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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.

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2 Answers 2

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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):

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

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From the second order Taylor expansion (variational principle) of a value of an option V around the underlying S, we have: $$ \Delta V = { \partial V \over \partial S } \Delta S + { 1 \over 2 } { \partial^2 V \over \partial S^2 } \Delta S^2 + f(t,\sigma,r) $$ For simplification I put the rest of the terms in f.

By definition: $$ \delta := { \partial V \over \partial S } $$ $$ \gamma := { \partial \delta \over \partial S } = { \partial^2 V \over \partial S^2 } $$ So the variational principle in this case is: $$ \Delta V = \delta \Delta S + { 1 \over 2 } \gamma \Delta S^2 + f(t,\sigma,r) $$ We re-arrange the terms to add another S to get the dollar values (assuming S != 0 which might not be always the case - ask the CL traders from 21 Apr 2020 why that matters): $$ \Delta V = \delta S { \Delta S \over S } + { 1 \over 2 } \gamma S^2 { \Delta S^2 \over S^2 } + f(t,\sigma,r) $$ Now, we define delta and gamma dollars as follows: $$ \Delta_{DV$} := k * \delta S $$ $$ \Gamma_{DV$} := k * { 1 \over 2 } \gamma (S * 1 \%)^2 $$ To verify that definition, we replace it back in the variational principle after multiplying with k (the contract multiplier for the contract - how many \$ of cash corresponds to one point in S). The dollar value of the position is just: $$ k * \Delta V = \Delta_{DV$} { \Delta S \over S } + \Gamma_{DV$} * { 1 \over 1\%^2 } * { \Delta S^2 \over S^2 } + ... $$

The correctness of the definition can be verified with a few examples:

  1. If we have a 5% increase in S then the delta contribution to the option portfolio value in dollars is $ 5\% * \Delta_{DV$} $. Check.
  2. If we have a 1% increase in S then the gamma contribution to the option portfolio value in dollars is $ \Gamma_{DV$} $. As per narrative definition in many sources. Check.
  3. If we have a 5% increase in S then the gamma contribution to the option portfolio value in dollars is $ 25 * \Gamma_{DV$} $.
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  • $\begingroup$ How is this related to a variance swap? You seem to only look at vanilla options and standard delta and gamma? $\endgroup$
    – user70573
    Jan 8 at 21:03
  • $\begingroup$ I did not make any assumption about the type of option contract at all - so it should be valid for any type of exotic options or swaps or any other derivative. Specifically, I assume you refer to convexity and the fact that you need it to price a variance swap properly. For that, you would need to include additional terms that I omitted in f to account for that convexity. Vomma, vanna, volga, etc. But then again, instead of doing second and third order effects, you can just do monte carlo - it's probably faster and more correct. $\endgroup$
    – Dorian B.
    Jan 8 at 22:12

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