A Black–Scholes Approach to Dollar Gamma Variance

Summary

The document asks for a closed-form variance of cash gamma in the Black–Scholes model, defining the quantity as the underlying price squared times option gamma. It refines the target to the expectation of the squared cash-gamma exposure, while noting a known expectation for the unsquared exposure under the stated zero-rate assumption.

The response proposes treating dollar gamma through the second derivative of option value with respect to strike, scaled by strike squared, and applying an instantaneous volatility relation for a claim whose value follows the underlying price process. It points to a separate discussion for the strike-derivative identity. However, the response does not carry out the requested expectation calculation, give a closed formula for the variance, or explain the conditions needed to use the proposed relation. Its method is therefore a suggested route rather than a complete derivation.

Key ideas

• The question seeks the variance of underlying-price-scaled option gamma in Black–Scholes with zero rates.
• The response relates dollar gamma to a strike second derivative of option value.
• It proposes using an instantaneous volatility expression for the option claim as a route to analyze dollar gamma.
• The response does not derive the requested squared expectation or provide a closed variance formula.

Tags

# Variance of cash gamma (or dollar gamma)


# 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)$

## Answer by user34971 (score 2)

https://quant.stackexchange.com/a/46438

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.