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

Benjamin Joseph, Gregoire Loeper, Jan Obloj

TL;DR

The paper develops a non-parametric, semimartingale optimal transport approach to calibrate local volatility models with stochastic interest rates, aiming to reproduce market prices of European options while staying close to a reference model. It introduces a discounted-density framework with $Y_t=\exp(-\int_0^t r_s ds)$ and a convex cost $F$, and proves a duality with a second-order nonlinear HJB equation that governs the dual optimizer. A key novelty is the discounted-density transformation, which enables a two-dimensional Markovian reduction even in the presence of stochastic rates, and a discounted superposition principle that links marginals to augmented processes. The method is illustrated in a sequential calibration setup where a Vasicek rate model is fixed and a time-, state-, and rate-dependent local volatility is inferred to fit European option prices; the authors also discuss a sequel addressing simultaneous joint calibration. Overall, the work provides a rigorous variational framework for calibrating rate and equity components with path-independent constraints, highlighting both the potential and computational challenges of high-dimensional HJB-based solvers in finance.

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.

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

TL;DR

The paper develops a non-parametric, semimartingale optimal transport approach to calibrate local volatility models with stochastic interest rates, aiming to reproduce market prices of European options while staying close to a reference model. It introduces a discounted-density framework with and a convex cost , and proves a duality with a second-order nonlinear HJB equation that governs the dual optimizer. A key novelty is the discounted-density transformation, which enables a two-dimensional Markovian reduction even in the presence of stochastic rates, and a discounted superposition principle that links marginals to augmented processes. The method is illustrated in a sequential calibration setup where a Vasicek rate model is fixed and a time-, state-, and rate-dependent local volatility is inferred to fit European option prices; the authors also discuss a sequel addressing simultaneous joint calibration. Overall, the work provides a rigorous variational framework for calibrating rate and equity components with path-independent constraints, highlighting both the potential and computational challenges of high-dimensional HJB-based solvers in finance.

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.
Paper Structure (7 sections, 4 theorems, 28 equations)

This paper contains 7 sections, 4 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Problem formulation
  4. First Markovian reduction
  5. A 'discounted density' transformation and superposition principle
  6. Second Markovian reduction
  7. The dual problem

Key Result

theorem 1

Let $\mathbb P\in {\mathcal{P}}^1$ be a candidate model. There exist jointly measurable versions of the conditional expectations, $\alpha_t(x,y)=\mathbb E^{\mathbb P}_{t,x,y}[\alpha_t^{\mathbb P}]$, $\beta_t(x,y)=\mathbb E^{\mathbb P}_{t,x,y}[\beta_t^{\mathbb P}]$, $\mathop{}\!\mathrm{d} t\times \ma such that $(X_t,Y_t)\sim \mathbb P\circ (X_t,Y_t)^{-1}$ for all $t\in [0,T]$, and where $W^{\mathbb

Theorems & Definitions (14)

  • definition thmcounterdefinition
  • theorem 1
  • definition thmcounterdefinition
  • remark thmcounterremark
  • remark thmcounterremark
  • definition thmcounterdefinition
  • proposition thmcounterproposition
  • proof : of Proposition \ref{['prop: localised problem']} (Part I)
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • ...and 4 more