Replicating an Up-and-Out Barrier Payoff with Vanilla Options
Summary
The document explains the distinction between matching an up-and-out call’s terminal payoff and replicating its full path-dependent behavior. If the underlying has not touched the upper barrier, a long call at the lower strike combined with a short call at the barrier produces the desired capped payoff at expiration. In the example, the strikes are 50 and 60, and the option maturity is 0.75 years.
That vanilla call spread alone does not cancel its payoff when the underlying crossed the barrier earlier: at expiration above 60, both calls are in the money, but their combined payoff is zero; the key failure occurs when the barrier was touched before expiration and the underlying later finishes between the strikes. The response says additional options at earlier maturities are needed to account for such path events, while cautioning that a static replication using only vanilla options may not be possible. It supplies an intuition, not a complete hedge construction or proof.
Key ideas
- A long call at the lower strike and a short call at the barrier form a capped terminal payoff.
- That call spread matches the surviving barrier option payoff if the barrier has not been touched.
- A barrier option depends on the path taken before expiration, not only the final underlying price.
- Earlier-maturity options may be needed to account for barrier hits before expiration.
- The response doubts that vanilla options alone can provide a static replication.
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Full text
# Artifically creating Barrier option
# Artifically creating Barrier option
I am looking for some insights on how a Barrier option payoff can be replicated either with Plain vanilla options.
In Chapter 25, Hull has provided one such example to replicate a Up and Out Barrier option with boundary at 60 and strike at 50 and maturity 0.75 years.
He suggested portfolio of call options where first 2 call options has strike 50 and 60 and bother are maturing at 0.75. The remaining options have maturities in other time points like 0.50 and 0.25.
I wonder how this strategy can replicate Barrier option payoff at 0.75? For example if at 0.75, underlying's price is > 60, then Barrier option has zero value. However the replicating portfolio, which has call options with strikes 50 and 60, will be ITM both.
Could you please help how exactly this works?
Any insight and/or online reference can be very helpful.
Thanks for your time.
## Answer by NoIdeaWhatIamDoing (score 1)
https://quant.stackexchange.com/a/81816
i will drop my 5 cents, and hope it help.
you can replicate an up and out barrier option, by buying a cap spread and a digital that has the same payoff of the cap spread's profit.
Lets say, notional 10m, eurusd underlying, bought call strike 1.05, sold call strike 1.10, sold digital that pays 500k. During, cap spread you 'win', but if you reach the digital your profit is 0.
## Answer by MrLCh (score 0)
https://quant.stackexchange.com/a/79318
Unfortunately I could not find the reference to the replication in my version of Hull, but I will try to explain the replication argument that matches up with what you describe.
Let's first look at the payoff of the option $C_{uo}$ at expiration and assume that the asset price $S$ has not hit the barrier:
$$ C_{uo}(S) = \begin{cases} 0, S < 50 \text{ or } S>60 \\ S - 50, \text{ else } \end{cases} $$
This payoff can be replicated by buying a call option with strike 50 and selling a call option with strike 60.
Now you have to have a look at the cases where the asset is above the barrier before expiration: The up and out option becomes worthless. This is why you need to hold options at earlier points in time to replicate this payoff (although I don't think it can be replicated (at least not static) using only vanilla options).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.