# Consistent Time-Homogeneous Modeling of SPX and VIX Derivatives

## TL;DR

This paper develops a method to recover a consistent stochastic volatility model for SPX and VIX derivatives from market data, ensuring the models are aligned and non-negative, with analysis of solution uniqueness.

## Contribution

It introduces a novel inverse problem approach to derive a stochastic volatility function from market models of VIX futures, ensuring consistency with SPX derivatives.

## Key findings

- Method for recovering stochastic volatility functions from market models.
- Conditions for uniqueness of the inverse solution.
- Illustrations of potential negativity and inconsistency issues.

## 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.