I have been going through Euan Sinclair's book and there it is written that the Gamma of an option is not symmetric with respect to ATM price because it is scaled by the underlying price.
What does it mean to be scaled by the underlying price? Also, can someone give an intuitive answer explaining why Gamma is high just before the ATM price?
On a different topic, can someone explain what does N(d1) signify in the BSM equation? I understand N(d2) gives the probability that an option will end in the money in a risk neutral world but have not been able to satisfactoraly understand N(d1).
1) I don't have the book with me and it's sorta difficult to understand the meaning of the statement without more context.
2) If you think of gamma as the rate of change of delta wrt to the change in the underlying, it should make sense that ATM is where gamma is highest. ATM is where delta can change the most (just imagine a situation where you're right before expiry and the price of the underlying fluctuates around the strike).
3) Here you go, from a paper I obtained by googling: "[N(d1)] ...factor by which the present value of contingent receipt of the stock, contingent on exercise, exceeds the current value of the stock" http://www.ltnielsen.com/wp-content/uploads/Understanding.pdf
I also thought that Gamma should be the maximum at ATM. But in the book it is written that the maximum is not exactly at the Strike Price but before the strike price given by the formula:
Gamma_max = Sexp(-(b+3/2(sigma^2))T)
where, S : Underlying Price b: can take various values depending on the underlying
So I'm trying to understand why is it maximum just before the Strike Price and not at the Strike Price.
Also, I tried to read that paper and could not get my head around the language used in there. After reading it, according to me N(d1) is the factor by which the PV of the underlying, if exercised, will exceed the current price of the underlying. Is that correct?
Yeah, so, again, I am not sure what specific argument the book makes, but I suspect it has something to do with the famous old niggle that the delta of an ATMF option isn't precisely 0.5 (due to the properties of the lognormal distribution). You can find lots of stuff written about this. Therefore, because your "peak delta" isn't precisely at ATMF, your peak gamma won't be either. However, I am not sure whether this is the reasoning given by Sinclair.
That's what the paper says... I have no reason to doubt the veracity of that statement, but then I haven't spent a lot of time thinking about it.
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