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. 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 and simulation of the volatility process is exact.

For pricing SPX products, we show that the Quintic model is part of a larger class of stochastic volatility model where the volatility is driven by a linear function of the signature of a Brownian motion enhanced with the running time. For this larger class of models, we develop pricing and hedging methodologies using Fourier inversion techniques on the characteristic function which is known up to an infinite-dimensional Riccati equation. We illustrate our method on numerical examples for fast pricing, hedging and calibration of vanilla and path-dependent options in several classes of Markovian and Non-Markovian models.

Based on joint works with Louis-Amand Gérard, Camille Illand & Shaun Li.

Getting here