The quintic Ornstein-Uhlenbeck volatility model that jointly calibrates SPX & VIX smiles

TL;DR

This paper introduces a simple yet effective quintic Ornstein-Uhlenbeck volatility model that jointly calibrates SPX and VIX smiles with minimal parameters, offering efficient pricing and simulation methods.

Contribution

The paper presents a novel quintic polynomial volatility model that achieves joint SPX-VIX smile calibration with only six parameters and maintains tractability for pricing and simulation.

Findings

Successfully fits SPX and VIX smiles jointly

Provides efficient pricing for VIX derivatives

Enables exact simulation of the volatility process

Abstract

The quintic Ornstein-Uhlenbeck volatility model is a stochastic volatility model where the volatility process is a polynomial function of degree five of a single Ornstein-Uhlenbeck process with fast mean reversion and large vol-of-vol. The model is able to achieve remarkable joint fits of the SPX-VIX smiles with only 6 effective parameters and an input curve that allows to match certain term structures. We provide several practical specifications of the input curve, study their impact on the joint calibration problem and consider additionally time-dependent parameters to help achieve better fits for longer maturities going beyond 1 year. Even better, the model remains very simple and tractable for pricing and calibration: the VIX squared is again polynomial in the Ornstein-Uhlenbeck process, leading to efficient VIX derivative pricing by a simple integration against a Gaussian density; simulation of the volatility process is exact; and pricing SPX products derivatives can be done efficiently and accurately by standard Monte Carlo techniques with suitable antithetic and control variates.