Pricing S&P 500 Index Options under Stochastic Volatility with the Indirect Inference Method

Abstract

This paper studies the price of S&P 500 index options by using Heston's (1993) stochastic volatility option pricing model. The Heston model is calibrated by a two-step estimation procedure to incorporate both the information from time-series asset returns and the information from cross-sectional option data. In the first step, the recently developed simulation-based 'indirect inference method' is used to estimate the structural parameters that govern the asset return distribution; in the second step, the risk premium, λ, the spot variance, vt, and the correlation coefficient between the asset return and its volatility, ρ, are estimated by a nonlinear least-squares method that minimizes the sum of the squares of the error between the cross-sectional option price and the corresponding model price. The model performance is assessed by directly comparing the computed option model price with the market price. We find that both the Black–Scholes model and the Heston model overprice the out-of-the-money options and underprice the in-the-money options, but the degree of the bias is different. The Heston model significantly outperforms the Black–Scholes model in almost all moneyness-maturity groups. On average, the Heston model can reduce pricing errors by about 25%. However, pricing bias still exists in the Heston model. In particular, the Heston model always overprices short-term options, indicating that some other factors, such as the random jump, may also be needed to explain the option price.