Pricing Volatility Derivatives Using Space Scaled Levy Processes

The VIX index measures the one-month risk-neutral forward volatility of the S&P500 (SPX) index. While Lévy processes such as the CGMY process can price options on the underlying stock or index, they implicitly assume a constant forward volatility. This makes them unsuitable for pricing options on VIX. We propose a model within the one dimensional Markovian framework for pricing VIX and SPX options simultaneously. We introduce space dependence of volatility by scaling the CGMY process with a leverage function. The resultant process can consistently price options on SPX and VIX of a given maturity. We also perform surface calibrations of options on the two indices separately. We explore the properties of the implied distribution of the SPX from both indices and conclude that the VIX index under-weighs small jumps as compared to large jumps as well as the skewness of the SPX index .