Option Valuation with Conditional Heteroskedasticity and Nonnormality

Abstract

We provide results for the valuation of European-style contingent claims for a large class of specifications of the underlying asset returns. Our valuation results obtain in a discrete time, infinite state space setup using the no-arbitrage principle and an equivalent martingale measure (EMM). Our approach allows for general forms of heteroskedasticity in returns, and valuation results for homoskedastic processes can be obtained as a special case. It also allows for conditional nonnormal return innovations, which is critically important because heteroskedasticity alone does not suffice to capture the option smirk. We analyze a class of EMMs for which the resulting risk-neutral return dynamics are from the same family of distributions as the physical return dynamics. In this case, our framework nests the valuation results obtained by Duan ( 1995 ) and Heston and Nandi ( 2000 ) by allowing for a time-varying price of risk and nonnormal innovations. We provide extensions of these results to more general EMMs and to discrete-time stochastic volatility models, and we analyze the relation between our results and those obtained for continuous-time models.