YouTube T4-15: Option gamma

[Nicole Seaman]

Gamma is the rate of change of delta; aka, the second partial derivative with respect to a change in the stock price. Because call option delta is the cumulative normal distribution function (CDF), gamma has the shape of the normal probability density function (pdf)! When gamma is high, the delta-hedge is fragile. When gamma is low (i.e., when the option is deeply in- or out- of the money), delta is not very responsive to stock price changes and the delta-hedge is more robust.

David's XLS is here: https://trtl.bz/2XbNws1

 

[Pedro Cazorla]

Hi! I have one question. 90% of sources on the internet says that Gamma is at its highest when the option is ATM. However, in the video, gamma is at its highest when the option is OTM. Can you please explain the difference?

 

[David Harper CFA FRM]

Hi Pedro, Peak gamma is near ATM so those statements are only approximately true, and realistically peak gamma tends to be when the option is slightly OTM. The formula is [N'(d1)*exp(-qT)]/[S*σ*sqrt(T)] where N'(d1) is the normal pdf, so a few factors "infect this bias." For one thing, d1 <> 0 when ATM due to the drift (risk free rate) and variance. Also, the bias amplifies with longer terms, so it's a safer approximation as the option maturity tends to zero, but in most of my realistic simulations (typical inputs), gamma peaks when slightly OTM. Hope that's helpful,