An analytical approximation to the option formula for the GARCH model

Abstract

This article derives an analytical approximation to the option formula for a spot asset price whose conditional variance equation follows a nonlinear asymmetric GARCH (NGARCH) process. The approximate option formula, which is just a volatility adjustment in comparison to the Black–Scholes (BS) formula, is very simple and provides the volatility term structure of spot asset prices. Also, the formula shows that the most characteristic feature of an NGARCH model appears in the vega of a European option, which depends on both the spread between the long-run variance and the current one and a parameter reproduced from the stationary property of the conditional variance. This methodology can be easily extended to an option formula for the generalized GARCH process.