We want to define the Gamma Swap of some underlying asset Xfor the long and short sides to have the following PnL, respectively:
A long position pays the funding fee and has the PnL∝(x−x0)2 , where x0 is the entry price of the position;
A short position collects the funding fee and has the PnL∝−(x−x0)2 , where x0 is the entry price of the position;
Mathematically, we can split (x−x0)2 into the following two parts, which can be tracked by long Power Perps and short Perpetual Futures, respectively.
(x−x0)2=[x2−x02]−[2x0(x−x0)] Due to the Gamma premium, the theoretical price of 1 unit of power perp is 1−hTx2 , where T is the funding period and h=r+σ2 (r is the risk-free interest rate and σ is the volatility). Therefore, we define Gamma Swap as follows:
1GammaSwap=1PowerPerp−1−h0T2x0PerpFutures where h0 is the value of h at the entry point. That is, we define 1 unit of Gamma Swap has the PnL equivalent to the portfolio of long 1 Power Perp and short 1−h0T2x0 Perpetual Futures. Per this definition, Gamma swap has the following theoretical value:
Pgamma=1−hT1(x2−x02)−1−h0T2x0(x−x0) When volatility stays still at h=h0 , we have:
Pgamma=1−hT1(x−x0)2 At the entry point, a unit of Gamma Swap has zero Delta, while the perp component and the power component have the exact opposite non-zero Delta. To long 1 power and short 1−h0T2x0 perps separately, one needs to post margin for the power component (mostly for its positive Delta) as well as for the perp component (for its negative Delta), respectively. These two parts of margin are purely a waste of capital since the two corresponding parts of Delta are not wanted and canceled out.
By default, a Gamma swap is traded with the current price as the entry price. However, one can also enter a Gamma swap position specifying the entry price xs to gain the Gamma exposure around the specified point, i.e. to have Pnl∝(x−xs)2 . A Gamma swap traded with a specified entry price is treated per the following definition:
1GammaSwap(xs)=1PowerPerp−1−h0T2xsPerpFutures To trade Gamma swap, we need a composite trading venue that contains two (virtual) primitive trading venues for power perps and perpetual futures, respectively. This paper only introduces the concept of Gamma swap and the implementation of the composite trading venue will be discussed in a subsequent paper. Here we just assume we already have such a composite trading venue that has real-time Mark prices for the (virtual) power perps and perpetual futures, denoted as Mpower and Mperp respectively.
Hereafter, we use i to represent the index price of the underlying, which is different from the general term of price x in the previous conceptual discussion. Just as how perpetual futures and power perps are traded (with orderbook or AMM), the index price i is an external input to the trading venue. With the composite trading venue giving Mpower and Mperp , it is straightforward to define the mark price of Gamma swap as follows:
Mgamma=Mpower−1−h0T2i0Mperp+1−h0Ti02 where i0 is the entry price (or the specified entry price). Note the third item on RHS is a constant to make the expression financially meaningful.
Fgamma=Fpower+(1−hT2i−1−h0T2i0)Fperp where(i0,h0) are the (i,h) as of the position entry point, and
Fpower=Fperp=TMpower−i2TMperp−i Notice that the coefficient of Fperp is a bit complicated, constantly varying with i . Essentially, this can be understood as every unit of Gamma swap has −1−h0T2i0 “real futures” as well as 1−hT2i “virtual futures”, and hence totally (1−hT2i−1−h0T2i0) effective futures.
The Greeks are the motivation to design Gamma Swap. We look into them in this section. To avoid confusion, in this paper the word “gamma” refers to the derivative Gamma swap, while the Greek letter Γ refers to the Greek.
Since1GammaSwap=1PowerPerp−1−h0T2x0PerpFutures , we have:
Δgamma=Δpower−1−h0T2x0Δperp=1−hT2x−1−h0T2x0 Especially, at the entry point, we have Δgamma=0 . This is exactly what we want from the Gamma swap: pure Γ without Δ .
However, when the price changes while h stays still (h=h0) , we have:
Δgamma=1−h0T2(x−x0) That is, when x drifts away from x0, the Gamma swap will linearly build a non-zero Δ proportional to (x−x0). This is the Δ “built” by the Γ , which is the micro-process of how the Gamma swap works. Note that Γ is the second-order differential, which would only affect PnL by its integral, i.e., the first-order differential, Δ .
Please note that a Gamma swap traded with a specified entry price xs has a non-zero Δ at the beginning:
Δgamma(ent.price=xs)=1−hT2(x0−xs) So, if wanting to obtain Γ together with a non-zero Δ, one can trade Gamma swap with a properly specified entry price.
The Γ of Gamma swap is as follows:
Γgamma=Γpower−1−h0T2i0Γperp=1−hT2 which is a constant as long as volatility stays still. This is also what we want from the Gamma swap: almost constant Γ .
The previous analyses assume volatility stays unchanged. So how would varying volatility affect Gamma swaps? It’s easy to see that a Gamma swap has the same Vega as power’s. So we only look into how volatility affects Δgamma and Γgamma .
∂σ∂Δgamma=(1−hT)22xT∂σ∂h=(1−hT)24xTσ A small change in volatility, δσ , would cause:
δΔgamma=σδσ⋅(1−ht)24xTσ2≈σδσ4Tσ2x For small T, e.g. 1 day or 1 week, this is usually a negligible amount.
∂σ∂Γgamma=(1−hT)24σT ΓgammaδΓgamma=σδσ⋅1−hT2σ2T which is around 4% of (δσ/σ) for T=1week , or around 0.5% for T=1day .
Therefore, we can conclude that varying volatility does not substantially affect the constancy of a Gamma swap’s Γ , which is a very important feature for the use cases.
In summary, the analysis of the Greeks shows Gamma swap gives exactly what we want: (almost) constant and pure Γ without Δ .
One possible margin solution for Gamma swap could be the “Greek-based margin”. That is, with Δgamma and Γgamma , we can estimate the change of the Gamma swap value due to the underlying price change from x to x+δx , with a second-order Taylor expansion:
δPgamma≈Δgammaδx+21Γgammaδx2 And the margin system works as follows: the trader is required to post as collateral the possible loss associated with a specific risk scenario, e.g. δx/x=±5% .