Why Up-and-Out Option Delta Can Turn Negative Near Its Barrier
Summary
The document explains why an up-and-out call can have negative delta when the underlying approaches its upper knock-out barrier. Near the barrier, a rise in the underlying has two opposing effects: it increases the ordinary call payoff conditional on survival, but also raises the chance that the option will be knocked out. The barrier effect can dominate, reducing the option's value as spot rises and producing negative delta.
The discussion uses the Black–Scholes setting and a near-expiry intuition: a short option may gain value as the underlying approaches the barrier, while a short position in the underlying can offset that change. It also clarifies that delta is the price sensitivity used for hedging, not the probability of finishing in the money. For vanilla options those quantities can appear similar, but a barrier condition can separate them substantially. The explanation is qualitative and does not give a full derivation or quantify how delta varies across barrier distance, expiry, or model assumptions.
Key ideas
- An up-and-out call's value can fall as spot rises near its upper barrier because knock-out risk increases.
- The barrier-related loss in value can outweigh the usual positive call payoff sensitivity and make delta negative.
- A short underlying position can hedge a short option's changing value in the described near-expiry intuition.
- Delta measures price sensitivity and is not the same quantity as the probability of finishing in the money.
- The discussion is qualitative and does not establish the behavior for every maturity or model.
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Full text
# Up And Out, negative delta close to barrier # 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 ## Answer by Paul (score 2) https://quant.stackexchange.com/a/78703 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 ## Answer by James M.Shihua (score 0) https://quant.stackexchange.com/a/85300 I think you reverse the cause and effect of delta hedging. In your comment: "isn’t delta also supposed to be number of the underlying to hold in order to hedge the option?" This is true because when underlying goes up $1, your option goes up $Delta. So you short Delta shares of underlying to offset your risk. However, it is not because of your delta hedge strategy that makes delta positive or negative. On the other hand, the probability of a vanilla option ending up in the money is N(d2) not Delta=N(d1). For a vanilla call or put, they can be quite similar. But in this up-and-out example, the probability ITM and delta can be quite different.
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