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I want to hedge a barrier option, say a knock-out call with strike K and barrier B out-of-the-money. My idea was to start from the payoff diagram of this option, and try to accomodate it with vanilla options, as it can be done for instance in the (approximate) replication of a digital call by means of vanilla call options. However in this case it seems that this method fails simply because ofr such an option the payoff diagram is simply the same of a vanilla call... so what can be done??

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  • $\begingroup$ What's with the passive aggressive response? $\endgroup$ – Olaf Apr 16 '16 at 14:35
  • $\begingroup$ @RandomGuy - you have good instinct for solving the problem - you use vanillas (possible different strikes / expirations) to get as close to your hedge as you can, and then you're left with some residual path dependent risk that you're left with. $\endgroup$ – onlyvix.blogspot.com Apr 16 '16 at 18:48
  • $\begingroup$ @onlyvix.blogspot.com, thanks but do you know how to do it precisely? Can you provide a geometric construction of the payoff of the replicating contract, and can you just explicitly state which options precisely compose the replicating portfolio? $\endgroup$ – RandomGuy Apr 17 '16 at 11:17
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there are a number of ways to do this. You do have to make some modelling assumptions, however. eg continuity, BS model holds, or log stock price process is independent of level.

The most common way is to take the pay-off and geometrically reflect in the barrier. (i.e. pass to log coordinates and reflect). i.e. write the function as $f(x)$ where $x= \log S_t,$ now find a function $g$ such that $$ g(x) = f(x) $$ for $x>\log B$ and $$ g(x) = -f(2B-X) $$ for $x < \log B.$

Then statically replicate the $g$ behind the barrier at maturity. (Look up put-call symmetry.)

The resulting contract has close to zero value at the barrier. You then take contracts wholly paying off behind the barrier with varying maturity eg puts struck at $B$ and use them to cancel the value on the barrier iteratively starting at the end and stepping back towards 0. Note this is highly model dependent and requires continuity, however.

There is extensive discussion of these topics in my book the Concepts and Practice of Mathematical Finance.

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  • $\begingroup$ Sorry, I don't understand your answer. What does exactly mean "take the pay-off and geometrically reflect in the barrier"?? $\endgroup$ – RandomGuy Apr 17 '16 at 11:15

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