Consistent time‐homogeneous modeling of SPX and VIX derivatives

Abstract

This paper shows how to recover a stochastic volatility model (SVM) from a market model of the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore are better suited for pricing VIX futures and VIX 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. Therefore, consistent modeling for both SPX and VIX should involve an SVM that can be obtained by inverting the market model. This paper's main result is a method for the recovery of a stochastic volatility function by solving an inverse problem where the input is the VIX function given by a market model. Analysis will show conditions necessary for there to be a unique solution to this inverse problem. The models are consistent if the recovered volatility function is non-negative. Examples are presented to illustrate the theory, to highlight the issue of negativity in solutions, and to show the potential for inconsistency in non-Markov settings.

The Rough Hawkes Heston Stochastic Volatility Model

“…For VIX futures and Exchange-Traded products these features are studied in [10]. The complexity of volatility markets is also exemplified by the difficulty to jointly model the behavior of the volatility smiles of vanilla options written on the underlying and its volatility index, see for instance [6,64,68]. This longstanding puzzle is known as the S&P 500 (SPX)/VIX calibration puzzle.…”