A multidimensional Hilbert transform approach for barrier option pricing and survival probability calculation

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.