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Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models

Author

Listed:
  • Eduardo Abi Jaber

    (CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

  • Shaun Xiaoyuan Li

    (UP1 - Université Paris 1 Panthéon-Sorbonne)

  • Xuyang Lin

    (X - École polytechnique - IP Paris - Institut Polytechnique de Paris)

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 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.

Suggested Citation

  • Eduardo Abi Jaber & Shaun Xiaoyuan Li & Xuyang Lin, 2024. "Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models," Working Papers hal-04567783, HAL.
    • Handle: RePEc:hal:wpaper:hal-04567783
    Note: View the original document on HAL open archive server: https://hal.science/hal-04567783
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