Paper: arXiv 2302.00434

Authors: Christoph Reisinger, Maria Olympia Tsianni

Abstract

In this paper, we study the Euler–Maruyama scheme for a particle method to approximate the McKean–Vlasov dynamics of calibrated local-stochastic volatility (LSV) models. Given the open question of well-posedness of the original problem, we work with regularised coefficients and prove that under certain assumptions on the inputs, the regularised model is well-posed. Using this result, we prove the strong convergence of the Euler–Maruyama scheme to the particle system with rate 1/2 in the step-size and obtain an explicit dependence of the error on the regularisation parameters. Finally, we implement the particle method for the calibration of a Heston-type LSV model to illustrate the convergence in practice and to investigate how the choice of regularisation parameters affects the accuracy of the calibration.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical proof of the strong convergence of the Euler-Maruyama particle scheme for regularised McKean-Vlasov equations in LSV calibration. While mathematically sophisticated and novel in its analytical treatment, the empirical validation is limited to illustrative implementation rather than extensive empirical backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Prove convergence of EM particle scheme for regularised LSV McKean-Vlasov models"] --> B["Inputs & Assumptions: Model inputs, regularisation parameters (epsilon, delta), step-size h"]
    B --> C["Well-Posedness Proof: Establish well-posedness of the regularised McKean-Vlasov dynamics"]
    C --> D["Theoretical Analysis: Prove strong convergence of Euler-Maruyama scheme with rate O(h^1/2)"]
    D --> E["Numerical Implementation: Apply particle scheme to calibrate a Heston-type LSV model"]
    E --> F["Outcomes: Explicit error bounds on parameters & validated practical calibration accuracy"]