in scarcity. It all began in 2008, when Satoshi Nakamoto (an anonymous person or group of people) published a white paper about the Bitcoin. Nakamoto then released Bitcoin to the public. In 2010, the very first Bitcoin purchase was made: an Internet user exchanged 10,000 Bitcoins for two pizzas. At today’s prices, that would be the equivalent of about 500 million euros: that’s a lot of money for a pizza. By late 2010, dozens of other cryptocurrencies started popping out as more and more people started to mine and exchange cryptos. It grew in legitimacy when it became accepted as a means of payment by major companies, such as WordPress, Microsoft or Tesla. As of May 2021, the cryptos’ market cap is $2 trillion.

How do cryptos work?

There are several concepts that you should know about in order to get how cryptos work. Cryptocurrencies use cryptographic protocols, or extremely complex code systems that encrypt sensitive data transfers, which make cryptos them virtually impossible to break, and thus to duplicate or counterfeit the protected currencies. These protocols also mask the identities of cryptocurrency users.Then the crypto’s blockchain records and stores all prior transactions and activity, validating ownership of all units of the currency at all times. Identical copies of the blockchain are stored in every node of the cryptocurrency’s software network — the network of decentralized server farms, run by miners, that continually record and authenticate cryptocurrency transactions. The term “miners” relates to the fact that miners’ work literally creates wealth in the form of brand-new cryptocurrency units. Miners serve as record-keepers for cryptocurrency communities, using vast amounts of computing power, often manifested in private server farms owned by mining collectives that comprise dozens of individuals. The scope of the operation is quite similar to the search for new prime numbers, which requires tremendous amounts of computing power. Miners’ work periodically creates new copies of the blockchain, adding recent, previously unverified transactions that aren’t included in any previous blockchain copy — effectively completing those transactions. Each addition is known as a block, which consist of all transactions executed since the last new copy of the blockchain was created. Sincce the cryptocurrencies’ supply and value are controlled by the activities of their users and highly complex protocols built into their governing codes, not the conscious decisions of central banks or other regulatory authorities, which is why cryptos are said to be decentralized. Although mining periodically produces new cryptocurrency units, most cryptocurrencies are designed to have a finite supply — a key guarantor of value. Generally, this means miners receive fewer new units per new block as time goes on. For instance, if current trends continue, observers predict that the last Bitcoin unit will be mined sometime around 2150.

Why are cryptocurrencies so successful?

You may be wondering why crypto-currencies are gaining so much momentum today. With no intrinsic value, and no commodity to fall back on, economically speaking it makes no sense for this market to reach such an astronomical price. There are two rationales that often come up in the argument for cryptocurrencies. On the one hand, the anonymity via cryptography provided by blockchain technology: as there is very little regulation in this industry yet, one can end up with astronomical amounts of money without necessarily having to pay taxes on it, as there is no centralized body to follow what is going on. The second reason is more sociological: since there are people mining and trading cryptocurrencies, the logic is that they must have value. The consequence is that other people join the rush, and so on until it becomes a global phenomenon. You could call it a crowd movement, or a 21st century digital gold rush.

But these two reasons don’t necessarily answer the question of why Bitcoin and all these other cryptocurrencies are valuable. To get a clear answer, we need to go back to the basics of economics: any value applied to a commodity or currency is subjective. That is, if we, as individuals, see value in it, the commodity in question has value. The snowball effect resulting from a group of people’s growing interest in a commodity is at the origin of any bubble, and from that point of view cryptos are a massive bubble. Which does not mean that it is a bad investment: after all, a bubble is a bubble when it blows up, but it might never happen.

Summary

To sum up, if you want to invest in cryptocurrencies, there are a couple of things you should consider. First, if you’re aiming for the long-term (if you believe cryptocurrencies will keep increasing in value as “deflationary currencies”) or the short-term (pure speculation). Second, you should examine the specific characteristics of the cryptos and see which best fits you in terms of anonymity, growth potential and liquidity. Last but not least, follow the latest regulation announcements on cryptos, such as central banks or governments comments on cryptos, which are a pretty good indicator of the crypto’s evolution on both the long and short term.

About the author

Article written in July 2021 by Alexandre VERLET (ESSEC Business School, Master in Management, 2017-2021).

