Conditional Probability of Hitting Barrier

A model is developed for evaluating the conditional probability of hitting an upper barrier before a lower barrier, and vice versa, for a tied down geometric Brownian motion with drift. The method produces an analytical value for this probability, assuming that the barrier levels are constant and continuously monitored.

We consider the conditional probability that the upper barrier level is crossed during the interval

An analytical value for this conditional probability is provided. The derivation is based, in part, on an application of Theorem, which gives an analytical value for a similar conditional probability but with respect to standard Brownian motion

For standard Brownian motion, consider the conditional probability that the upper barrier level is crossed during the interval

Also for standard Brownian motion consider the probability that the upper barrier level is crossed during the interval [0,T], and for a smaller time than for which the lower barrier level is crossed, and that WT lies in an interval I

Given its similarity to the result for hitting an upper barrier before a lower barrier, we would like to recommend that this approach be considered for use in a future implementation of this method to price an actual deal.

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