# Question on Barrier Option and Skew

If you bought an Equity Call Option with a Down-and-In Barrier, are you Long Skew or Short Skew? Please provide explanation as well. Thanks.

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You're going to have to provide a more rigorous definition of "skew" if you want a meaningful answer. There are many metrics that characterize skew, such as 25-delta minus 75-delta vol, and the choice of metric will affect the result. (It's also quite possible, depending on your metric, that some of these barrier options will be "long" it and some will be "short" it). –  Brian B Nov 3 '13 at 19:02
Is this a homework question? –  chrisaycock Nov 4 '13 at 3:57
I am willing to help if you, the OP, work through this together: First thought, and I like to hear your input on is: Imagine you are a trader and just sold a down-and-in call to a client (lets make it easy and lets say the barrier is set equal to the strike of the underlying option). How would you hedge the risk at the initiation of the trade? Explain why it makes sense to buy a vanilla put at initiation and why you are hedged as long as the barrier is not broken. Also, think and explain how you need to adjust the hedge if/when the barrier is breached. –  Matt Wolf Nov 4 '13 at 4:21
Also, google and read up on put-call symmetry and report back, unless you by then already got a clue about the skew risk of your barrier option. –  Matt Wolf Nov 4 '13 at 4:26
@Matt Wolf: At initiation, if spot goes up, the trader is hedged because the option has not Knocked In yet (no payoff to client). However, if spot goes down and hit the barrier, the client would get the value of a vanilla call option. In order to hedge this risk, the trader needs to enter a transaction that would give positive value when spot is below the barrier. So buying a vanilla put at initiation makes sense. When barrier is breached, the option would behave like a vanilla call option, so the trader would be buying underlying stock to delta hedge the option sold to client. –  chengcj Nov 4 '13 at 4:47

If spot moves down towards Knock-In Barrier, I want high vol (to knock the barrier). Therefore, I'm long low strike (long skew).

To summarise:

Down-and-In Barrier Option = Long Skew

Down-and-Out Barrier Option = Short Skew

Up-and-In Barrier Option = Short Skew

Up-and-Out Barrier Option = Long Skew

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vanilla = down and out + down and in Vanilla is not skew sensitive.

The down and out can be modelled as a vanilla plus the vanilla pay-off geometrically reflected in the barrier. Its price will go up if the strike vol goes up or the reflected strike vol goes down. So we want skew if that means the high strike implied vols going up and the low strikes ones going down.

Opposite for the down and in.

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