2022/07/06 by Svetlana Boyarchenko, Boyarchenko, Svetlana, Sergei Levendorskiı̌ +1
Economics, Econometrics and Finance · #42A38 #42B10 #44A10 #60-08 #65G51 #65R10 #91G20 #91G60 #Computational Finance (q-fin.CP) #FOS: Economics and business #FOS: Mathematics #Numerical Analysis (math.NA) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2207.02858
openalex publication_date 2022/07/06 · openalex created_date 2023/02/13 · openalex updated_date 2026/07/28
We prove simple general formulas for expectations of functions of a random walk and its running extremum. Under additional conditions, we derive analytical formulas using the inverse Z-transform, 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 a new efficient numerical realization of the inverse Z-transform, the sinh-acceleration technique and simplified trapezoid rule. The program in Matlab running on a Mac with moderate characteristics achieves the precision E-10 and better in several dozen of milliseconds, and E-14 - in a fraction of a isecond.