I am working in delta-gamma hedging with machine learning. I guess I have to predict gamma (since predicting gamma tells you how delta will behave) but I don't know why is it needed. I think that a mathematical framework can help me have good understanding of the context.

Suppose we currently have a call option with a current delta of $0.6$ at time $t$. At $t=t+1$, if the stock price goes up by $1$, then the option's value goes up by $0.6$ and we need to short $60$ shares and that's it I have a delta neutral portfolio at that time, and I keep doing this at each time step with the new deltas so that I always have a zero delta portfolio and there is nothing to predict and no need for gamma. By the way, when delta hedging the delta of the portfolio is constant equal to 0 and since gamma is the derivative of delta, isn't gamma forced to be 0 too as the derivative of a constant process ?

Of course I know I am wrong and my reasoning above has many issues so I tried to set up a mathematical framework for this. Let $S_t, D_t, G_t$ be respectively the stock price, Delta and Gamma of a call option at time $t$. These quantities are known at $t$.

Let $S'_t, D'_t, G'_t$ be similar quantities for another option on a same underlying, to be able to do neutral gamma hedging. What we want is at time $t+1$, $P_{d,t+1}=D_{t+1}+yD'_{t+1}-x=0$ and $P_{g,t+1}=G_{t+1}+yG'_{t+1}=0$ where $P_{d,t+1}$ is the portfolio's delta, $P_{d,t+1}$ its gamma, and $N_{t+1}$ the number of shares we short sell (that have a delta of 1). If the rebalancing is done before time $t+1$ I would understand why we are interested in predicting. Is it possible to buy a portion $y$ of an option ?

Quant

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