This paper proposes a multidimensional Hilbert transform approach for pricing discretely monitored multi-asset barrier options and computing joint survival probability in multivariate exponential Lévy asset price models. We generalize the univariate Hilbert transform method of Feng and Linetsky (Math Financ 18(3), 337–384, 2008) for single-asset barrier options and the well-known Sinc approximation theory of Stenger (Numerical methods based on sinc and analytic functions. Springer, New York, 1993) for computing the one-dimensional Hilbert transform to any dimension. We prove that, for Lévy processes with joint characteristic functions having an exponentially decaying tail, the error of our method decays exponentially in some power of the number of terms used in the expansion for each dimension. Numerical experiments demonstrate the efficiency of our method in the two-dimensional and three-dimensional problems for some popular multivariate Lévy models.

本文提出了一种多维希尔伯特变换方法，用于定价离散监控的多资产障碍期权并计算多元指数 Lévy 资产价格模型中的联合生存概率。我们推广了 Feng 和 Linetsky 的单变量 Hilbert 变换方法 (Math Financ 18(3), 337–384, 2008) 用于单一资产障碍期权和著名的 Stenger Sinc 逼近理论（基于 sinc 和解析函数的数值方法. Springer, New York, 1993) 用于计算一维希尔伯特变换到任何维度。我们证明，对于具有指数衰减尾部的联合特征函数的 Lévy 过程，我们的方法的误差在每个维度的展开中使用的项数的某个幂中呈指数衰减。