This article first describes the nature of an option gamma as a second partial derivative of the option value with respect to the underlying price. The article also shows the analytical formulas for the gamma of plain-vanilla European-style call or put options, based on the Black/Scholes (1973) model. This is illustrated by means of an example. Moreover, major determinants of the gamma of an option are pointed out (e.g. the maturity and the price of the underlying), which directly leads to the coverage of the gamma surface. Here, the impact of the moneyness and maturity of the option on its gamma are elaborated. In addition, the gamma in the Merton option pricing model with dividends is explicitly determined. Afterwards, the article describes how to calculate the gamma of a portfolio of derivatives, complemented by a numerical example. The technical terms "gamma-long", "gamma-short" and "gamma-neutral" are defined. The final part deals with the use of gamma for risk management (e.g. the "delta-gamma-normal method" of the Value-at-Risk). This part also shows how to make a portfolio delta-gamma neutral, again including a short numerical example.
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Black/Scholes modelEuropean-style call optionEuropean-style put optionGreek variablesTaylor seriesdelta-gamma-normal methodgamma surfacepartial derivativeportfolio gammarisk management