Index Option Pricing Models with Stochastic Volatility and Stochastic Interest Rates

Abstract

While the stochastic volatility (SV) generalization has been shown to improve the explanatory power compared to the Black-Scholes model, the empirical implications of the SV models on option pricing have not been adequately tested. The purpose of this paper is to first estimate a multivariate SV model using the efficient method of moments (EMM) technique and then investigate the respective effect of stochastic interest rate, systematic volatility and idiosyncratic volatility on option prices. We compute option prices using both underlying historical volatilities obtained through reprojection and volatilities implied from observed option prices and gauge each model’s performance through direct comparison with observed market option prices. Our results suggest: (i) While theory predicts that the short-term interest rates are strongly related to the systematc volatility of the consumption process, our estimation results suggest that the short-term interest rate fails to be a good proxy of the systematic factor; (ii) While allowing for stochastic volatility of stock returns can in general reduce the pricing errors and allowing for asymmetry or “leverage effect” in the SV models does help to explain the skewness of the volatility “smile”, allowing for stochastic interest rate has minimal impact on option prices in our case; (iii) Similar to Melino and Turnbull (1990), our empirical findings strongly suggest the existence of a non-zero risk premum for stochastic volatility of stock returns. Allowing for non-zero risk-premium of stochastic volatility and based on implied volatility, the SV models can largely reduce the option pricing errors, suggesting the importance of incorporating the information in the options market in pricing options.