A Finite Volume - Alternating Direction Implicit Approach for the Calibration of Stochastic Local Volatility Models

TL;DR

This paper introduces a novel finite volume method combined with an ADI scheme for efficiently calibrating stochastic local volatility models by solving non-linear forward Kolmogorov equations without PDE transformation.

Contribution

It develops a new FV discretization that preserves mass and handles nonsmooth coefficients, improving calibration accuracy and efficiency for SLV models.

Findings

Mass conservation property of the FV method

Effective handling of nonsmooth PDE coefficients

Enhanced computational efficiency with ADI scheme

Abstract

Calibration of stochastic local volatility (SLV) models to their underlying local volatility model is often performed by numerically solving a two-dimensional non-linear forward Kolmogorov equation. We propose a novel finite volume (FV) discretization in the numerical solution of general 1D and 2D forward Kolmogorov equations. The FV method does not require a transformation of the PDE. This constitutes a main advantage in the calibration of SLV models as the pertinent PDE coefficients are often nonsmooth. Moreover, the FV discretization has the crucial property that the total numerical mass is conserved. Applying the FV discretization in the calibration of SLV models yields a non-linear system of ODEs. Numerical time stepping is performed by the Hundsdorfer-Verwer ADI scheme to increase the computational efficiency. The non-linearity in the system of ODEs is handled by introducing an inner iteration. Ample numerical experiments are presented that illustrate the effectiveness of the calibration procedure.