Inflation and the economic crisis of the 1970s and 1980s

July 31, 2021 by Alexandre VERLET

Inflation and the economic crisis of the 1970s and 1980s

In this article, Alexandre VERLET (ESSEC Business School, Grande Ecole Program – Master in Management, 2017-2021) goes back on the inflation issue of 1970’s/1980’s and the lessons it teaches us for the 2020’s.

In the developed capitalist countries, the fight against inflation became the top priority of economic policy in the 1970s. Georges Pompidou’s famous formula: “better inflation than unemployment” was buried for good. Inflation can be defined as the continuous and self-sustaining rise in the general price level. It is the result of a monetary struggle conducted by the various economic agents to maintain or increase their income or their capital: it has winners and losers. For economic decision-makers, inflation is a “sweet poison”: on the one hand, it is a factor of growth (by stimulating investment and consumption, and at the same time favoring production and employment); on the other hand, it is a danger for this same growth if the rise in prices gets out of hand (trade deficit, capital flight, ruin of savers). By what mechanisms does the inflationary growth of the 1960s give way to the rapid stagflation of 1973-1986?

Low inflationary growth was at the heart of the virtuous circle of the Trente Glorieuses

The Second World War and the post-war period were times of great inflationary pressure due to the large-scale expenditure by governments to finance the war effort, economic reconstruction and the establishment of the welfare state. France struggled with the problems of currency and price stability. Germany had the lowest inflation of the OECD countries since the monetary reform of 1948 and the priority given to a strong currency. Some countries, such as France, had chronic inflation. The debate raged in the years 1945-1952: a man like Mendès-France resigned from the government in 1945 to protest against monetary and budgetary laxity, stating that “distributing money to everyone without taking it from anyone is to maintain a mirage… “(extract from his letter of resignation, June 6, 1945). The growth of the 1950s and 1960s was generally not very inflationary in the developed countries: the Bretton Woods agreements ratified the stability of exchange rates around the dollar, the only reference currency convertible into gold. However, it was not until 1958 that European currencies regained their convertibility. Wartime periods remained inflationary: the Korean War (1950-53), for example, during which there was a rise in the price of raw materials, an increase in public spending in the United States and an increase in the circulation of dollars. From the beginning of the 1950s, once reconstruction had been completed, to the beginning of the 1960s, inflation fluctuated between 1 and 4% per year in the industrial countries. Moreover, Keynesian economic policies aimed to stimulate demand through deficit spending, which created inflation, and then to contain the pressure of demand when tensions were too great, so that inflation was limited. The alternation of stimulus (inflation) and austerity (deflation) took the form of the stop-and-go policy that characterized Great Britain and the United States in the 1950s. Consequently, in a period of full employment, a certain amount of “natural” unemployment is accepted in order to avoid too much pressure on wages and therefore on prices, as demonstrated by the British economist A.W. Philllips (Economica Journal, 1958). Inflation is in this perspective a lesser evil: it is seen as a painless way of financing growth: in fact, it works in favor of companies that go into debt, it has a favorable effect on their financial profitability. In a country such as France, it makes it possible to arbitrate social conflicts by defusing profit/wage tensions (the government negotiates both wage increases and low-cost credit).

The 1960s: the “inflationary spiral” begins to get out of control

From 1961-62 onwards, the developed industrial countries experienced an acceleration in price increases: a significant and lasting rise in inflation, from 3 to 5% until the early 1970s. During this period, there was no significant reduction in unemployment and even a slight increase in the number of job seekers: is this the end of the jobless era? In any case, the Phillips curve seemed to apply more and more poorly to the economic situation. There are several causes for this. Firstly, the growing importance of budget deficits: due to the use of deficit spending in the Keynesian logic; due to the implementation of the welfare state and social programs: for example, in the United States, the New Frontier programs of J.F. Kennedy and the Great Society of L. Johnson. Secondly, the deterioration of the international monetary system: devaluation of the pound sterling, crisis of the dollar at the end of the 1960s. Lastly, the wage increases outstripped productivity gains, which were slowing down: the “crisis of Fordism”: in other words, inflation through wage costs.

The 1973 and 1979 oil shocks

As seen previously, the 1970’s inflation is a consequence of economic phenomena already observed in the 1960’s. However, the two oil shocks were game changers. This time we are talking about cost inflation: the cost of energy supply is at stake, with the price of a barrel of oil multiplying by more than 11 in 1973 and 1979. This explains why inflation continues even when demand is lacking, when there is stagflation and part of the production capacity is unused. During classical crises, overproduction results in a general fall in the price level and a collapse of production, as shown by the Great Depression of the 1930s. On the contrary, during the crisis of the 1970s, prices rose continuously after the two oil shocks of 1973 and 1979, while production was very unstable (after a collapse in 1973-1974, it picked up again in 1975-1976). Inflation was now high: from an average of around 5% per year in the early 1970s, it rose to double-digit figures between 1973 and 1975, and again between 1979 and 1982.

