On the Weak Error for Local Stochastic Volatility Models

ArXiv ID: 2506.10817 “View on arXiv”

Authors: Peter K. Friz, Benjamin Jourdain, Thomas Wagenhofer, Alexandre Zhou

Abstract

Local stochastic volatility refers to a popular model class in applied mathematical finance that allows for “calibration-on-the-fly”, typically via a particle method, derived from a formal McKean-Vlasov equation. Well-posedness of this limit is a well-known problem in the field; the general case is largely open, despite recent progress in Markovian situations. Our take is to start with a well-defined Euler approximation to the formal McKean-Vlasov equation, followed by a newly established half-step-scheme, allowing for good approximations of conditional expectations. In a sense, we do Euler first, particle second in contrast to previous works that start with the particle approximation. We show weak order one for the Euler discretization, plus error terms that account for the said approximation. The case of particle approximation is discussed in detail and the error rate is given in dependence of all parameters used.

Keywords: Local Stochastic Volatility, McKean-Vlasov Equation, Euler Discretization, Particle Methods, Conditional Expectations, Equities / Derivatives

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper involves advanced stochastic analysis, McKean-Vlasov equations, and detailed weak error proofs with extensive theoretical derivations, justifying a high math score. It lacks empirical backtesting or real-world data validation, relying instead on theoretical error analysis and simple toy examples, resulting in a lower empirical rigor score.

  flowchart TD
    A["Research Goal:<br>Weak Error Analysis for Local Stochastic Volatility Models"] --> B["Key Methodology:<br>Well-defined Euler Approximation of McKean-Vlasov Equation"]
    B --> C["Computational Process:<br>Half-step Scheme for Conditional Expectations"]
    C --> D["Data/Input:<br>Formal McKean-Vlasov Equation"]
    D --> E["Data/Input:<br>Particle Method for Calibration"]
    E --> F["Key Outcome 1:<br>Weak Order 1 Convergence for Euler Discretization"]
    E --> G["Key Outcome 2:<br>Explicit Error Rate Analysis for Particle Approximation"]
    F --> H["Conclusion:<br>Well-posedness established via numerical error control"]
    G --> H