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

Paper: arXiv 2212.10917

Authors: Eduardo Abi Jaber, Camille Illand, Shaun, Li

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.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 7.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly novel yet tractable stochastic volatility model using a quintic polynomial of an OU process to solve the classic joint SPX-VIX calibration problem. The mathematical development is rigorous and analytically elegant, supported by strong calibration performance against real market option smiles.

Research Flowchart

  flowchart TD
    A["Research Goal: Jointly calibrate SPX & VIX smiles efficiently with few parameters"] --> B["Inputs: SPX/VIX term structures & 6-parameter Quintic OU process"]
    B --> C["Model Design: Polynomial (deg 5) function of fast mean-reverting OU process"]
    C --> D["Pricing VIX Derivatives: Exact evaluation via Gaussian density integration"]
    C --> E["Pricing SPX Derivatives: Monte Carlo with antithetic & control variates"]
    D --> F["Key Outcome: Remarkable joint SPX-VIX fit and exact volatility simulation"]
    E --> F