The economic consequences of inflation

The crisis is industrial and commercial: companies’ profits collapse because of rising costs; their international competitiveness is severely damaged because of the relative rise in prices. The crisis is social: the unemployment curve follows that of inflation, but without showing any real inflection between 1973 and 1982: it calls into question the Phillips curve analysis, as there is a simultaneous rise in unemployment and inflation. The number of unemployed in the OECD rose from 10.1 million in 1970 to almost 33 million in 1983, which roughly corresponds to a tripling. European countries seem to be particularly affected: unemployment has multiplied by almost 4 in the same period. The crisis is also financial. On a national scale, part of the population is ruined by rapid inflation (savers, rentiers, farmers, employees), while another part makes significant gains (speculators). On an international scale, the debt of Third World countries literally exploded: from 130 billion dollars in 1973 to more than 660 billion dollars in 1983. Currencies tend to depreciate, which causes a generalized rise in prices: galloping inflation becomes global (Mexico for example). What’s more, Keynesian policies further reinforced the symptoms that had been combated, and were strongly criticized by the monetarist movement. Double-digit inflation makes Keynesian anti-crisis policies ineffective. For example, with an inflation rate of 13.5% in 1980 in France, the inflationary policy of President F. Mitterrand had disastrous effects on the competitiveness of French firms: it wiped out their margins, caused them to lose market share and finally penalized foreign trade. The fight against inflation became the main objective of monetarist policies. For Mr. Friedman, it is necessary to return to Phillips’ interpretation: it applies in a transitory way in the history of capitalism, when economic agents cannot predict or anticipate the rate of inflation. It is no longer a question of explaining inflation by the state of the labor market, but the opposite: it is the inflation anticipated by consumers that explains the tensions on the labor market; he shows that Keynesian recipes increase inflation through money creation without any effect on employment (because consumers anticipate it, consume less, which translates into a reduction in employment among producers). More inflation leads to more unemployment and, in an open economy, a decrease in the competitiveness of companies. The 1970’s crisis sheds light on how inflation works and to what extent the Phillips curve model can be applied to real-world situations. This useful to remember in a time when inflation is coming back for the first time in thirty years.

About the author

Article written in July 2021 by Alexandre VERLET (ESSEC Business School, Grande Ecole Program – Master in Management, 2017-2021).

Throwback to the Karlsruhe vs ECB fight, one year ago

July 31, 2021 by Alexandre VERLET

Throwback to the Karlsruhe vs ECB fight, one year ago

In this article, Alexandre VERLET (ESSEC Business School, Grande Ecole Program – Master in Management, 2017-2021) explores the legal thriller of 2020, in which the opposition between German orthodoxy and the flexible monetary policy came to light.

On May 5th 2020, the Constitutional Court of Karlsruhe released a decision. It stated that the government debt purchase programme performed by the European Central Bank was not in line with the ECB’s mandate under the European treaties. This decision came at a critical time when the ECB implemented a second large-scale quantitative easing programme in response to the Covid-19 economic crisis. You might have just heard the story in the media, but such an event is a key element of 2020’s economic and financial news, as the future of the ECB and the quantitative easing programme are at stake here.

Which part of the ECB’s action is being criticised by the German constitutional court?

