A Multidimensional Hilbert Transform Approach for Barrier Option Pricing and Survival Probability Calculation

This paper proposes a novel approach for pricing discretely monitored multi-asset barrier options and computing joint survival probability in multivariate exponential Levy asset price models. We calculate the Fourier transform of appropriately dampened value functions recursively using multidimensional Hilbert transform, which can be approximated using multidimensional Sinc expansions. We prove that, for Levy processes whose joint characteristic functions have 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. Various numerical examples confirm the computational efficiency of our method in the two-dimensional case for popular Levy models and it outperforms the two-dimensional Fourier-cosine algorithm. Our method can also handle the survival probability calculation efficiently for the three-dimensional Black-Scholes model