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).

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).

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).

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

Related posts on the SimTrade blog

Useful Resources

About the author

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

Related posts on the SimTrade blog

Useful resources

Academic research

About the author

Call-Put Parity

Introduction

Call-put parity relation

Demonstration

Application

Implication

Example of application of the call-put parity

Related posts on the SimTrade blog

Useful resources

Academic research

About the author

Option Greeks – Theta

Introduction

Intrinsic and time value of an option contract

Calculating Theta for call and put options

Excel pricer to calculate the theta of an option

Example for calculating theta

Related Posts on the SimTrade blog

Useful resources

About the author

Option Greeks – Vega

Introduction

Calculating the vega for call and put options

Example for calculating vega

Excel pricer to calculate the vega of an option

Related posts ont he SimTrade blog

Option pricing and Greeks

Useful resources

About the author

Option Greeks – Gamma

Introduction

Calculating gamma for call and put options

Excel pricer to calculate the gamma of an option

Delta-gamma hedging

Limitations of gamma hedging

Related posts in the SimTrade blog

Useful resources

About the author

Option Greeks – Delta

Introduction

Delta formula

Call option

Put option

Delta as a function of the price of the underlying asset