FRM Exam Part I · Exotic Options

Barrier Options: Knock-In and Knock-Out Explained

Updated 11 October 2026 · Fact-checked

A barrier option is an option whose payoff depends on whether the underlying price touches a set barrier before expiry. Knock-ins start to exist when the barrier is hit; knock-outs cease to exist. To solve questions, identify the barrier type, check the path, then apply in-out parity: knock-in + knock-out = vanilla.

Understand Barrier Options

A barrier option is a path-dependent option. It is a normal call or put with one extra condition: a barrier level H. Whether the option pays depends on whether the asset price touches H at any time before expiry.

There are two families. A knock-out option exists at the start and is cancelled if the price hits the barrier. A knock-in option does not exist at the start and only comes alive if the price hits the barrier. Barrier options are cheaper than vanilla options because they can lose value (knock-out) or may never activate (knock-in).

The name tells you the direction. Up means the barrier is above the starting price. Down means it is below. So an up-and-out call is cancelled if the price rises to the barrier. A down-and-in put activates if the price falls to the barrier. Combining up/down, in/out and call/put gives eight standard types.

Some options pay a rebate if they are knocked out, or if they never knock in. Read the question for it.

The key hedging problem is the barrier itself. Near the barrier, the value and delta of a barrier option can change sharply. For example, an up-and-out call has a high delta just below the barrier is not the usual pattern: its value drops toward zero as the price nears H, so its delta becomes large and negative, and gamma can be large and change sign. Delta hedging is hard and costly there. Dealers often use static options replication (a portfolio of vanilla options that mimics the barrier option's payoff on the barrier) or shift the barrier slightly when pricing.

Key formulas to remember

In-out parity: Knock-in + Knock-out = Vanilla option; Applies when both have the same type (call or put), strike, expiry and barrier, and the same rebate treatment (no rebate). Holding both gives exactly the vanilla payoff.
Knock-in value from parity: Knock-in = Vanilla − Knock-out; Use this to find one price when you know the other two.
Barrier price bound: Knock-in ≤ Vanilla and Knock-out ≤ Vanilla; Without rebates, barrier options are never worth more than the matching vanilla option, since both values are non-negative.
Payoff of knock-out call (barrier never hit): max(S_T − K, 0); If the barrier is hit at any time, payoff is 0 (plus any rebate).
Payoff of knock-in call: max(S_T − K, 0) if barrier hit; otherwise 0; Same logic for puts with max(K − S_T, 0).
Barrier starting position: Down: H < S₀. Up: H > S₀; If the barrier is already breached at the start, a knock-out is worthless and a knock-in is a vanilla option.

How to solve Barrier Options questions

Use this routine for any barrier option question. It handles payoffs, classification and parity.

1Read the name: direction (up or down), then in or out, then call or put.
2Compare the barrier H with the starting price S₀ to confirm direction. Note any rebate.
3Check the price path: did the underlying touch H at any time before expiry, not just at maturity?
4Apply the rule: knock-out pays the vanilla payoff only if H was never hit; knock-in pays it only if H was hit.
5Compute the vanilla payoff at expiry: max(S_T − K, 0) for a call, max(K − S_T, 0) for a put.
6For price questions, use in-out parity: knock-in + knock-out = vanilla. Solve for the unknown.
7For hedging questions, ask where delta and gamma become unstable (near H) and name the fix: static replication, or hedging with shifted barriers.

Quickest way: Barrier in 30 seconds

When to use it: Use when a multiple-choice question gives a price path or two option prices and asks for a payoff or a missing price.

If the question gives prices, go straight to parity: in + out = vanilla.
If it gives a path, ask one question: did it touch H? Yes: knock-in lives, knock-out dies. No: the reverse.
Then compute the vanilla payoff only if the option is alive.
Eliminate options that make a barrier option worth more than the vanilla.

Common mistakes in Barrier Options

Treating a barrier option as alive or dead based only on the final price.
Vanilla options depend only on S_T, so students carry that habit over.
Fix: Barrier options are path dependent. Check whether the price touched H at any time before expiry.
Confusing 'up' and 'down' with in or out.
The names pack three features into one phrase.
Fix: Read in order: up/down gives barrier position vs S₀; in/out gives activation or cancellation; then call/put.
Applying in-out parity to options with different strikes, barriers or rebates.
The rule is remembered as a slogan without its conditions.
Fix: Parity holds only for matching call/put type, strike, expiry and barrier, and no rebate.
Thinking a barrier option can cost more than the vanilla.
Students focus on the extra features rather than the extra conditions.
Fix: Without rebates, the barrier option's payoff is never above the vanilla payoff, so its price is not above either.
Saying barrier options are easy to delta hedge because they are cheaper.
Low premium is confused with low risk.
Fix: Delta and gamma can jump near the barrier. Hedging is hard there, and static replication is a common alternative.

Worked examples

Example 1

A European down-and-in call on a stock has strike $100 and barrier $85. The stock starts at $100. Over the life of the option the stock falls to $82 and then rises to $115 at expiry. What is the payoff? What would the payoff be for a down-and-out call with the same terms?

Show the solution

The barrier $85 is below the start of $100, so this is a valid down barrier.
The price touched $82, which is below $85, so the barrier was hit.
The knock-in call is activated. Its payoff is max(115 − 100, 0) = $15.
The knock-out call was cancelled when the barrier was hit, so its payoff is 0.
Check with parity: 15 + 0 = 15, the vanilla payoff max(115 − 100, 0).

Answer: Down-and-in call pays $15; down-and-out call pays $0.

Example 2

A vanilla European put is worth $6.40. An up-and-out put with the same strike and expiry, and a barrier above the current price, is worth $4.90. There is no rebate. What is the value of the matching up-and-in put?

Show the solution

Same type, strike, expiry and barrier with no rebate, so in-out parity applies.
Knock-in + knock-out = vanilla.
Knock-in = 6.40 − 4.90 = 1.50.
Sanity check: 1.50 is positive and below the vanilla value of 6.40.

Answer: The up-and-in put is worth $1.50.

Exam tips

Parity questions are quick marks. Check the conditions (same strike, expiry, barrier, no rebate), then subtract.
For path questions, mark the barrier on a simple sketch of the path and ask whether it was touched.
For hedging questions, link the difficulty to the barrier: delta and gamma become unstable and discontinuity risk is high close to H. Static replication is a standard remedy.
Remember that barrier options are cheaper than the matching vanilla. Reject answers that say otherwise.

Practice questions from Exotic Options

Barrier Options: frequently asked questions

What is the difference between up-and-out and down-and-in options?

An up-and-out option has a barrier above the starting price and is cancelled if the price rises to it. A down-and-in option has a barrier below the starting price and only comes into existence if the price falls to it. The words give direction first, then whether it activates or cancels.

What is in-out parity for barrier options?

A knock-in plus a knock-out with the same type, strike, expiry and barrier equals the matching vanilla option. Exactly one of the two is alive at expiry, so together they replicate the vanilla payoff. It assumes no rebates.

Why are barrier options hard to hedge?

Near the barrier, the option value changes sharply with a small price move, so delta and gamma can become large and unstable. Frequent rebalancing is costly. Dealers may use static replication with vanilla options or price with a shifted barrier.

Why are barrier options cheaper than vanilla options?

A knock-out can be cancelled before expiry and a knock-in may never activate. Both features can only remove payoff compared with a vanilla option. So the premium is lower, assuming no rebate.