What's the intuition behind barrier option pricing and when are knock-ins cheaper than vanillas?
I'm reviewing exotic options for FRM Part I and barrier options keep coming up. I understand the basic payoff — the option activates or deactivates when the underlying hits a barrier. But I'm struggling with the pricing intuition. Why would someone buy a knock-in call instead of a regular call? And how does the in-out parity work?
Barrier options are among the most widely traded exotics, especially in FX and commodity markets. Let's build the intuition systematically.
Types of Barrier Options:
| Type | Barrier Direction | What Happens |
|---|---|---|
| Down-and-out call | Barrier below spot | Option dies if spot falls to barrier |
| Down-and-in call | Barrier below spot | Option activates if spot falls to barrier |
| Up-and-out put | Barrier above spot | Option dies if spot rises to barrier |
| Up-and-in put | Barrier above spot | Option activates if spot rises to barrier |
In-Out Parity:
This is the most elegant relationship in barrier options:
Knock-In + Knock-Out = Vanilla
So if a vanilla call costs $5.20 and the down-and-out call costs $4.10, the down-and-in call must cost $1.10. This makes sense: between them, the knock-in and knock-out cover all possible paths of the underlying.
Why Buy a Knock-In?
Knock-in options are cheaper because they only activate in specific scenarios. Consider a portfolio manager at Crestview Capital who believes EUR/USD will first dip to 1.05 (triggering the barrier) before rallying to 1.15. A down-and-in call with barrier at 1.05 and strike at 1.08 might cost 40% less than the equivalent vanilla call, because it only has value along paths that touch 1.05.
Pricing Intuition:
The further the barrier is from the current spot (and thus less likely to be triggered), the cheaper the knock-in and the closer the knock-out is to the vanilla price.
Common Exam Traps:
- A down-and-out call with a barrier above the strike is practically worthless — if the spot falls to the barrier, the option is already out of the money AND gets knocked out.
- Rebate options pay a fixed amount when the barrier is hit, adding complexity.
- Barrier options have discontinuous delta near the barrier — hedging them is much harder than vanillas.
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