What is the reasoning/meaning behind the second derivative of a put option
closed as off-topic by Daneel Olivaw, LocalVolatility, Alex C, Helin, skoestlmeier Feb 25 at 7:17
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It is the rate at which the price of the option changes with respect to the change of the delta (the rate of change with respect to the underlying). As by design, options are non-linear in order to provide protection (limit loss) as well as provide some exposure to the underlying, their value will change its sensitivity to changes in the underlying. Due to curvature, so will this sensitivity to changes in the underlying. The second derivative is a measure of this change in sensitivity. It is a measure of realized volatility and is commonly referred to as gamma, among the option “greeks.”
As for a put option, if you are long the put option you are short delta and long gamma. If you are short the put, you are long delta and short gamma.
It's called Gamma one of the option Greeks.