How a Down-and-Out Call’s Delta Changes Near Its Barrier
Summary
The document corrects a claim about the delta of a down-and-out call as the underlying approaches its lower barrier. It explains that delta generally falls toward zero near the barrier, since reaching the barrier knocks out the option. For options close to expiry, this decline can be smooth; for longer-dated options, delta may instead drop more abruptly near the barrier.
The response points to a published graph as supporting evidence, but does not reproduce its model assumptions or give numerical values. Another answer cautions that the relationship depends on the strike, barrier, and time to expiry. The discussion is qualitative, so it should not be treated as a complete pricing rule or as a substitute for calculating the option’s Greeks under specified contract terms and market assumptions.
Key ideas
- A down-and-out call’s delta tends toward zero as the underlying approaches the knock-out barrier.
- The shape of the delta decline depends on time to expiry, with smoother behavior near expiry and a sharper drop for longer maturities.
- Strike and barrier placement also affect the option’s delta profile.
- The document offers qualitative guidance and a reference graph, but no pricing inputs or numerical analysis.
Tags
Full text
# Delta of a Down and Out Call # Delta of a Down and Out Call I came across some graphs depicting the delta of a down-and-out call. They show that, if the risk free rate of return is 0, the delta is constant at 1. However, if the rate of return is for example 5%, the delta rises as the stock price approaches the barrier. I can't figure out why. ## Answer by Matt Wolf (score 0, accepted) https://quant.stackexchange.com/a/7030 You can find an accurate delta graph on page 62 of the following document: http://www.ederman.com/new/docs/insoutbarriers1.pdf. What you wrote is definitely incorrect. With a down and out call delta drops as the stock price approaches the barrier, it reaches zero smoothly as it approaches the barrier for close to expiration options and exhibits much more of discontinuous jump from values around 0.5 down to 0 at the barrier level for longer dated options. ## Answer by Strange (score -1) https://quant.stackexchange.com/a/7031 I am not sure if any numeric comment about the dynamics could be made without knowing the relative strike and the barrier. In general, however, you would expect delta to approach zero as you are approaching the barrier and delta approaching one as you go away from the barrier and deep into the money. In general, CDO and PUO are fairly easy to manage, since as the underlying approaches the barrier, the delta is already fairly low and the discontinuity is small. For obvious reasons, the discount to vanilla is fairly small and these products trade far less frequently then CUO/PDO and PDI/CUI.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.