Volatility Dynamics for the S&P500: Evidence from Realized Volatility, Daily Returns and Option Prices

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 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 fitting 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, and it provides a more realistic volatility term structure and implied volatility smirk.

Keywords: Stochastic volatility, option valuation, particle filtering, skewness, kurtosis, mean reversion

JEL Classification: G12