Barrier Options

Every payoff we have priced so far — vanilla calls, puts, even American options — depends only on where the underlying ends up, or at worst on when you choose to exercise. A barrier option is the first contract in this course whose payoff depends on the whole path: whether the price ever touches a trigger level H along the way. That path-dependence is exactly why desks reach for Monte Carlo simulation to price them — you cannot read the payoff off the terminal price alone, so you need to simulate (or otherwise track) the entire trajectory.

Barriers are enormously popular on FX and commodity desks precisely because they are cheaper than the vanilla option covering the same strike and maturity. A corporate treasurer who wants protection against the yen weakening past 150, but is happy to give up that protection if it ever strengthens dramatically past 130 first, is paying for less insurance — and the market charges less for it.

The taxonomy: in, out, up, down

A barrier option is built from three independent choices, giving eight variants in total:

Call or put — the usual choice of direction.
Knock-out or knock-in — an out option starts alive and dies (pays zero, forever) the instant the barrier is touched; an in option starts dead and only springs to life — becoming an ordinary vanilla option — if the barrier is touched.
Up or down — whether the trigger barrier H sits above the current spot S_0 (an up-barrier) or below it (a down-barrier).

So an up-and-out call behaves exactly like a normal call (S_T - K)^+ — right up until the price ever touches a barrier H > S_0, at which point it dies and pays nothing, no matter how the price finishes. A down-and-in put is worthless unless the price falls to touch a barrier H < S_0 at some point during its life — only then does it "switch on" and become an ordinary put (K - S_T)^+.

Worked example: an up-and-out call

Take a call struck at K = 100, and add an up-and-out barrier at H = 130. If the price wanders up to 120 and stays below 130 the whole way, the option behaves exactly like a vanilla call — it pays 20 at maturity. But if the price so much as touches 130 at any point — even for an instant, on day 3 of a one-year contract — the option is dead. It does not matter that the price might drift back down to a very profitable-looking 115 by expiry: once knocked out, always knocked out.

The chart below draws the idealized terminal payoff — what the option pays as a function of where it finishes, assuming the barrier was never touched along the way. It is the ordinary hockey stick, sliced off to zero above H. Slide the barrier down and watch how much of the upside gets cut away — the closer H sits to the strike, the less of the vanilla payoff survives.

Read this chart with one big caveat: it is not the whole story. Two paths can both finish at S_T = 115, yet one pays 15 (it never touched 130) and the other pays 0 (it spiked to 135 on day 3 before drifting back down). The true price is an expectation over all paths, most of which this simple picture cannot show — which is exactly why the number underneath the chart has to come from simulating whole paths, not from a formula in S_T alone.

In–out parity, and why barriers are cheap

Every path either touches the barrier at some point, or it never does. Add up an in option and the matching out option — same strike, same barrier, same maturity, same call-or-put — and every path pays off exactly as the vanilla option would: the touched paths are covered by the in, the untouched paths by the out. That gives a clean parity relation, with no model assumptions required at all:

C_{\text{in}} + C_{\text{out}} = C_{\text{vanilla}}

For matching strike, barrier and maturity: knock-in price + knock-out price = vanilla price.
Both barrier prices are therefore non-negative and each is at most the vanilla price.
The parity holds call-for-call and put-for-put — it never mixes a call with a put.

That identity is also the cleanest way to see why a barrier option is cheaper: it is a vanilla option with a chunk of its value carved off and handed to the other side of the parity relation. An out-option gives up the states of the world where the barrier is touched; an in-option gives up the states where it isn't. Removing entire branches of the payoff tree can only remove value, never add it — so both C_{\text{in}} and C_{\text{out}} sit strictly below C_{\text{vanilla}}.

Because the buyer is paying for exactly the protection they need, and not a cent more. An airline hedging jet fuel might genuinely believe an extreme spike is unlikely, or might already have other hedges that kick in above a certain level — so paying full vanilla premium for protection against a scenario they consider remote (or already covered) is wasteful. Selling away that tail in exchange for a lower premium is a rational trade, not a mistake. The same logic runs the other way for a knock-in: a buyer who only wants protection to activate if things get bad enough pays nothing for protection they don't need until the market actually moves against them.

It is tempting to assume a barrier option's price moves the same direction as volatility that every vanilla option's does — more volatility, more value. That is not always true near the barrier. For an out-option trading close to its knock-out level, more volatility mostly means the barrier is now more likely to be hit, killing the option — an effect that can outweigh the usual benefit of extra optionality. Out-barrier options can therefore have negative vega in some regions, the one place in this course where "more uncertainty always helps the option holder" quietly breaks down.

Yes, a lot. A continuously monitored barrier is watched every instant, so even a fleeting intraday spike triggers it. A discretely monitored barrier — checked only at, say, each day's close — can miss a spike that reverses before the close print. Discrete monitoring therefore makes knock-out less likely, pushing the out-option's price up (closer to the vanilla price) and the in-option's price down. Real-world contracts almost always specify discrete monitoring for exactly this reason — it is easier to verify from an official closing price, and it is slightly more generous to the option holder of an out-barrier.