The Reflection Principle

A barrier option knocks out the moment the price touches a level, whether or not it's still there at expiry. Pricing it means answering "what's the probability the maximum of the path over $[0,T]$ exceeded some level $b$?" — a question about the whole path's extreme value, not just its endpoint. That sounds like it needs tracking every possible path shape. It doesn't: a symmetry trick collapses it into a question about the endpoint alone.

The analogy: mirror everything that overshoots

Brownian motion, once it touches a level, is equally likely to continue upward or bounce back down from that instant on — by the strong Markov property, the future beyond a hitting time forgets everything except where it currently is, and up-versus-down from there are symmetric coin flips. So take every path that touches the barrier $b$ at some point and then ends up below $b$ by time $T$. For each one, there's an equally likely path that touches $b$ at the same instant and then, from that point on, does exactly the mirror-image motion — ending up above $b$ by the same distance instead. Reflecting every such path across the barrier pairs them up one-to-one with paths that end up above $b$. That pairing is the entire trick.

Writing it down

Let $M_T = \max_{0 \le t \le T} W_t$ be the running maximum of Brownian motion up to time $T$. The reflection principle states

$$P(M_T \ge b) = 2\,P(W_T \ge b), \qquad b > 0.$$

In words: the probability the path ever reached level $b$ equals exactly twice the probability that it happens to be above $b$ at the final time $T$ alone. Every path ending above $b$ certainly touched $b$ along the way (since it started at 0), and the reflection argument shows there's an equal-probability mirror path, touching $b$ and ending an equal distance below it, for each one — so "touched $b$ and ended above it" and "touched $b$ and ended below it" are equally likely, and together they make up the full event $\{M_T \ge b\}$.

Since $W_T \sim \mathcal{N}(0,T)$, this gives a closed form:

$$P(M_T \ge b) = 2\left(1 - \Phi\!\left(\frac{b}{\sqrt{T}}\right)\right),$$

where $\Phi$ is the standard normal CDF. In words: a question about the entire path's extreme value has been reduced to a single lookup in a normal distribution table.

Sample paths above and pick a level in your head. Some paths that stay below your level the whole time count as "never touched." Some cross it and end up above; the reflection principle's claim is that exactly as many paths cross it and drift back down to end below it — you can see the rough symmetry by eye across many redraws, even though any single path's ending is unpredictable.

Worked example 1: probability of ever touching a level

For $T=1$ and barrier $b=1$: $P(M_1 \ge 1) = 2(1-\Phi(1/\sqrt{1})) = 2(1-\Phi(1))$. Using $\Phi(1) \approx 0.8413$: $P(M_1 \ge 1) = 2(1-0.8413) = 2(0.1587) = 0.3174$. So there's about a 31.7 percent chance standard Brownian motion ever reaches level 1 within one unit of time — nearly four times the roughly 8 percent chance it happens to be above 1 exactly at $t=1$ (computed as $1-\Phi(1)=0.1587$, then halved isn't needed — that's already $P(W_1\ge1)$ itself). The gap between 15.9 percent (ending above) and 31.7 percent (ever touching) is entirely paths that spiked above 1 at some point and drifted back down by $t=1$ — invisible if you only looked at the endpoint.

Worked example 2: pricing the probability behind a barrier option

A stock modeled as driftless Brownian motion (for simplicity) starts effectively at 0 in log terms, and a down-and-out barrier sits at $b=-0.5$ (in log-price units), with $T=0.25$ years to expiry and volatility scaled so the process has variance $T$. $P(\text{knocked out}) = P(M_T^{\downarrow} \le -0.5)$, using the mirror-image version of the formula for a minimum: $2\Phi(-0.5/\sqrt{0.25}) = 2\Phi(-1.0) = 2(1-0.8413) = 0.3174$. So there's a 31.7 percent chance this barrier knocks the option out before expiry — the same reflection-principle formula, just applied to a running minimum instead of a maximum, is the core building block inside every closed-form barrier option pricing formula.

What this means in practice

• Closed-form barrier option prices (up-and-out, down-and-in, and their combinations) are built directly from reflection-principle probabilities like the one above, layered with the risk-neutral drift and discounting.
• First-passage-time distributions — "how long until the price first hits a stop-loss level" — follow from differentiating the reflection-principle formula with respect to the barrier or time, giving the density of the hitting time directly (see First Passage Times).
• The classic ballot problem and related combinatorial brainteasers ("in an election counted vote by vote, what's the probability the winner was always ahead") are the discrete random-walk version of exactly this same mirroring argument.