# Probability the stock price (following geometric Brownian motion) hits the upper boundary U before there is a retracement from the max by amount R?

I am looking for the probability that the stock price/Geometric Brownian Motion hits the upper boundary U, before there is a retracement (from the maximum price) that exceeds amount R. In other words, for my application the lower bound isn't a constant boundary at L, it is a fixed distance R that is measured from the maximum price.

Thank you for considering this problem!

• Just one observation : L must vary between $S_0 - R$ and $U-R$ so you can bound the probability accordingly using the known formulae for fixed U and L
– dm63
Feb 12, 2022 at 15:27
• @dm63 Thank you for your suggestion. A lower bound on the probability I am looking for would help a lot. Unfortunately the bound in the suggestion above is an upper bound... Feb 14, 2022 at 2:53
• @dm63 I am interested in finding the probability stock price reaches objective U before a "trailing stop-loss" is triggered by the price retracing a fixed amount R from its current maximum. If the maximum price is achieved at time t*, then the probability I am looking for is the probability that the stopping time of the maximum being above U is smaller than the stopping time of the minimum (defined on time interval t* ≤ s ≤ t being above maximum - R. Feb 14, 2022 at 2:54
• @dm63 Let t* be the earliest time at which Bs has attained its maximum thus far: t* ≡ min{s ∈ [0, t]|Bs = max{Br |r ∈ [0, t]}}. Thus Bt* is the maximum attained by time t. Let t** be the latest time at which the distance between Bt* and the minimum on time interval (t*, t) does not exceed amount R. t** ≡ max{t ∈ [0, T ]| Bt* - min{Bq |q ∈ [t*, t]} < R} }. Let t***≡ min{s ∈ [0, t]|Bt* > U}. I am looking for the P(t*** < t**). Feb 14, 2022 at 2:55
• Yes I understand what you are searching for and I do not know how to calculate that exactly- but I did provide a 2 sided bound in my comment , not a one sided.
– dm63
Feb 14, 2022 at 3:11