Using the Reflection Principle to Find a Brownian Stock’s Barrier-Crossing Probability

Summary

The document asks for the chance that a stock modeled as driftless Brownian motion with volatility scale σ reaches an upper barrier before a fixed horizon. It outlines a reflection-principle method: first calculate the probability that the normally distributed terminal price lies above the barrier, then double that tail probability to obtain the probability of crossing at any point before the horizon.

This is a concise conceptual answer rather than a full derivation. It gives no numerical example and leaves the normal-tail expression implicit, so readers must supply the distance to the barrier and the time-scaled volatility themselves. The result relies on the stated zero-drift Brownian model and an upper barrier above the starting price; it does not address drift, jumps, changing volatility, or other price dynamics.

Key ideas

• The terminal stock price is normally distributed under the stated driftless Brownian model.
• The reflection principle relates the probability of hitting the upper barrier by the horizon to the terminal probability above it.
• The crossing probability is twice the terminal upper-tail probability in this setup.
• The method assumes continuous Brownian paths and does not cover drift or jumps.

Tags

# Probability of Stock breaching barrier


# Probability of Stock breaching barrier












If a stock has a process: $dS(t) = sigma*dB(t)$, where $B(t)$ is a standard Brownian motion, and current stock price is $S(0)$. There is a barrier $H>S(0)$. What is the probability that the stock price will breach $H$ before time $T (T>0)$?

## Answer by mbison (score 2, accepted)

https://quant.stackexchange.com/a/20762

You can use the reflection principle to solve this problem. First you calculate the probability that the Stock will be above the barrier at time T. i.e. $P(S_T > H) = blabla$ (which is easy to calculate since it is just normally distributed). Then via the reflection principle you know that the probability of crossing the barrier between time 0 and T is 2x as big, i.e. $2 blabla$.

https://en.wikipedia.org/wiki/Reflection_principle_(Wiener_process)

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