Efficient evaluation of expectations of functions of a L\'evy process and its extremum

TL;DR

This paper develops simple formulas for calculating expectations of functions of a Lévy process and its extremum, combining analytical and numerical methods for efficient computation of probabilities and option prices.

Contribution

It introduces new analytical formulas and efficient numerical techniques, including sinh-acceleration, for evaluating expectations related to Lévy processes and their extrema.

Findings

Formulas enable quick computation of distributions and maxima.

Numerical methods achieve high precision in milliseconds.

Applications include option pricing and probability calculations.

Abstract

We prove simple general formulas for expectations of functions of a L\'evy process and its running extremum. 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 power of a stock for its maximum 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.