1) An option $V$ under the Black-Scholes model is perfectly hedged when it is delta hedged continuously with the underlying $S$. When the hedging time is discrete, the delta $\Delta$ needs to take into account the gamma $\Gamma$. One way to do this is to Taylor expand the hedge portfolio $V(t,S)-\Delta S$ over a finite time interval $\delta$ and minimize the variance var$[\delta(V(t,S)-\Delta S)]$. We can obtain, $$\Delta = V_S+cSV_{SS}\delta t+O((\delta t)^2)$$ for some constant $c$.
For the discrete time hedging, is it better to introduce another option $H$ with weight $w$ on $S$ to form the portfolio $\Pi:=V-wH-\Delta S$ to hedge $V$, as $H$ has its gamma to hedge that of $V$? I Taylor-expand the $\Pi$ over $\delta t$ and have obtained a result with a complicated expression for $(\Delta,w)$ which I am not entirely sure of. Are there any references on this topic of "gamma hedging"?
2) Thanks to Quantuple's pointer to Chapter 1 of Lorenzo Bergomi's book Stochastic Volatility Modeling, it is clear now that the gamma comes into the hedge for a purpose that is very distinct from what I described above. It is first and foremost a hedge against the stochasticity of the difference between the realized variance and the implied variance, rather then coming into the delta hedging as a first-order correction for the finiteness of the hedging time interval. Here is the follow-up question.
Suppose the option $V$ has $2$ underlyings $S_1$ and $S_2$ which are correlated with each other with correlation $\rho$ and we have two hedging options $H_1$ with only underlying $S_1$ and $H_2$ with only underlying $S_2$. The volatility of $S_1$ is $\sigma_1$ and that of $S_2$ is $\sigma_1$. I would like to determine the impact of $\rho$ on hedging coefficients $w_1$ of $H_1$ and $w_2$ of $H_2$. Is the following approach correct?
We delta hedge all the options. We assume the volatilities are stochastic. Let $\langle\cdot\rangle:=\mathbf E[\cdot]$. \begin{align} \text{p&l} = &\frac12S_1^2\frac{\partial^2(V-w_1H_1)}{\partial S_1^2}(\sigma_1^2-\sigma_{1,\text{imp}}^2)+\frac12S_1^2\frac{\partial^2(V-w_2H_2)}{\partial S_2^2}(\sigma_2^2-\sigma_{2,\text{imp}}^2) \nonumber\\ &+S_1S_2\frac{\partial^2V}{\partial S_1\partial S_2}(\rho\sigma_1\sigma_2-\langle\rho\sigma_1\sigma_2\rangle) \nonumber\\ &+\text{functional of }(V-w_1H_1-w_2H_2)\text{ the volatilities, realized and implied}, \end{align} where "imp" on the subscript denotes that the variable is implied from the option price. Ignore the last term for now. Denote $\alpha_1:=S_1^2\frac{\partial^2}{\partial S_1^2}(V-w_1H_1)$ and similarly for $\alpha_2,\, v_{1,2}:=\rho\sigma_1\sigma_2-\langle\rho\sigma_1\sigma_2\rangle,\, v_1:=\sigma_1^2-\sigma_{1,\text{imp}}^2$ and similarly for $v_2$. It is reasonable to assume $\langle v_1\rangle=\langle v_2\rangle=\langle v_{12}\rangle=0$. We have \begin{equation} \text{var[p&l]} = \frac14\alpha_1^2 \langle v_1^2\rangle+\alpha_1\alpha_{1,2}\langle v_1v_{1,2}\rangle+\frac14\alpha_2^2 \langle v_2^2\rangle+\alpha_2\alpha_{1,2}\langle v_2v_{1,2}\rangle+ \alpha_{1,2}^2\langle v_{1,2}^2\rangle. \end{equation} Minimizing the variance above can be viewed geometrically as the vector $\alpha_{1,2}v_{1,2}$ projecting on the hyperplane spanned by the vectors $\{\frac12\alpha_1v_1,\,\frac12\alpha_2v_2\}$. So $\alpha_{1,\min} = -2\frac{\langle v_1v_{1,2}\rangle}{\langle v_1^2\rangle}\alpha_{1,2}$ and $\alpha_{2,\min} = -2\frac{\langle v_2v_{1,2}\rangle}{\langle v_2^2\rangle}\alpha_{1,2}$ minimizes var[p&l]. \begin{equation} w_{1,\min} = \frac{\Gamma_1+2\frac{\langle v_1v_{1,2}\rangle}{\langle v_1^2\rangle}\frac{S_2}{S_1}\Gamma_{1,2}}{\Gamma_{H_1}} \end{equation} and exchanging $1$ and $2$ on the subscript gives the expression for $w_{2,\min}$.
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