What's the deal with Barrier Options?

Barrier options offer several interesting benefits to professional traders with increased cost-effectiveness and complex hedging needs, and they can be statically replicated with vanilla options. Today we look at the basic payoff structure of an “up-and-out Call”, and replication with vanilla options so to avoid OTC transactions.

Payoff structure of an Up-and-Out Call Option
Examples are from Static Options Replication by Derman, Ergene, and Kani.

Given that

Underlying	100
Strike	100
Barrier	120
Rebate	0
Time to Expiration	1 Year
Dividend Yield	5%
Expected Volatility	25%
Risk-Free Rate	10%

We get the following option values

Up-and-Out Call	0.656
Vanilla Call	0.114

Payoff diagram

Replication with vanilla options

The key to replication is about matching theoretical payoffs at various future underlying values so that the portfolio behaves like the barrier option. See the Goldman Sachs Research Note for detailed notes. I will leave the replication for the example Up-and-Out Call here,

Quantity	Strike	Expiration	Values
0.16253	120	2	0.0001
0.25477	120	4	0.018
0.44057	120	6	0.106
0.93082	120	8	0.455
2.79028	120	10	2.175
-6.51351	120	12	-7.14
1	100	12	6.67
		Total Cost	2.284