#09-06
Models for S&P500 Dynamics: Evidence from Realized Volatility, Daily Returns, and Option Prices
Peter Christoffersen, Kris Jacobs and Karim Mimouni, December 2008
Abstract: Most recent empirical option valuation studies build on the affine square root (SQR) stochastic volatility model. The SQR model is a convenient choice, because it yields closed-form
solutions for option prices. However, relatively little is known about the resulting biases.
We investigate alternatives to the SQR model, by comparing its empirical performance with
that of five different but equally parsimonious stochastic volatility models. We provide empirical evidence from three different sources: realized volatilities, S&P500 returns, and an
extensive panel of option data. The three sources of data we employ all point to the same
conclusion: the SQR model is severely misspecified. The best of the alternative volatility
specifications is a model with linear rather than square root diffusion for variance, which
we refer to as the VAR model. This model captures the stylized facts in realized volatilities,
it performs well in fittting various samples of index returns, and it has the lowest option
implied volatility mean squared error in- and out-of-sample. It fits the option data better
than the SQR model in several dimensions: it improves the fit of at-the-money options, it
provides a more realistic volatility term structure and implied volatility smirk.
Keywords: Stochastic volatility; option valuation; particle filtering; skewness; kurtosis; mean reversion.
JEL classifications: G12
