Efficient inverse $Z$-transform and pricing barrier and lookback options with discrete monitoring

TL;DR

This paper develops efficient analytical and numerical methods for pricing barrier and lookback options with discrete monitoring, leveraging inverse Z-transform, Wiener-Hopf factorization, and advanced numerical techniques.

Contribution

It introduces a new efficient numerical realization of the inverse Z-transform combined with sinh-acceleration and trapezoid rule for fast option pricing.

Findings

Achieves high precision (E-10 and better) in milliseconds

Provides analytical formulas for expectations involving random walks and extrema

Demonstrates practical implementation in Matlab with rapid computation

Abstract

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.