Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models

Abstract

We consider the Fourier-Laplace transforms of a broad class of polynomial Ornstein-Uhlenbeck (OU) volatility models, including the well-known Stein-Stein, Schöbel-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 system of Riccati equations. Next, under some non-vanishing conditions on the Fourier-Laplace transforms, we establish an existence result for such Riccati equations and show how the Fourier-Laplace functional can be seen as the limit of a sequence of entire functions. On the practical side, we develop a numerical scheme to solve the stiff infinite-dimensional system of 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.

arXiv comment46 pages, 8 figures