It's true that even if cost of carry is zero, ATM options do not have a delta of exactly 0.5, due to the lognormal distribution. But for purposes of the discussion in this thread, an option with volatility of .5 is ATM and behaves like an ATM in that its gamma drops when IV rises. Under most circumstances, options sufficiently far OTM that their gamma rises when IV rises have a much smaller delta (or else they're deep ITM and have a much higher delta).
relation between ivol and gamma
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It's true that even if cost of carry is zero, ATM options do not have a delta of exactly 0.5, due to the lognormal distribution. But for purposes of the discussion in this thread, an option with volatility of .5 is ATM and behaves like an ATM in that its gamma drops when IV rises. Under most circumstances, options sufficiently far OTM that their gamma rises when IV rises have a much smaller delta (or else they're deep ITM and have a much higher delta).
How about this: could delta of ATM call (no carry) be say 0.75? If yes, could the delta of an OTM call option immediately next to the ATM be higher than 0.5?
My point is that while an approximate understanding of delta is enough, a complete understanding is the answer to avoid what might appears as paradoxes/contradictions/etc. Gamma then becomes easier to understand (a person understand goemetric interpretation of derivatives and can depict them in their minds).
Vast majority think that ATM call detla is 0.5 (if carry is zero), and those who know otherwise may not capture the whole picture.
Yes, if you jack up volatility and time remaining enough, your scenario could happen. But raising time and volatility like that also vastly expands the number of strikes that are "at the money" in the sense that a rise in volatility decreases their gammas.
I think you'll find that however you do it, an option with a delta of .5 will have more gammas as volatility decreases and less gammas as volatility increases.
Imo a trader causes him/her self problems by _only_ understanding the greeks from a trader mentality. If you unwound the definitions of what the sensitivities [delta, gamma, etc] meant in terms of their mathematical definitions in terms of partial derivatives (first order, second order, etc) with respect to price, time etc, the intuition goes way up, imo.
Then doing a surface plot and starring at it ties the loose ends together.
Then doing a surface plot and starring at it ties the loose ends together.
DMO:
Are you still puzzled with OTM deltas and the number 0.5?
Glad to hear you have played with volty and observed some of its effects on deltas for ATM options, but I am not clear whether you played with OTM options as well, and the results of your experiments. My intuition tells me that OTM options can have a delta higher than 0.5 , and even going much higher than that. The limit is of course 1.
As for gamma, it has rather to do with the distance from strike price to (ATM strike + threshhold). Shrink it, and gamma rises, increase it and gamma decreases.
Could someone give the threshold expression? It is a nice expression and should be easy to remember (if the maths done in my head are correct)?
I think we'll have to invent some vocabulary, as we're getting into concepts that are pretty interesting though not often discussed. Shall we call it the "positive/negative vomma threshold?" I assume that's the threshold you're talking about. No, I don't know how to calculate it. I'd be happy to learn though.
check at plus one std.