Handbook question 17.7

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Hi David, kindly see the below question:

Trader A purchases a down-and-out call with a strike price of USD 100 and a barrier at USD 96 from trader b. Both traders need to unwind their delta hedge at the barrier. Which trader is more at risk if there is a price gap (discontinuity) that prevents them from exiting the trade at the barrier?

a. Trader A has bigger risk.
b. B has bigger risk.
c. both have the same risk.
d. neither has any risk because both are hedged.

Would you kindly explaine the answer:
Answer is b. Each trader replicates dynamically the down-and-out call as a hedge. Trader b sold the option, so needs to replicate a long position in this call. The hedge ratio for a down-and-out call resembles the usual one except that is has an abrupt discontinuity, dropping to 0 below barrier. When the price jumps down below the barrier, trader b will be stuck with a large loss.


My question: Trader b is selling the barrier call. When the price jumps below the barrier, trader b should have a gain. Why does the answer say trader b has a large loss?

Thanks a lot!

 

[David Harper CFA FRM]

Hi handbookandbt,

You'd be right except they are delta hedging.
The barrier changes the particulars of the delta hedge but we don't need to worry about that here. Keep in mind the basic delta hedge:
long call options = positive position delta (= positive percentage delta between 0 and 1 * quantity of options), which is delta hedged with a short position in shares; and
short call options = negative position delta (= positive percentage delta between 0 and 1 * quantity of options) which is delta hedged with a long position in shares

As the barrier is down (the exotic option dies if the price drops to 96), the price discontinuity implied is an abrupt DROP in the price. So, Trader B has written (short) the barrier option but delta hedges with a long position in shares. So the question is looking for the loss on the hedge (long shares) in the case of an abrupt drop in share price.

I hope that helps, David

 

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Thank you David!