---
description: >-
  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

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Conditional Probability of Hitting Barrier

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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.

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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

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![](file:///C:/Users/Xiao/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png)

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Also let

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· _Hu_ and _Hd_ (where _H H u d_ > ) respectively denote constant upper and lower barrier levels,

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·  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](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.
