A Finite Volume - Alternating Direction Implicit Approach for the Calibration of Stochastic Local Volatility Models
This work provides a robust numerical method for calibrating SLV models in quantitative finance, addressing challenges with nonsmooth coefficients and conservation properties.
The paper proposes a finite volume discretization combined with an alternating direction implicit scheme for calibrating stochastic local volatility models, achieving accurate and efficient solutions for nonsmooth PDE coefficients while conserving total numerical mass.
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.