{"id":4934,"date":"2021-08-08T21:15:38","date_gmt":"2021-08-08T20:15:38","guid":{"rendered":"https:\/\/www.simtrade.fr\/blog_simtrade\/?p=4934"},"modified":"2026-02-25T21:13:20","modified_gmt":"2026-02-25T21:13:20","slug":"option-greeks-gamma","status":"publish","type":"post","link":"https:\/\/www.simtrade.fr\/blog_simtrade\/option-greeks-gamma\/","title":{"rendered":"Option Greeks – Gamma"},"content":{"rendered":"

\"Akshit<\/a><\/p>\n

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

Introduction<\/h2>\n

Gamma is a type of option Greek which is used to compute the sensitivity or rate of change of delta (\u0394) 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 (\u0393), the gamma is defined by<\/p>\n

\"Formula<\/p>\n

Where (\u0394) is the delta of the option and S<\/i> the price of the underlying asset.<\/p>\n

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<\/p>\n

\"Formula<\/p>\n

Where V<\/i> is the value of the option and S<\/i> the price of the underlying asset.<\/p>\n

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.<\/p>\n

The gamma as a function of the price of the underlying asset for a call option is given below.<\/p>\n

Figure 1. Gamma of a call option.
\n\"Gamma
\nSource: computation by the author (Model: Black-Scholes-Merton).<\/p>\n

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.<\/p>\n

Calculating gamma for call and put options<\/h2>\n

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:<\/p>\n

\"Formula<\/p>\n

Where,N\u2019<\/i>d<\/i>1<\/sub> represents the first order derivative of the cumulative distribution function of the normal distribution given by:<\/p>\n

\"First_<\/p>\n

and d<\/i>1<\/sub> is given by:<\/p>\n

\"Formula<\/p>\n

Where S<\/i> is the price of the underlying asset (at the time of valuation of the option), \u03c3 the volatility in the price of the underlying asset, T<\/i> time to option\u2019s maturity, K<\/i> the strike price of the option contract and r<\/i> the risk-free rate of return.<\/p>\n

Excel pricer to calculate the gamma of an option<\/h2>\n

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.<\/p>\n

\"Download<\/a><\/p>\n

Delta-gamma hedging<\/h2>\n

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\u2019s value against sharp price movements in the price of the underlying asset.<\/p>\n

\"Formula<\/p>\n

Limitations of gamma hedging<\/h2>\n

The limitation of gamma hedging includes the following:<\/p>\n