Problem with the concept of Dollar Gamma

Question

I was reading up on variance swaps and encounter the notion of Dollar Gamma, which is defined as the change in dollar value of the Dollar Delta (Δ * S) for a 1% change in spot (S). The formula for Dollar Gamma is then given as $Γ = Γ/100 * S^2.

However, as I tried to prove it, I ran into an issue. The proof is below.

$Δ = Δ * S

$Δ' = Δ' * S' (Function 1)

$Δ' = (Δ + Γ * S/100) * (S + S/100)

$Δ' = Δ * S + Γ/100 * S^2 + [1% * (Δ * S + Γ/100 * S^2)] (Function 2)

Clearly from Function 2, the change in Dollar Delta is not only the Dollar Gamma as per definition but also the part in the square brackets. However, if in Function 1, the new Delta is multiplied by S instead of S', then the definition would be correct. So my question is whether the new Dollar Delta is calculated by multiplying the new Delta (Δ') with the previous spot (S) or my proof in Function 2 is correct but in practice traders just ignore the terms in the square brackets because they're negligible?

Please forgive any formatting errors for the proof.

Thanks.

Answer 1

Many people get confused by this, including on derivatives trading desks... what you want is to know how much dollar delta you will have to rebalance after a small move to get an idea of the quantum of monetary risk you’re running and be able to compare across stocks with differents spots (your gamma shares will not let you do meaningful comparisons), or alternatively a quick way to calcultate your gamma P&L for a given (small) move.

The answer to the first question is to take the number of shares that your delta changes by, times the spot. It’s important not to take the difference in cash delta i.e. do not take something like $\Delta_+S_+ - \Delta_-S_-$ because this only represents a monetary risk scaling due to price fluctuations, not a convexity risk: a simple stock has no gamma, but this formula would show you a change in cash delta when spot changes which in itself is perfectly reasonable (you have more dollar risk on your holding when spot is high) but is not indicative of a gamma risk.

What spot should you multiply this by ? I find it best to multiply by the starting spot, that way it is straightforward to appoximately calculate your expected gamma P&L from the move: half your cash gamma times the percent price change squared (divided by 1% if that is your scaling factor):

$PL_{\Gamma_{\$}}\approx \frac{1}{2}\frac{\Gamma_{\$}}{1\%}.(\frac{dS}{S})^2$