Unit Gamma and Dollar Gamma in Variance Swaps

Summary

The note clarifies the distinction between unit gamma and dollar gamma for a variance swap. Unit gamma is expressed with an inverse-square dependence on the underlying price, while dollar gamma multiplies gamma by the square of the underlying price. This price scaling cancels the inverse-square term, explaining how dollar gamma can remain constant even when unit gamma varies with the underlying price.

The explanation addresses a question about why a gamma formula appears to lack price terms in its numerator. It supplies the relationship between the two measures, but does not derive the variance swap replication formula or discuss assumptions, hedging implementation, or empirical evidence. The result should therefore be read as a focused clarification of notation and scaling, rather than a full treatment of variance swap construction.

Key ideas

• Unit gamma can vary inversely with the square of the underlying price.
• Dollar gamma is calculated by multiplying unit gamma by the square of the underlying price.
• The price terms cancel in the dollar gamma expression, producing the stated constant-dollar objective.
• The explanation clarifies the formula but does not derive the full variance swap strategy.

Tags

# constant dollar Gamma for variance swap


# constant dollar Gamma for variance swap












Here is the result of Gamma for Variance swap. I think $S_t$ is the only variable in $\sigma^2_{Exected,t}.$ Then how could we get the presentation of Gamma: $$\dfrac{1}{S_t^2},$$ there should be some $S_t$ terms in numerator.

## Answer by Newquant (score 1)

https://quant.stackexchange.com/a/75948

That’s the formula for unit gamma. The dollar gamma formula (the objective that the variance swap is trying to create as constant) is given by: $$\Gamma * S^2$$ so the numerator contains the $S^2$ term to cancel the denominator.

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