On the Distribution of First Exit Time for Brownian Motion with Double Linear Time-Dependent Barriers

Lin Xu,Dongjin Zhu

doi:10.1155/2013/865347

Abstract

This paper focuses on the first exit time for a Brownian motion with a double linear time-dependent barrier specified by y=a+bt, y=ct, (a>0, b<0, c>0). We are concerned in this paper with the distribution of the Brownian motion hitting the upper barrier before hitting the lower linear barrier. The main method we applied here is the Girsanov transform formula. As a result, we expressed the density of such exit time in terms of a finite series. This result principally provides us an analytical expression for the distribution of the aforementioned exit time and an easy way to compute the distribution of first exit time numerically.

Highlights

In financial investment affairs, investors are exposed to credit risk, due to the possibility that one or more counterparts in a financial agreement will default
This paper focuses on the first exit time for a Brownian motion with a double linear time-dependent barrier specified by y = a + bt, y = ct, (a > 0, b < 0, c > 0)
We are concerned in this paper with the distribution of the Brownian motion hitting the upper barrier before hitting the lower linear barrier

Summary

Introduction

Investors are exposed to credit risk, due to the possibility that one or more counterparts in a financial agreement will default (cf [1]). Given a distribution for the default time, it is usually impossible to find a closed-form expression for the corresponding time-dependent barrier, in derivatives pricing, such as pricing barrier options or lookback options, which involve crossing certain levels (cf Chadam et al [5], Merton [6], and Metwally and Atiya [7]), or pricing American options [8], which entail evaluating the first passage time density for a time varying boundary Such applications are typically applied to large portfolios involving thousands of securities, and, in addition, thousands of iteration runs are needed on each security in the portfolio to calibrate the model parameters.

Preliminaries and Problem to Be Investigated

Distribution of First Exit Time of BM from Double Linear Barrier

Conclusions