# What to predict in delta-gamma hedging?

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 ?

I don't think you can buy a portion of a option, it makes much more sense to size up the position. For example, if you need to hedge 1 $$C$$ with 0.4 $$C'$$, it is more feasible to long (short) 100 $$C$$ (40 $$C'$$). Then again, it really depends on what is offered on the platform or your broker.
• For your first comment, I allow $y$ to be negative i.e. shorting the prime option if the system says so Commented Apr 21 at 0:15