The Maximum of Brownian Motion and the Reflection Principle

The distribution of a standard Brownian motion X at a positive time t is, by definition, centered normal with variance t. What can we say about its maximum value up until the time? This is Xt = sups ≤ tXs, and is clearly nonnegative and at least as big as Xt. To be more precise, consider the probability that the maximum is greater than a fixed positive value a. Such problems will be familiar to anyone who has looked at pricing of financial derivatives such as barrier options, where the payoff of a trade depends on whether the maximum or minimum of an asset price has crossed a specified barrier level.

This can be computed with the aid of a symmetry argument commonly referred to as the reflection principle. The idea is that, if we reflect the Brownian motion when it first hits a level, then the resulting process is also a Brownian motion. The first time at which X hits level a is τ = inf{t ≥ 0: Xt ≥ a}, which is a stopping time. Reflecting the process about this level at all times after τ gives a new process

Reflected Brownian motion
Figure 1: Reflecting Brownian motion when it hits level a.

Continue reading “The Maximum of Brownian Motion and the Reflection Principle”