First Passage Times

A trader sets a stop-loss ten ticks below entry. A bond defaults the instant a firm's asset value first drops below its debt. A barrier option knocks out the moment the underlying first touches a strike. All three questions have the same shape: given a random path, when does it first reach a specific level, and how likely is that to happen at all? First passage time theory answers exactly this.

An analogy: the drunk near the cliff edge

Picture someone stumbling randomly left and right, starting ten steps from a cliff edge. You don't care about their whole wandering path — you care about one number: how many steps until they first reach the edge, if they ever do. Some wanderers reach it fast, some wander for ages first, some (if there's a gentle breeze pushing them away from the edge) might never reach it at all. First passage time is the random variable "steps until first arrival," and its distribution captures all of this variability in one object.

The mathematics, one symbol at a time

Let $W_t$ be a Brownian motion (see Brownian Motion) starting at $W_0 = 0$, and fix a level $b > 0$. The first passage time is

$$T_b = \inf\{ t \geq 0 : W_t = b \},$$

read as: the smallest time $t$ at which the path first equals $b$. For a driftless Brownian motion, $T_b$ is finite with probability 1 — the path is guaranteed to eventually touch any level — but its expected value is infinite, because occasionally the wait is astronomically long. Its density is the inverse Gaussian:

$$f_{T_b}(t) = \frac{b}{\sigma\sqrt{2\pi t^3}} \exp\left(-\frac{b^2}{2\sigma^2 t}\right),$$

in words: the chance of first hitting $b$ right around time $t$ is largest for moderate $t$ and tapers off (slowly) for very large $t$ — that heavy tail is exactly why the average wait is infinite even though hitting is certain. Add a drift $\mu$ (the path trends toward or away from $b$) and, if $\mu > 0$, the expected hitting time becomes finite: $\mathbb{E}[T_b] = b/\mu$ — distance divided by speed, just like everyday travel time.

Worked example 1: gambler's ruin with a stop-loss

A trader's P&L moves +$1 or −$1 each tick with equal probability, starting at $0, with a stop-loss at −$5 and a take-profit at +$10. Using the classical gambler's-ruin formula, the probability of hitting the upper barrier $a=10$ before the lower barrier $-b=-5$ (symmetric random walk) is $b/(a+b) = 5/15 = 1/3$. So there's roughly a 33% chance the trade hits the profit target before the stop, and 67% it stops out first — despite the two outcomes looking "50/50" on the surface, the asymmetric barrier distances drive the real odds.

Worked example 2: expected time to hit a barrier with drift

A strategy's cumulative P&L behaves like a Brownian motion with drift \mu = \2 per day and needs to first reach \20 to trigger a scale-up rule. Using $\mathbb{E}[T_b] = b/\mu$: $\mathbb{E}[T_{20}] = 20 / 2 = 10$ days. If the drift were flat ($\mu=0$), the same $20 target would still be hit eventually with certainty, but the expected wait would be infinite — a useful reminder that "will happen" and "happens soon" are very different claims.

Drag the drift and volatility above and watch how often, and how quickly, a sample path first crosses a horizontal barrier — zero drift still crosses eventually, but the wait stretches out unpredictably.

What this means in practice

First passage times underpin stop-loss and take-profit sizing, barrier and knock-out option pricing, and structural credit models where default is defined as assets first falling below liabilities. The gambler's-ruin ratio in example 1 is the same logic desks use to size stop distances relative to targets, and the $b/\mu$ formula in example 2 is why adding drift (an edge) to a strategy shrinks expected time-to-target, not just improves the odds of getting there.