Strong existence and uniqueness of a calibrated local stochastic volatility model

Paper: arXiv 2406.14074

Authors: Scander Mustapha

Abstract

We study a two-dimensional McKean-Vlasov stochastic differential equation, whose volatility coefficient depends on the conditional distribution of the second component with respect to the first component. We prove the strong existence and uniqueness of the solution, establishing the well-posedness of a two-factor local stochastic volatility (LSV) model calibrated to the market prices of European call options. In the spirit of [Jourdain and Zhou, 2020, Existence of a calibrated regime switching local volatility model.], we assume that the factor driving the volatility of the log-price takes finitely many values. Additionally, the propagation of chaos of the particle system is established, giving theoretical justification for the algorithm [Julien Guyon and Henry-Labordère, 2012, Being particular about calibration.].

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper provides rigorous mathematical proofs for the strong existence, uniqueness, and propagation of chaos for a calibrated two-factor local stochastic volatility (LSV) McKean-Vlasov SDE system. While highly theoretical with advanced mathematical machinery, it lacks empirical backtesting or real market data implementation.

Research Flowchart

  flowchart TD
    A[Research Goal: Prove strong existence & uniqueness of a 2-factor calibrated LSV model] --> B[Inputs: European call option market prices & finite-state volatility factor]
    B --> C[Formulate 2D McKean-Vlasov SDE with conditional distribution dependency]
    C --> D[Establish mathematical well-posedness following Jourdain & Zhou approach]
    D --> E[Prove propagation of chaos for the interacting particle system]
    E --> F[Outcomes: Theoretical validation of Guyon & Henry-Labordère calibration algorithm]