Title: 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.