An efficient numerical method for American options and their Greeks under the two-asset Kou jump-diffusion model

Abstract

In this paper we consider the numerical solution of the two-dimensional time-dependent partial integro-differential complementarity problem (PIDCP) that holds for the value of American-style options under the two-asset Kou jump-diffusion model. Following the method of lines (MOL), we derive an efficient numerical method for the pertinent PIDCP. Here, for the discretization of the nonlocal double integral term, an extension is employed of the fast algorithm by Toivanen (2008) in the case of the one-asset Kou jump-diffusion model. For the temporal discretization, we study a useful family of second-order diagonally implicit Runge-Kutta (DIRK) methods. Their adaptation to the semidiscrete two-dimensional Kou PIDCP is obtained by means of an effective iteration introduced by d'Halluin, Forsyth & Labahn (2004) and d'Halluin, Forsyth & Vetzal (2005). Ample numerical experiments are presented showing that the proposed numerical method achieves a favourable, second-order convergence behaviour to the American two-asset option value as well as to its Greeks Delta and Gamma.

Figures (6)

• Figure 1: Sample spatial grid for $m=50$, $K=100$, $S_{\rm max} = 500$.
• Figure 2: The value, Delta and Gamma functions of an American put-on-the-average option for $t=T$ and parameter set given by Table \ref{['paramset']}.
• Figure 3: In grey: the early exercise region of an American put-on-the-average option for $t=T$ and parameter set given by Table \ref{['paramset']}. In blue: the region of interest.
• Figure 4: Temporal discretization errors of the DIRKa-P, DIRKb-P, DIRKc-P, DIRKd-P methods for $m=100$. Option value (top left), $\Delta_1$ (middle left), $\Delta_2$ (bottom left), $\Gamma_{11}$ (top right), $\Gamma_{12}$ (middle right), $\Gamma_{22}$ (bottom right).
• Figure 5: Temporal discretization errors of the DIRKa-P, DIRKb-P, DIRKc-P, DIRKd-P methods for $m=200$. Option value (top left), $\Delta_1$ (middle left), $\Delta_2$ (bottom left), $\Gamma_{11}$ (top right), $\Gamma_{12}$ (middle right), $\Gamma_{22}$ (bottom right).