A finite volume – alternating direction implicit approach for the calibration of stochastic local volatility models

Abstract

ABSTRACTCalibration of stochastic local volatility (SLV) models to their underlying local volatility model is often performed by numerically solving a two-dimensional nonlinear forward Kolmogorov equation. We propose a novel finite volume discretization in the numerical solution of general 1D and 2D forward Kolmogorov equations. The finite volume method does not require a transformation of the partial differential equation (PDE). This constitutes a main advantage in the calibration of SLV models as the pertinent PDE coefficients are often non-smooth. Moreover, the finite volume discretization has the crucial property that the total numerical mass is conserved. Applying the finite volume discretization in the calibration of SLV models yields a nonlinear system of ordinary differential equations (ODEs). Numerical time stepping is performed by the Hundsdorfer–Verwer alternating direction implicit scheme to increase the computational efficiency. The nonlinearity 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.