I was reading a lot about the idea that long positions (call or put) have positive gamma, and short positions (call or put) have negative gamma. But I couldn't understand why. In "Bunds and Bund Futures" book, the author argues as: "The reason for this is that as the futures price rises the delta of a long call (or long put) option will become more positive (or less negative). For short call (or short put) options, the deltas will become more negative (or less positive) as the futures price increases." But why!? Why "as the futures price rises the delta for a long put option will become less negative"? I would really appreciate some feedback about this.
It works in the Black-Scholes world assuming other factors are constant. For the Black-Scholes, you can check online the formula for Gamma. It is positive obviously if you are long. In reality this may not hold. A simple example would be the case when the implied vol is correlated with spot price. In this case, the "Gamma" would not be as simple as the plain Black-Scholes since there will be "mixed-derivative" terms when calculating second derivatives.