Karel J. in 't Hout

Tinne Haentjens, Karel J. in 't Hout

In this paper we investigate the effectiveness of Alternating Direction Implicit (ADI) time discretization schemes in the numerical solution of the three-dimensional Heston-Hull-White partial differential equation, which is semidiscretized by applying finite difference schemes on nonuniform spatial grids. We consider the Heston-Hull-White model with arbitrary correlation factors, with time-dependent mean-reversion levels, with short and long maturities, for cases where the Feller condition is satisfied and for cases where it is not. In addition, both European-style call options and up-and-out call options are considered. It is shown through extensive tests that ADI schemes, with a proper choice of their parameters, perform very well in all situations - in terms of stability, accuracy and efficiency.

Massimiliano Moda, Karel J. in 't Hout, Michèle Vanmaele et al.

We propose a numerical method for the valuation of European-style options under two-asset infinite-activity exponential Lévy models. Our method extends the effective approach developed by Wang, Wan & Forsyth (2007) for the 1-dimensional case to the 2-dimensional setting and is applicable for general Lévy measures under mild assumptions. A tailored discretization of the non-local integral term is developed, which can be efficiently evaluated by means of the fast Fourier transform. For the temporal discretization, the semi-Lagrangian theta-method is employed in a convenient splitting fashion, where the diffusion term is treated implicitly and the integral term is handled explicitly by a fixed-point iteration. Numerical experiments for put-on-the-average options under Normal Tempered Stable dynamics reveal favourable second-order convergence of our method whenever the exponential Lévy process has finite-variation.