Let us assume we are in the Black-Scholes model. Is there a closed formula for the variance of the cash-gamma? I define cash gamma as $CG = S_t^2 * \Gamma(t,S_t)$, assuming interest rates are 0 to simplify.
Edit. More precisely, I would like to compute $E( S_t^4 \Gamma^2(t,S_t) )$. We already know that $ E( S_t^2 \Gamma(t,S_t) ) = S_0^2 \Gamma(0,S_0)$