GARCH Option Valuation: Theory and Evidence

Abstract

Black and Scholes assumed stock volatility was a constant and known parameter, but the evidence for stochastically time-varying volatility is overwhelming. Most stochastic volatility models introduce a second stochastic process for variance, driven by shocks that are (negatively) correlated with the return process. But this makes volatility a latent variable and the volatility risk factors cannot be hedged, so they should logically carry risk premia. Estimating the variance equation requires sophisticated econometric methods whose small sample properties may be questioned. Generalized autoregressive conditional heteroskadasticity (GARCH) also offers a specification in which volatility is random, but both volatility and returns are driven by the same shocks, so the variance equation is fitted directly on the observed returns.

This article presents a comprehensive review of option valuation under GARCH, covering both theory and empirical estimation. The basic GARCH specification captures time-varying volatility, and the authors review modifications that incorporate known characteristics of real-world return processes, including asymmetric response to up and down shocks, reversion toward a slowly time-varying long-run mean, non-affine and/or non-Gaussian shocks, and return processes subject to multiple shocks, including jumps.

GARCH models are typically fitted to series of stock returns, but the authors strongly recommend using both stock returns and option data in model fitting. This helps tie the statistical variance estimates to the markets for variance-dependent securities and exploits information from many additional prices available in the options market. They describe practical suggestions for optimizing and review methods for pricing options under GARCH, including early exercise of American options. The article ends with a series of Appendices giving Matlab code for option pricing under alternative GARCH specifications.