# Variance of cash gamma (or dollar gamma)

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 $$E( S_t^2 \Gamma(t,S_t) ) = S_0^2 \Gamma(0,S_0)$$

Let $$F$$ be a claim (an option), then in the Black-Scholes model and assuming zero interest rates the SDE for the claim is $$dF = \frac{\sigma S}{F} F_S F dW$$ where the subscript $$S$$ denotes the partial derivative with respect to $$S$$. So the instantaneous volatility of $$F$$ is $$\frac{\sigma S}{F} F_S$$

The dollar gamma is equal to $$K^2 C_{KK}$$, where $$C_{KK}$$ is a butterfly centered at strike $$K$$. Hence you can write $$F = K^2 C_{KK}$$ and find the instantaneous volatility for the dollar gamma.

Why dollar gamma is equal to $$K^2 C_{KK}$$ can be found in the following thread:

Expectation of Gamma times S$^2$ in Black-Scholes model

• thank you for rhe answer but i am not sure to understand. i would like to compute E( St^4 gamma(t,St)^2 ). we already know that E( St^2 gamma(t,St) ) = S0^2 gamma(0,S0) – user40929 Jul 4 '19 at 7:36
• Ah, well, your question did not originally state that you want to calculate $E [S^4 \Gamma^2]$, so I thought you'd like to calculate the variance of the instantaneous change in dollar gamma for eg short horizon risk management purposes. To calculate what you want to do is not hard in Black-Scholes: simply perform a numerical integration as you know what the BS density is for $S$. I am not aware of a simple analytical expression for what you'd like to calculate. – ilovevolatility Jul 4 '19 at 9:06
• thank you. how do you do that when you dont have a closed formula for rhe Gamma? is there a way to provide an approximation of this expectation? thank you for your help – user40929 Jul 4 '19 at 10:29
• There is a closed form formula for Gamma in the Black Scholes model. Just google it. – ilovevolatility Jul 4 '19 at 12:35
• i meant for general payoff (other than calls or put) there is no formula – user40929 Jul 4 '19 at 13:57