On the First Exit Time of Geometric Brownian Motion from Stochastic Exponential Boundaries

Abstract

This article deals with the boundary crossing probability of a geometric Brownian motion (GBM) process when the boundary itself is a GBM process. An exact formula is obtained for the probability that the first exit time of $$ S\left( t \right) $$ from the stochastic interval $$ \left[ {H_{1} \left( t \right),H_{2} \left( t \right)} \right] $$ is greater than a finite time $$ T $$ using a partial differential equation approach. Applications and numerical results are provided. The possibility of an extension to higher dimension is also discussed. In particular, the steps to obtain the probability that $$ S_{1} \left( t \right) $$ , $$ S_{2} \left( t \right) $$ and $$ S_{3} \left( t \right) $$ remain above $$ S_{4} \left( t \right) $$ , $$ \forall 0 \le t \le T $$ , are outlined, while pointing out that the entailed numerical issues make the relevance of an analytical approach questionable.