Complex discontinuities of the square root of Fredholm determinants in the Volterra Stein-Stein model

ArXiv ID: 2503.02965 “View on arXiv”

Authors: Unknown

Abstract

Fourier-based methods are central to option pricing and hedging when the Fourier-Laplace transform of the log-price and integrated variance is available semi-explicitly. This is the case for the Volterra Stein-Stein stochastic volatility model, where the characteristic function is known analytically. However, naive evaluation of this formula can produce discontinuities due to the complex square root of a Fredholm determinant, particularly when the determinant crosses the negative real axis, leading to severe numerical instabilities. We analyze this phenomenon by characterizing the determinant’s crossing behavior for the joint Fourier-Laplace transform of integrated variance and log-price. We then derive an expression for the transform to account for such crossings and develop efficient algorithms to detect and handle them. Applied to Fourier-based pricing in the rough Stein-Stein model, our approach significantly improves accuracy while drastically reducing computational cost relative to existing methods.

Keywords: Fourier-Based Pricing, Volterra Stein-Stein Model, Fredholm Determinant, Rough Stochastic Volatility, Numerical Instability, Derivatives (Options)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper delves deeply into operator theory, Fredholm determinants, and analytic continuation in complex analysis, making the math dense and advanced. While it discusses numerical algorithms and cites computational improvements, it focuses on theoretical derivation and methodological correction rather than providing backtests, datasets, or implementation-heavy empirical validation.

  flowchart TD
    A["Research Goal<br>Analyze discontinuities in complex square root<br>of Fredholm determinants in Volterra Stein-Stein model"] --> B["Input Data<br>Fourier-Laplace transform of log-price<br>and integrated variance (analytical)"]

    B --> C["Key Methodology<br>Characterize determinant crossing behavior<br>on negative real axis"]

    C --> D["Computational Process<br>Develop detection algorithms for crossings<br>Derive corrected transform expression"]

    D --> E["Application<br>Implement efficient Fourier-based pricing<br>in rough Stein-Stein model"]

    E --> F["Key Outcomes<br>1. Resolved numerical instabilities<br>2. Improved accuracy vs. existing methods<br>3. Drastically reduced computational cost"]