The Reiner-Rubinstein barrier option formula
A single closed form prices all eight barrier variants under Black-Scholes — here is where it comes from, and where it stops being exact.
A barrier option behaves like a vanilla until the underlying touches a preset level — the barrier — at which point it either springs into existence (knock-in) or disappears (knock-out). Reiner and Rubinstein (1991) gave a closed-form price for all of them under Black-Scholes, and it is still the analytics a structuring desk reaches for first.
Eight variants, one toolkit
The family is down/up × in/out × call/put = eight contracts. The formula assembles each one from a small set of building blocks — standard-normal terms in the usual d1/d2 style, plus an extra parameter that carries the barrier level. Two constants do the heavy lifting: a drift term μ = (b − ½σ²)/σ² that captures the cost of carry, and λ, which sets how the barrier reflects the distribution.
The reflection principle
The intuition is a mirror. To value a knock-out you need the probability that the path never touches the barrier; the reflection principlefor Brownian motion computes exactly that by subtracting a "reflected" image of the payoff across the barrier. The (H/S) power terms in the formula are that reflection — the image option that cancels the paths which would have breached the level.
In-out parity — a free consistency check
A knock-in plus the matching knock-out equals the plain vanilla: in + out = vanilla. If the barrier can only either knock you in or leave you out, owning both must reproduce the unconditional payoff. It means you only ever need to price one side; the other comes for free by subtraction, and it is the quickest way to sanity-check an implementation.
Where it stops being exact
The closed form assumes continuous monitoring, constant volatility and geometric Brownian motion. Real contracts monitor the barrier discretely (daily or at fixings), which makes a knock-out slightly less likely to trigger — the Broadie-Glasserman-Kou correction shifts the barrier by about 0.58·σ·√Δt to compensate. And because it is a single-vol model, it ignores the volatility smile, which matters a lot for a payoff whose whole value hinges on hitting one level. For those cases you move to a local- or stochastic-vol model, or Monte Carlo.
Try itOpen the Barrier pricer, price a down-and-out call, then check that adding the down-and-in call gives back the plain vanilla — in-out parity, live.Go deeper · ProBarrier hedging, pin risk and gap risk are worked through in the Coach.