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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. Although single-factor stochastic volatility models can capture the slope of the smirk, 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 stochastic volatility 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.
Key words: stochastic correlation; stochastic volatility; equity index options; multifactor model; persistence; affine; out-of-sample
History: Received March 2, 2007; accepted June 16, 2009, by David A. Hsieh, finance. Published online in Articles in Advance September 11, 2009.
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1. Introduction
An extensive empirical literature has documented the empirical biases of the Black and Scholes (1973) option valuation model for the purpose of the valuation of equity index options. Most prominently among these biases, observed market prices for out-of-the-money put prices (and in-the-money call prices) are higher than Black-Scholes prices. This stylized fact is known as the volatility "smirk." Implied volatilities for atthe- money options also contain a term structure effect that cannot be explained by the Black-Scholes model.
Perhaps the most popular approach to modeling the smirk is the use of stochastic volatility models that allow for negative correlation between the level of the stock return and its variance.1 This negative correlation captures the stylized fact that decreases in the stock price are associated with larger increases in variance than similar stock price increases (Black 1976, Christie 1982). This stylized fact...