Up And Out, negative delta close to barrier

For Up And Out options, is there an intuition as to why delta becomes negative as spot approaches the barrier. Thinking in terms of replicating portfoliio I would have assumed delta is always non-negative (since there is positive (even if very small) probability of ending up below the barrier at T) approaching zero close to the barrier since there will be nothing to hedge once the barrier is crossed

I.e if you short an up and down barrier option why you need to short an option when close to barrier?

Picture from here

• I think you may be mixing up the value of the option and the delta. Indeed the option has a (small) positive value becasue "there is positive (even if very small) probability of ending up below the barrier at T". But the Delta is negative because a further rise in S makes that probability and the value smaller. $\Delta V / \Delta S$ is negative. Commented Mar 18 at 11:46
• Yeah it makes perfect sense to me if I think of it simply as a derivative wrt price of underlying. But isn’t delta also supposed to be number of the underlying to hold in order to hedge the option? I would have expected to long a very small amount of underlying close to the knock out barrier to hedge a short option paying out S - K at expiry with very small probability and 0 if barrier is crossed. Does it mean that delta being the number of underlying in replicating portfolio to hedge is wrong?
– Paul
Commented Mar 18 at 13:28

1 Answer

Answering my own question in case anyone else is confused about it. Think close to expiry under BS model, if you short an option you are making money as it approaches the barrier and keeps losing it's value. To hedge that you can short the underlying to offset delta gain of the short option. If underlying keeps increasing in price you gain from shorted option and lose from shorted underlying. If underlying price decreases, you lose value from short option (because it's less likely to hit barrier and will be exercised at S-K at T) but gain value from shorting the underlying, again both offset each other

The reason why it was confusing to me is because vanila options either go up (Call) or down (Put) in value as underlying price increases. Up and Out barrier imposes another condition that can devalue the option as underlying goes up in value and approaches knock out barrier so there are two forces at work putting presurre on the option value in opposite directions