Efficient evaluation of expectations of functions of a Lévy process and its extremum

Abstract

We prove a simple general formula for the expectation of a function of a Lévy process and its running extremum, which is more convenient for applications than general formulas in the first version of the paper. The derivation of explicit formulas in applications significantly simplifies. Under additional conditions, we derive analytical formulas using the Fourier/Laplace inversion and Wiener-Hopf factorization, and discuss efficient numerical methods for realization of these formulas. As applications, the cumulative probability distribution function of the process and its running maximum and the price of the option to exchange the maximum of a stock price for a power of the price are calculated. The most efficient numerical methods use the sinh-acceleration technique and simplified trapezoid rule. The program in Matlab running on a Mac with moderate characteristics achieves the precision E-7 and better in several milliseconds, and E-14 -in a fraction of a second.