Sensitivity of boundary crossing probabilities of the Brownian motion

Sercan Gür,Klaus Pötzelberger

doi:10.1515/mcma-2019-2031

Abstract

Abstract The paper analyzes the sensitivity of boundary crossing probabilities of the Brownian motion to perturbations of the boundary. The first- and second-order sensitivities, i.e. the directional derivatives of the probability, are derived. Except in cases where boundary crossing probabilities for the Brownian bridge are given in closed form, the sensitivities have to be computed numerically. We propose an efficient Monte Carlo procedure.

Highlights

Events are often triggered when a stochastic process reaches a specified site at a specified time
The distribution of the one-sample Kolmogorov-Smirnov statistic which is the first passage time of the process defined as the difference between the empirical and true distribution of a random sample, to a constant boundary, the CUSUM-procedure for structural change in the fluctuation test framework are some of these areas
A Parisian option is a type of barrier option which can only be exercised if the underlying value process reaches a predefined barrier level but remains for a certain prescribed time below this level

Summary

Introduction

Events are often triggered when a stochastic process reaches a specified site (or set of sites) at a specified time This phenomenon has been studied extensively in the literature and it occurs in many scientific disciplines including finance, economics, statistics, biology, engineering, medicine etc. The evaluation of barrier options, which are financial instruments whose payoff depends on whether or not the underlying price process crosses the barrier level, is another field where boundary crossing probabilities are involved. Apart from these financial settings, the boundary crossing probabilities often appear in different areas of statistics. Other applications in credit risk and life insurance can be found in Moraux (2002) and Chen and Suchanecki (2007), respectively

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