Quintic Ornstein-Uhlenbeck Model for Joint SPX and VIX Smile Calibration

Summary

This document describes a stochastic volatility model in which volatility is a fifth-degree polynomial of a single, fast mean-reverting Ornstein-Uhlenbeck process with substantial volatility of volatility. It is designed to fit SPX and VIX volatility smiles jointly, using a small set of effective parameters and an input curve to match selected term structures. The authors also examine alternative input-curve specifications and time-varying parameters for improving fits at maturities beyond one year.

The model’s computational design supports practical pricing: squared VIX is polynomial in the Ornstein-Uhlenbeck state, enabling VIX derivative pricing by integration against a Gaussian density. Volatility can be simulated exactly, while SPX derivatives can be priced with Monte Carlo methods using variance-reduction techniques. The excerpt reports strong joint fitting capability but provides no calibration data, error measures, or comparison with competing models, so the quality and generality of the fits cannot be assessed here.

Key ideas

• Volatility is modeled as a fifth-degree polynomial of a single Ornstein-Uhlenbeck process.
• The model is designed to calibrate SPX and VIX smiles jointly with a compact parameterization.
• Alternative input curves and time-dependent parameters can help match term structures and longer maturities.
• The polynomial form of squared VIX supports efficient pricing of VIX derivatives.
• Exact volatility simulation and variance-reduced Monte Carlo are proposed for pricing applications.

Tags

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


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









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.

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