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.

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.

Let St denote the price at time equal to t of an underlying security. Furthermore assume that the process {S t |t Î[0,+¥) } satisfies, under some measure P , the stochastic differential equation

Also let

· Hu and Hd (where H H u d > ) respectively denote constant upper and lower barrier levels,

· t 1 = inf { } inf t ³ 0|,S ³ H t u and t { } 2 = inf t ³ 0|S £ H t d respectively denote the first hitting times of the barrier levels Hu and Hd (here we assume that H S H d u < < 0 ), and

· T (where T > 0 ) denote a length of time.

We consider the conditional 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,

given that S y T = , that is,

An analytical value for this conditional probability is provided in [Myint, 1997]. The

derivation is based, in part, on an application of Theorem 4.2 in [Anderson, 1960] (see

page 175), which gives an analytical value for a similar conditional probability but with

respect to standard Brownian motion (see https://finpricing.com/lib/FxForwardCurve.html)

We first introduce some notation. Specifically let

and

denote first hitting times, respectively from below and from above, of the constant barrier level g .

Next let

· {W t } t | Î[0,+¥) denote standard Brownian motion under a probability measure P ,

And

· g 1 and g 2 (where g 1 > 0 and g 2 < 0 ) respectively denote constant upper and lower

barrier levels.

For standard Brownian motion, consider the conditional 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, given that W y T = , that is,

From Theorem 4.2 in [Anderson, 1960] (with d d 1 2 = = 0 ), this conditional probability is equal to

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 , that is,

From Bayes’ Theorem and (1), this probability is equal to

For the process {S t } t | Î[0,+¥) , consider the conditional probability that the lower barrier level is crossed during the interval [0,T], and for a smaller time than for which the upper barrier level is crossed, given that S y T = , that is,

FP chooses to obtain this conditional probability by considering the identity

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.