Stochastic Volatility Option Pricing

Abstract

This paper examines alternative methods for pricing options when the underlying security volatility is stochastic. We show that when there is no correlation between innovations in security price and volatility, the characteristic function of the average variance of the price process plays a pivotal role. It may be used to simplify Fourier option pricing techniques and to implement simple power series methods. We compare these methods for the alternative mean-reverting stochastic volatility models introduced by Stein and Stein (1991) and Heston (1993). We also examine the biases in the Black-Scholes model that are eliminated by allowing for stochastic volatility, and we correct some errors in the Stein and Stein (1991) analysis of this issue. Empirical evidence on underlying asset prices and on their derivatives strongly suggests that asset price volatility is stochastic. The pricing of options under this condition is, therefore, an important problem. In this paper, we study alternative option pricing methods that account for stochastic volatility. We are particularly concerned with power series methods, which we show to be quite accurate and easy to implement for alternative volatility specifications. We point out their relation with recently proposed exact methods. This permits a precise assessment of the approximation involved in the simple power series approach, and of the impact of stochastic volatility on option prices. Following Garman (1976), stochastic volatility option prices must satisfy a bivariate fundamental partial differential equation (PDE) in the two state variables, security price and volatility. Since volatility is not spanned by assets in the econ? omy, the volatility risk may not be eliminated by arbitrage methods. Therefore, its market price explicitly enters into the PDE. The associated risk premium is *Owen Graduate School of Management, Vanderbilt University, Nashville, TN 37203, and Universita di Siena, Facolta di Scienza Economiche e Bancarie, 53100 Siena, Italy, respectively. Part of this research was supported from a grant from the Dean's Fund and the Financial Markets Research Center at the Owen School of Management, Vanderbilt University. Some of this research was completed while Roma was visiting the Anderson Graduate School of Management ofthe University of California at Los Angeles. We thank participants of the Finance, Accounting, and Economics Workshop at the Owen School of Management, Vanderbilt University for useful comments. We appreciate the suggestions of Roger Huang, Craig Lewis, Ron Masulis, and Walter Torous. We thank Joanne Ball for editorial