Let's say there is a $1 stock, with say 1 day to expiration. The 1.5 strike call, is probably a 0 delta at this point; however, a 1 point increase would mean the stock would be at trading at 2 dollars; thus the 1.5 strike call would now almost be guaranteed to finish in the money right, and would seemingly now have close to a 100 delta. It seems then, that it's gamma was 100 in this scenario, making it higher than an ATM option. Where am I wrong?
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under the black scholes assumptions it is nearly impossible to move 50% in 1 day. A 1day move will be of size $\sigma \sqrt{1/252}$ (assuming 252 trading days). So you are talking about something that is about a 40 stdev move (assuming $\sigma$ = 0.2).
that's why your gamma is 0.
