Barrier Options

A barrier option is an ordinary call or put with a trip-wire attached. You pick a barrier price level, and the option's existence depends on whether the underlying ever touches it. A knock-out option dies the moment the barrier is hit, all your rights vanish. A knock-in option is the opposite: it's dormant and only springs to life once the barrier is touched. Because whether the barrier gets hit depends on the path the price takes, not just where it ends up, barrier options are path-dependent, which is what makes them exotic.

Why would anyone want an option that can evaporate? Money. A knock-out option is cheaper than the plain vanilla version, sometimes far cheaper, because you're handing back some of the scenarios in which it might pay. If you're fairly sure the price won't crash to the barrier, you buy the knock-out, pocket the discount, and only lose out in the world you didn't expect anyway.

The eight flavours

Barriers come in a tidy grid: the barrier sits above the current price (an "up" barrier) or below it (a "down" barrier), the option knocks in or knocks out, and it's a call or a put. That's eight combinations, named like down-and-out call or up-and-in put. The names read literally: a down-and-out call is a call that dies if the price falls to a lower barrier.

In-out parity: the pricing shortcut

There's a lovely relationship that makes barrier options half as much work to price. Over the life of the trade, either the barrier gets hit or it doesn't, there's no third option. A knock-in pays exactly when the barrier is hit; a knock-out pays exactly when it isn't. So owning both, with the same strike, barrier, and expiry, is identical to owning a plain vanilla option:

$$\text{knock-in} + \text{knock-out} = \text{vanilla}.$$

This is in-out parity, and it means you only ever need to price one of the pair, subtract from the vanilla to get the other.

Worked example

Suppose a plain vanilla call on a stock is worth $8.00. You also price the matching down-and-in call, the one that only activates if the stock first falls to the barrier, and find it worth $1.50. By in-out parity, the down-and-out call must be worth

$$\text{down-and-out} = \text{vanilla} - \text{down-and-in} = 8.00 - 1.50 = 6.50.$$

So a trader who's confident the stock won't fall to the barrier can buy the down-and-out call for $6.50 instead of the $8.00 vanilla, saving $1.50, roughly 19% off. That saving is exactly the value of the scenarios (price crashing to the barrier) that they're willing to give up. If the barrier is set well below where the stock is likely to trade, the knock-out is only a touch cheaper than vanilla; set it close, and the discount is steep because the knock-out is far more likely to be triggered.

Where they get dangerous

Barrier options look tame but are treacherous to hedge, and the trouble concentrates right at the barrier. A knock-out has a discontinuous payoff: one tick above the barrier the option is alive and worth something; one tick below it's worth zero. That cliff makes the Greeks explode near the barrier, delta can be enormous and even flip sign, and gamma spikes, especially close to expiry. A dealer who's short a barrier option can find their hedge ratio swinging violently as the price hovers around the level.

Because of the path dependence, closed-form prices exist only under clean Black-Scholes assumptions; for realistic volatility they're usually priced by Monte Carlo (checking each simulated path against the barrier) or finite differences (imposing the barrier as a boundary on the grid).

Common pitfalls

• Confusing path with endpoint. A knock-out can finish deep in the money and still pay nothing if it touched the barrier at any earlier moment. Only the path matters for the trigger.
• Ignoring monitoring frequency. "Continuously monitored" and "monitored at the daily close" barriers price differently, discrete monitoring makes a knock-out slightly more valuable (harder to trip). Always check which one the contract specifies.
• Assuming vanilla intuition for the Greeks. Near the barrier, delta and gamma behave nothing like a vanilla option's. Sizing a hedge off vanilla Greeks will leave you badly exposed.
• Forgetting rebates. Some knock-out contracts pay a small rebate when they're knocked out. Leaving that cash flow out underprices the option.