Calibration of Local Volatility Models with Stochastic Interest Rates using Optimal Transport

Paper: arXiv 2305.00200

Authors: Benjamin Joseph, Gregoire Loeper, Jan Obloj

Abstract

We develop a non-parametric, semimartingale optimal transport, calibration methodology for local volatility models with stochastic interest rate. The method finds a fully calibrated model which is the closest, in a way that can be defined by a general cost function, to a given reference model. We establish a general duality result which allows to solve the problem by optimising over solutions to a second order fully non-linear Hamilton-Jacobi-Bellman equation. Our methodology is analogous to Guo, Loeper, and Wang, 2022 and Guo, Loeper, Obloj, et al., 2022a but features a novel element of solving for discounted densities, or sub-probability measures. As an example, we apply the method to a sequential calibration problem, where a Vasicek model is already given for the interest rates and we seek to calibrate a stock price’s local volatility model with volatility coefficient depending on time, the underlying and the short rate process, and the two processes driven by possibly correlated Brownian motions. The equity model is calibrated to any number of European options prices.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly sophisticated theoretical framework using semimartingale optimal transport, non-linear HJB equations, and duality results to calibrate local volatility under stochastic interest rates. However, it is predominantly a theoretical mathematical contribution with limited empirical validation or backtesting code provided.

Research Flowchart

  flowchart TD
    A["Research Goal: Calibrate local volatility model with stochastic interest rates via optimal transport"] --> B["Inputs: Reference model, Vasicek short rate model, and European option market prices"]
    B --> C["Formulate Semimartingale Optimal Transport Problem using discounted densities"]
    C --> D["Establish Duality Result: Convert primal problem to dual optimization"]
    D --> E["Solve fully non-linear Hamilton-Jacobi-Bellman (HJB) PDE"]
    E --> F["Outcome: Fully calibrated hybrid LV-Stochastic Interest Rate model closest to reference model"]