#09-05
The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work so Well
Peter Christoffersen, Steven Heston and Kris Jacobs, February 2009
Abstract: 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 fluctuate
largely independently. While single-factor stochastic volatility models can capture the slope of
the smile, they cannot explain such largely independent fluctuations 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 flexible modeling of the time variation in the smirk, the
model also provides more flexible 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 fit results from improvements in the modeling of the term structure
dimension as well as the moneyness dimension.
Keywords: Stochastic correlation; stochastic volatility; equity index options; multifactor model; persistence; affine; out-of-sample.
JEL classifications: G12