Basically, the quantitative easing programme and its public debt paybacks are being targeted in the legal decision. Let us recall what the ECB did exactly. On 22 January 2015, the European Central Bank announced the implementation of an Expanded Asset Purchase Programme (APP or EAPP). This programme provided for asset purchases amounting to €60 billion per month in the secondary market. It aims to provide monetary support to the economy at a time when the ECB’s key interest rates have reached their zero lower bound, by easing financing conditions for companies and households. Investment and consumption should ultimately contribute to a return of inflation to levels close to 2%, in line with the objective of the ECB’s mandate. It should be noted that this programme is distinct from the Outright Monetary Transactions (OMT) initiated in 2012, the primary objective of which is financial stability by reducing the cost of financing for euro area countries. On 4 March 2015, in the context of the EAPP, the ECB Governing Council established a substantial sub-programme of purchases of Member States’ securities, the Public Sector Purchase Programme (PSPP). The PSPP provides for each national central bank to purchase eligible securities from public issuers in its own country according to the capital key for the ECB’s capital subscription. This sub-programme is by far the largest component of the ECB’s unconventional quantitative easing (QE) policy, accounting for almost 84% of the ECB’s net purchases in July 2019, with the remainder split between the other three sub-programmes – the asset-backed securities purchase programme (ABSPP – 8%), the covered bond purchase programme (CBPP3 – 1%) and, since March 2016, the corporate sector purchase programme (CSPP – 7%). 90% of purchases under the CSPP are made in domestic sovereign bonds and 10% are allocated to supranational issuers (international organisations, development banks, etc.).

A highly political decision

The recent decision by the Constitutional Court in Karlsruhe echoes the case brought by German businessman Heinrich Weiss at the start of the PSPP in 2015, which accused the ECB of overstepping its mandate by financing eurozone states – particularly the less creditworthy ones. The Karlsruhe court referred the case to the Court of Justice of the European Union (CJEU) in 2017, which found that the PSPP did not infringe the ECB’s prerogatives. In its decision of 5 May 2020 , the Constitutional Court in Karlsruhe now considers itself competent to rule on the non-compliance of the ECB programme, to contradict the CJEU and to question the Bundesbank’s participation in the PSPP. There is a political significance to this decision, which reflects a real split between Germany and the ECB since the European sovereign debt crisis. The Karlsruhe decision should therefore be understood as the latest disagreement between the traditional German and ECB views, which have been increasingly diverging since 2011. Initially, the ECB’s structures were modelled on the Bundesbank, both in terms of its political independence and its hierarchical mandate. Price stability is the ECB’s primary objective, enshrined in Article 127 of the TEU. To achieve this, its strategy combines both quantitative monetary targeting – again a legacy of the Bundesbank – and inflation targeting. The TEU also provides that “without prejudice to the primary objective of price stability, the ESCB [European System of Central Banks] shall support the general policies in the Union”; the ECB’s mandate thus does not exclude the possibility of a policy whose secondary effects support the growth and employment objectives defined by the Member States. Indeed, the ECB’s bulletins and communiqués show that growth and employment are constant concerns, and the sovereign debt crisis and the arrival of Mario Draghi endorsed a broader interpretation of the ECB’s mandate.The growing divergence between the Bundesbank and the ECB was marked by the recurrent clashes between Mario Draghi, the new ECB head from 2011, and Jens Weidmann, who was appointed President of the Bundesbank in the same year. Jens Weidmann was appointed by Angela Merkel following the resignation of Axel Weber, who was known for his sharp criticism of the debt buyback programme for fragile eurozone states, which would have cost him the ECB presidency for which he was a candidate. Considered at the time of his nomination as less dogmatic than his predecessor, Jens Weidmann nevertheless continued the fight of his predecessor and systematically criticised the ECB’s debt buyback programmes. Shortly after Axel Weber’s resignation, Jürgen Stark, the ECB’s chief economist, had himself resigned in protest at the accommodating policy then being pursued by Jean-Claude Trichet, considering that “a fiscal stimulus would only increase the level of debt and therefore only increase these risks”.

What could explain the German exception?

Germany is the Euro Zone’s most powerful member, so it is one of the few countries that can actually rebel agains the ECB. But France never did, so there is a clearly a German exception linked to Germany’s economic culture and financial history. When the ECB announced the resumption of quantitative easing on 12 September 2019, the dissension became even stronger. Sabine Lautenschläger, one of the six members of the ECB’s Executive Board, resigned shortly after Christine Lagarde, recently appointed as successor to Mario Draghi, indicated that she wanted to continue her predecessor’s policy. In an interview with the German tabloid Bild in September 2019, the Bundesbank President openly criticised the resumption of quantitative easing; the article in question was also made famous by its illustration depicting Mario Draghi as Count Dracula ready to “suck the blood of German savers”. Interestingly, concern about the adverse effects of lower rates on savers is a constant in German concerns, as is the sovereign debt of fragile eurozone states. German households save on average more than 18% of their income in 2019, one of the highest rates in Europe. From 2015 to 2020, the real interest rate was -0.9% in the eurozone, which may explain German dissatisfaction with the negative rate effects partly caused by quantitative easing. German public opinion was therefore unfavourable to the resumption of the programme in September 2019. In addition to this, there were some high-profile decisions, such as the Munich Savings Bank, which at the same time decided to pass on these negative rates to some of its customers’ deposits. It is therefore these concerns combined with the trauma of the sovereign debt crisis that have pushed German opinion towards support for a more orthodox monetary policy, which the Karlsruhe ruling has materialised, and all the more so after the announcement of the large-scale €750bn Pandemic Emergency Purchase Programme on 18 March 2020 by the ECB.

