Modeling Volatility Risk in Equity Options Market: A Statistical Approach

This chapter provides a cross-sectional analysis of US option markets based on implied volatility data from August 2004 to August 2013. We analyse the implied volatility surface (IVS) for each security in the OptionMetrics database. We use implied volatility data across 13 deltas and four expiration dates. Employing methods from principal component analysis (PCA), and results from random matrix theory (RMT), we identify the significant eigenvalues of the correlation matrix of implied volatilities and conclude that, usually, three principal components suffice to reproduce the IVS. In this way we reduce dimensionality of the options market without loosing meaningful information. From this analysis we classify equities into those carrying mostly “systemic” risk and into those carrying mostly “idiosyncratic” risk.Based on the PCA results, we formulate a model which can be used to describe the dynamics of the joint statistics of the IVS of all US options, yet is compact and computationally feasible. Using 9 volatility points to represents each IVS, the model offers significant dimension reduction for each asset as well as for all assets in aggregate.We conclude with a PCA study of the correlation matrix of the entire cross-section equities and options market. We find that the the number of significant factors driving the US equities and options market are as follow:1. Equities in SPX: 15 significant factors (account for 55% of variance, out of an initial 440 variables).2. Equities and options with underlying in SPX: 84 significant factors (account for 55% of variance, out of an initial 4400 variables).3. Equities in OptionMetrics: 20 significant factors (account for 24% of variance, out of an initial 3,141 variables).4. All equities and options with underlying asset in OptionMetrics: 108 significant factors (account for 50% of variance, out of an initial 31,410 variables).