Journal of Computational Finance

Efficient numerical valuation of European options under the two-asset Kou jump-diffusion model

Karel in 't Hout and Pieter Lamotte

  • Extension of a technique by Toivanen (2008) to arrive at a highly efficient algorithm for evaluating the nonlocal double integral appearing in the two-dimensional Kou PIDE.
  • Study of seven contemporary operator splitting schemes of the implicit-explicit (IMEX) and the alternating direction implicit (ADI) kind which all treat the integral term in a convenient, explicit manner.
  • A von Neumann stability analysis of these schemes is provided pertinent to two-dimensional PIDEs.
  • Ample numerical experiments are presented yielding insight in their actual convergence behavior and mutual performance.

This paper concerns the numerical solution of the two-dimensional time-dependent partial integro-differential equation that holds for the values of European-style options under the two-asset Kou jump-diffusion model. A main feature of this equation is the presence of a nonlocal double integral term. For its numerical evaluation, we extend a highly efficient algorithm derived by Toivanen in the case of the one-dimensional Kou integral. The acquired algorithm for the two-dimensional Kou integral has an optimal computational cost: the number of basic arithmetic operations is directly proportional to the number of spatial grid points in the semidiscretization. For effective discretization in time, we study seven contemporary implicit–explicit and alternating-direction implicit operator splitting schemes. All these schemes allow for a convenient, explicit treatment of the integral term. We analyze their (von Neumann) stability. Through ample numerical experiments for put-on-the-average option values, we investigate the actual convergence behavior as well as the relative performance of the seven operator splitting schemes. In addition, we consider the Greeks Delta and Gamma.

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