Why an ATM Down-and-In Put Can Have Lower Delta

Summary

The document raises a question about why an at-the-money down-and-in put may show a smaller delta than a comparable vanilla put. The response cautions against treating that comparison as automatic: following a downward price move, the barrier may be crossed, in which case the option becomes equivalent to the put. Because the down-and-in option initially costs less than the vanilla put, this outcome alone does not establish that its delta must be lower.

A lower delta is plausible when a small underlying move produces only a limited change in the probability that the barrier is breached. The response therefore points to the sensitivity of barrier-hit probability as an important part of the intuition, alongside the option’s conditional payoff after activation. It does not derive a general delta formula or specify the market assumptions behind the quoted typical comparison. Its main lesson is that barrier-option sensitivities can depend on conditions, and an explanation should account for cases where the usual relationship reverses.

Key ideas

• A down-and-in put’s delta depends partly on the chance that the barrier is reached.
• After a barrier breach, the option behaves like the corresponding vanilla put.
• The option’s lower initial value does not by itself prove that its delta is smaller.
• Barrier-option intuition should allow for conditions where the usual delta comparison reverses.

Tags

# Delta of ATM barrier option


# Delta of ATM barrier option












Can you please share the intuition behind the delta of a ATM down & in PUT being less than a ATM plain vanilla put (usually around 0.3 instead of 0.5)?

## Answer by Arshdeep (score 0)

https://quant.stackexchange.com/a/79930

Best case, upon down move, the barrier is sure to be broken and so the new price is the put price. Old price is less than a put price. So it's not obvious that it is lesser.

It will be true if movement in probability of barrier broken is small enough, which is probably what we say will happen, but it is not obvious the other explanations take this in account.

A good explanation should be sensitive to when results overturn

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.