Consistent Pricing of S&P500 and VIX Options in Gatheral's Model

Abstract

In this paper we introduce a consistent framework for pricing options written on the S&P500 stock index and volatility index (VIX). Gatheral (2007, 2008) proposes a three-factor stochastic volatility model to achieve this goal. However, the non-affine structure of the model leads to analytical intractability so closed-form pricing formulae may not exist either for S&P500 options, or for VIX options. The Monte Carlo simulation method adopted in Gatheral is rather inefficient in terms of calculation and model calibration. This study proposes two analytical asymptotic formulae to efficiently price S&P500 options and VIX options, respectively, based on Gatheral's three-factor stochastic volatility model. By applying singular perturbation techniques, our formulae are obtained by solving a set of partial differential equation systems. We then rigorously justify the convergence of the asymptotic formulae. In addition, we present some numerical examples to demonstrate that our asymptotic formulae can achieve high efficiency and accuracy for a large class of options with relative short tenor.