Consistent Inter-Model Specification for Time-Homogeneous SPX Stochastic Volatility and VIX Market Models

Abstract

This paper shows how to recover stochastic volatility models (SVMs) from market models for the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore they are better-suited for pricing VIX futures and derivatives. But the VIX itself is a derivative of the S&P500 (SPX) and it is common practice to price SPX derivatives using an SVM. Hence, a consistent model for both SPX and VIX derivatives would be one where the SVM is obtained by inverting the market model. This paper's main result is a method for the recovery of a stochastic volatility function as the output of an inverse problem, with the inputs given by a VIX futures market model. Analysis will show that some conditions need to be met in order for there to not be any inter-model arbitrage or mis-priced derivatives. Given these conditions the inverse problem can be solved. Several models are analyzed and explored numerically to gain a better understanding of the theory and its limitations.