Applying the Wiener-Hopf Monte Carlo simulation technique for Levy processes to path functionals such as first passage times, undershoots and overshoots

TL;DR

This paper demonstrates the application of the Wiener-Hopf Monte Carlo method to efficiently simulate path functionals of Levy processes, improving accuracy over traditional methods in finance and insurance contexts.

Contribution

The paper extends the Wiener-Hopf Monte Carlo technique to path functionals of Levy processes, enabling efficient simulation of first passage times and related quantities for a broad class of processes.

Findings

WHMC outperforms plain Monte Carlo in approximating first passage times.

Applicable to a wide range of Levy processes including stable and meromorphic.

Provides practical examples illustrating the method's effectiveness.

Abstract

In this note we apply the recently established Wiener-Hopf Monte Carlo (WHMC) simulation technique for Levy processes from Kuznetsov et al. [17] to path functionals, in particular first passage times, overshoots, undershoots and the last maximum before the passage time. Such functionals have many applications, for instance in finance (the pricing of exotic options in a Levy model) and insurance (ruin time, debt at ruin and related quantities for a Levy insurance risk process). The technique works for any Levy process whose running infimum and supremum evaluated at an independent exponential time allows sampling from. This includes classic examples such as stable processes, subclasses of spectrally one sided Levy processes and large new families such as meromorphic Levy processes. Finally we present some examples. A particular aspect that is illustrated is that the WHMC simulation technique performs much better at approximating first passage times than a `plain' Monte Carlo simulation technique based on sampling increments of the Levy process.