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

2022/07/06 by Svetlana Boyarchenko, Boyarchenko, Svetlana, Sergei Levendorskiı̌ +1
Economics, Econometrics and Finance · #42A38 #42B10 #44A10 #60-08 #65G51 #65R10 #91G20 #91G60 #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2207.02793

openalex publication_date 2022/07/06 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28

Abstract

We prove simple general formulas for expectations of functions of a Lévy 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.

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