Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models

TL;DR

This paper develops a mathematical framework using Fourier-Laplace transforms for polynomial Ornstein-Uhlenbeck volatility models, providing solutions to complex Riccati equations and demonstrating practical applications in option and volatility swap pricing.

Contribution

It introduces a novel approach connecting Fourier-Laplace transforms with infinite dimensional Riccati equations in polynomial OU models, including new numerical schemes for practical financial derivatives pricing.

Findings

Efficient numerical scheme for Riccati equations

Accurate pricing of SPX options and volatility swaps

Successful calibration to real market data

Abstract

We consider the Fourier-Laplace transforms of a broad class of polynomial Ornstein-Uhlenbeck (OU) volatility models, including the well-known Stein-Stein, Sch\"obel-Zhu, one-factor Bergomi, and the recently introduced Quintic OU models motivated by the SPX-VIX joint calibration problem. We show the connection between the joint Fourier-Laplace functional of the log-price and the integrated variance, and the solution of an infinite dimensional Riccati equation. Next, under some non-vanishing conditions of the Fourier-Laplace transforms, we establish an existence result for such Riccati equation and we provide a discretized approximation of the joint characteristic functional that is exponentially entire. On the practical side, we develop a numerical scheme to solve the stiff infinite dimensional Riccati equations and demonstrate the efficiency and accuracy of the scheme for pricing SPX options and volatility swaps using Fourier and Laplace inversions, with specific examples of the Quintic OU and the one-factor Bergomi models and their calibration to real market data.