# Conditional Probability of Hitting Barrier 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 ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png) 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, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png) 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 ) We first introduce some notation. Specifically let ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png) and ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png) 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, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png) From Theorem 4.2 in \[Anderson, 1960] (with d d 1 2 = = 0 ), this conditional probability is equal to ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image012.png) 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, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image014.png) From Bayes’ Theorem and (1), this probability is equal to ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image016.png) 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, ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png) FP chooses to obtain this conditional probability by considering the identity ![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image020.png) 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. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://finwhite.gitbook.io/barrierhittingprobability/conditional-probability-of-hitting-barrier.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.