> For the complete documentation index, see [llms.txt](https://finwhite.gitbook.io/barrierhittingprobability/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://finwhite.gitbook.io/barrierhittingprobability/conditional-probability-of-hitting-barrier.md).

# 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

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

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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,

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png)

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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>)

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We first introduce some notation. Specifically let

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image006.png)

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and

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image008.png)

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

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Next let

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

And

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

barrier levels.

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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,

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image010.png)

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

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image012.png)

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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,

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image014.png)

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

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image016.png)

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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,

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image018.png)

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FP chooses to obtain this conditional probability by considering the identity

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image020.png)

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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.