Fast Analytic Option Valuation with GARCH

Abstract

One of the key ways in which asset returns in the real world depart from the lognormal diffusions assumed by Black and Scholes is that volatility has been found to be clearly time varying. Formal stochastic volatility models introduce a second Brownian motion variable to govern volatility changes, which leads to considerable additional complexity in solving the models and, importantly, gives up the dynamic market completeness that “no-arbitrage pricing” depends on. GARCH models occupy an intermediate position between models in which volatility is a fixed constant and those in which volatility is a second, unspanned, stochastic process, because the same Brownian motion determines both the returns and the evolution of volatility under GARCH. But current procedures for econometrically fitting a GARCH option pricing model are complex and computationally intensive. Heston and Nandi improved matters by developing a GARCH option pricing model whose characteristic function has a convenient form, but solving the model still requires considerable computation time. In this article, Mazzoni takes their approach further, starting from the cumulant-generating function and introducing a set of approximations that leads to analytic equations for an option’s value and its Greeks. The results are highly accurate, and the computation time is orders of magnitude faster than that of existing methods for pricing options under GARCH.