Is Karlsruhe right about the ECB not respecting its mandates?

Through the PSPP, the ECB fulfills its second objective of supporting Member States’ economic policies by enabling convergence of inflation rates but also convergence of the long-term interest rates of the euro area Member States and a more sustainable public debt path than in the absence of the PSPP (see above): in short, the fulfillment of the convergence criteria. In particular, the PSPP has led to a significant reduction in Member States’ sovereign bond yield spreads (see sovereign bond yield spreads graph): the standard deviation of different sovereign bond interest rates has fallen from almost 5% in 2015 (and 3% in 2016) to 1% in 2018 and 2019. So the ECB’s action via the PSPP and the PEPP seems today to live up to expectations from an economic point of view. Nevertheless, Karlsruhe considers that there is a potential violation of European treaties because the ECB and the European court of justice – which approved the PSPP in December 2018 – did not provide evidence that proportionality had been duly considered. Karlsruhe cites easier financing conditions for member states or the banking system, and ‘penalizing’ savers, but the court seems to ignore that the PSPP’s effects are not any different from those of other ECB instruments. Either the PSPP does not violate the proportionality principle, or all ECB instruments do. What’s more, if for instance, in considering an interest rate rise to counter inflationary pressures, the ECB found this would produce losses for bondholder or increase unemployment, then the ECB’s would be considering objectives that are explicitly out of its mandate. What the German court reproaches the ECB is rather paradoxical and the economic basis of the legal decision is weak , but it definitely show that there is a need for a redefinition of the ECB’s mandate. It could become a non-hierarchical dual mandate, more similar to the US Federal Reserve model. The ECB would thus have a clear function of pursuing a price stability objective combined with a full employment objective. The mandate could be materialized not by the definition of a monetary target, the evolution of the monetary aggregate M3 and inflation, as it currently is, but by the establishment of an implicit nominal anchor on the so-called “neutral” rate theoretically allowing the euro area to reach its potential growth. Nevertheless, such evolution of the ECB’s mandate will have to wait until the end of the Covid-19 crisis, and nothing has prevented the Bundesbank from implementing the ECB’s policy since Karlsruhe released its decision one year ago. Adding to that, the possible return of inflation might jeopardize the sustainability of further quantitative easing programmes. To be continued..

About the author

Article written in July 2021 by Alexandre VERLET (ESSEC Business School, Grande Ecole Program – Master in Management, 2017-2021).

Value at Risk

Introduction

Computational methods

Parametric methods

The historical distribution

Monte Carlo Simulations

Limitations of VaR

VaR doesn’t measure worst-case loss

VaR is not additive

VaR is only as good as its assumptions and input parameters

Different methods give different results

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Useful Resources

Academic research articles

About the author

Plain Vanilla Options

Introduction

Types of options

Call options

Put options

Types of exercise

American options

European options

Bermudan options

Moneyness

In-the-money options

Out-of-the-money options

At-the-money options

Option writers

Benefits

Option Trading

Option pricing

Use of options

Hedging

Speculation

Volatility

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Useful Resources

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Derivatives Market

Introduction

Main types of derivative contracts

Forwards and Futures

Options

Swaps

Exchange-traded vs Over-the-counter Derivatives Market

Exchange-traded derivatives markets

Over-the-counter markets

Types of market participants

Hedgers

Speculators

Arbitrageurs

Margin Traders

Criticism of derivatives

Conclusion

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Useful resources

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Linear Regression

Definition

Application in finance

Assumptions in the linear regression model

Continuity

Linearity

No outliers in data set

Homoscedasticity

Normally-distributed residuals

Ordinary Least Squares (OLS)

R-squared values

Useful Resources

Related Posts

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The Black-Scholes-Merton model

Introduction

History

Option pricing with BSM

The BSM formula

Example – Call and Put option pricing using Black-Scholes-Merton model

Conclusion

Limitations of the BSM model