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

ABSTRACT

Calibration 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.

Keywords:

• Stochastic local volatility
• forward Kolmogorov equation
• finite volume discretization
• ADI methods
• calibration

2010 AMS Subject Classifications:

• 35K55
• 97M30

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work by the first author has been supported financially by a PhD Fellowship of the Research Foundation – Flanders.