This paper concerns the numerical solution of the two-dimensional time-dependent partial integro-differential equation (PIDE) 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 (2008) in the case of the one-dimensional Kou integral. The acquired algorithm for the two-dimensional Kou integral has optimal computational cost: the number of basic arithmetic operations is directly proportional to the number of spatial grid points in the semidiscretization. For the effective discretization in time, we study seven contemporary operator splitting schemes of the implicit-explicit (IMEX) and the alternating direction implicit (ADI) kind. All these schemes allow for a convenient, explicit treatment of the integral term. By ample numerical experiments for put-on-the-average option values, the stability and convergence behaviour as well as the mutual performance of the seven operator splitting schemes are investigated. Moreover, the Greeks Delta and Gamma are considered.
翻译:本文涉及在两个资产库跳跃扩散模式下持有欧洲式选项值的二维时间依赖部分内分化方程式(PIDE)的数值解决方案。这个方程式的一个主要特征是存在非本地的双重整体术语。对于其数值评估,我们扩展了Toivanen(2008年)在单维库整体情况下得出的高效算法。二维库整体组合获得的算法具有最佳计算成本:基本算术操作的数量直接与半分化空间网格点的数量成比例。为了及时实现有效的离散,我们研究了7个当代操作者对隐显性(IMEX)和交替方向(ADI)类型的分解计划。所有这些计划都允许方便、明确地处理整体术语。通过大量数字实验,对平流选项值、稳定性和趋同行为以及7个操作者分解计划的共同表现进行了充分计算成本:基本算术操作的数量与半分解的空间网格点数量成正比。此外,还考虑了希腊三角洲和伽玛。