Paper: arXiv 2405.02170

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents advanced mathematical machinery, including infinite-dimensional Riccati equations and entire function theory, indicating high math complexity. It also includes real-world calibration examples, a numerical scheme, and a code implementation link, demonstrating solid empirical rigor.

Research Flowchart

  flowchart TD
  A["Research Goal: <br>Fourier-Laplace transforms for Polynomial OU Models"] --> B["Key Methodology: <br>Connection to Infinite-Dimensional Riccati Equation"]
  B --> C{"Data/Inputs: <br>SPX/VIX Market Data"}
  C --> D["Computational Process: <br>Solve stiff Riccati equation & <br>Fourier-Laplace Inversion"]
  D --> E["Key Outcomes: <br>Exponential Entire Discretization & <br>Efficient Pricing/Calibration"]

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and 28 more
  1. Signature SDEs from an affine and polynomial perspective — → 2302.01362
  2. The quintic Ornstein-Uhlenbeck volatility model that jointly calibrates \text SPX & VIX smiles (2023)
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2 of 36 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.