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).

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

Call option

Put option

Excel pricer to calculate the delta of an option

Delta Hedging

Limitations of delta hedging

Transaction cost

Illiquid Markets

Example for calculating delta

Related posts on the SimTrade blog

Option pricing and Greeks

Useful resources

Research articles

Books

About the author