The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work So Well

State-of-the-art stochastic volatility models generate a “volatility smirk” that explains why out-of-the-money index puts have high prices relative to the Black-Scholes benchmark. These models also adequately explain how the volatility smirk moves up and down in response to changes in risk. However, the data indicate that the slope and the level of the smirk ‡uctuate largely independently. While single-factor stochastic volatility models can capture the slope of the smile, they cannot explain such largely independent ‡uctuations in its level and slope over time. We propose to model these movements using a two-factor stochastic volatility model. Because the factors have distinct correlations with market returns, and because the weights of the factors vary over time, the model generates stochastic correlation between volatility and stock returns. Besides providing more ‡exible modeling of the time variation in the smirk, the model also provides more ‡exible modeling of the volatility term structure. Our empirical results indicate that the model improves on the benchmark Heston model by 24% in-sample and 23% out-of-sample. The better …t results from improvements in the modeling of the term structure dimension as well as the moneyness dimension. JEL Classi…cation: G12 Keywords: Stochastic correlation; stochastic volatility; equity index options; multifactor model; persistence; a¢ ne; out-of-sample. Christo¤ersen and Jacobs are with the Desautels Faculty of Management, McGill University. They are also a¢ liated with CIRANO and CIREQ and want to thank FQRSC, IFM and SSHRC for …nancial support. Christo¤ersen also thanks CREATES and CBS for their hospitality. Heston is with the R.H. Smith School of Business, University of Maryland. We would like to thank Chayawat Ornthanalai, Nick Pan, and Gregory Vainberg for expert research assistance, and Gurdip Bakshi, Haitao Li and Nour Meddahi for helpful comments. Any remaining inadequacies are ours alone. Correspondence to: Peter Christo¤ersen, McGill University, 1001 Sherbrooke Street, Montreal, Quebec, Canada H3A 1G5. Phone: 514-398-2869. Email: peter.christo¤[email protected].