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. © 2010 The Author.