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Introducing Gamma Swap

Introduction
This paper introduces a new type of derivative, Gamma Swap, inspired by Power Perpetuals. It is for the buyers (taking the long side) to gain the Gamma exposure of some underlying asset and the sellers (taking the short side) to earn the funding fees by providing that exposure.
We want to define the Gamma Swap of some underlying asset XXX
for the long and short sides to have the following PnL, respectively:
A long position pays the funding fee and has the PnL∝(x−x0)2PnL\propto(x-x_0)^2PnL∝(x−x0​)2
  , where x0x_0x0​
 is the entry price of the position;
A short position collects the funding fee and has the PnL∝−(x−x0)2PnL\propto-(x-x_0)^2PnL∝−(x−x0​)2
 , where x0x_0x0​
  is the entry price of the position;
Mathematically, we can split (x−x0)2(x-x_0)^2(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)]
(x-x_0)^2=[x^2-x_0^2]-[2x_0(x-x_0)](x−x0​)2=[x2−x02​]−[2x0​(x−x0​)]
Due to the Gamma premium, the theoretical price of 1 unit of power perp is x21−hT \frac{x^2}{1-hT}1−hTx2​
 , where TTT
  is the funding period and h=r+σ2h=r+\sigma^2h=r+σ2
  (rrr
 is the risk-free interest rate and σ\sigmaσ
  is the volatility). Therefore, we define Gamma Swap as follows:
1GammaSwap=1PowerPerp−2x01−h0TPerpFutures
1\text{GammaSwap}=1\text{PowerPerp}-\frac{2x_0}{1-h_0T}\text{PerpFutures}1GammaSwap=1PowerPerp−1−h0​T2x0​​PerpFutures
 where h0h_0h0​
 is the value of hhh
 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 2x01−h0T\frac{2x_0}{1-h_0T}1−h0​T2x0​​
  Perpetual Futures. Per this definition, Gamma swap has the following theoretical value:
Pgamma=11−hT(x2−x02)−2x01−h0T(x−x0)P_{gamma}=\frac{1}{1-hT}(x^2-x_0^2)-\frac{2x_0}{1-h_0T}(x-x_0)Pgamma​=1−hT1​(x2−x02​)−1−h0​T2x0​​(x−x0​)
When volatility stays still at h=h0h=h_0h=h0​
  , we have:
Pgamma=11−hT(x−x0)2P_{gamma}=\frac{1}{1-hT}(x-x_0)^2Pgamma​=1−hT1​(x−x0​)2
which gives exactly the PnL that we want in our motivation. Now we have successfully structured a new derivative, Gamma Swap, to provide the wanted risk exposure.
​
Derivative v.s. Portfolio
However, the equivalency brings us to the question: why do we need such a new derivative instead of simply holding a portfolio of futures and powers? This is because the latter has the following two disadvantages:
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  2x01−h0T\frac{2x_0}{1-h_0T}1−h0​T2x0​​
  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.
To construct such a portfolio, for the unwanted positive Delta and negative Delta, one is paying extra transaction fees and trading costs (slippage, funding fees, etc.).
Put simply, to obtain pure Gamma exposure by holding a portfolio of powers and perps, one has to allocate extra capital for margin as well as pay extra costs for 0 Delta. And Gamma Swap is introduced to directly provide the Gamma exposure without such wastes.
​
Trade with specified entry price
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 xsx_sxs​
 to gain the Gamma exposure around the specified point, i.e. to have Pnl∝(x−xs)2Pnl\propto(x-x_s)^2Pnl∝(x−xs​)2
 . A Gamma swap traded with a specified entry price is treated per the following definition:
1GammaSwap(xs)=1PowerPerp−2xs1−h0TPerpFutures1\text{GammaSwap}(x_s)=1\text{PowerPerp}-\frac{2x_s}{1-h_0T}\text{PerpFutures}1GammaSwap(xs​)=1PowerPerp−1−h0​T2xs​​PerpFutures
Mark Price and Funding
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 MpowerM_{power}Mpower​
  and MperpM_{perp}Mperp​
  respectively.
Hereafter, we use iii
 to represent the index price of the underlying, which is different from the general term of price xxx
  in the previous conceptual discussion. Just as how perpetual futures and power perps are traded (with orderbook or AMM), the index price iii
  is an external input to the trading venue. With the composite trading venue giving MpowerM_{power}Mpower​
  and MperpM_{perp}Mperp​
 , it is straightforward to define the mark price of Gamma swap as follows:
​
Mgamma=Mpower−2i01−h0TMperp+i021−h0TM_{gamma}=M_{power}-\frac{2i_0}{1-h_0T}M_{perp}+\frac{i_0^2}{1-h_0T}Mgamma​=Mpower​−1−h0​T2i0​​Mperp​+1−h0​Ti02​​
where i0i_0i0​
  is the entry price (or the specified entry price). Note the third item on RHS is a constant to make the expression financially meaningful.
With such a set-up, Gamma swap does not have a simple mark-price-based funding mechanism (like power perp’s). Instead, we define that the Gamma swap has composite funding consisting of power funding and perp funding:
Fgamma=Fpower+(2i1−hT−2i01−h0T)FperpF_{gamma}=F_{power}+\left(\frac{2i}{1-hT}-\frac{2i_0}{1-h_0T}\right)F_{perp}Fgamma​=Fpower​+(1−hT2i​−1−h0​T2i0​​)Fperp​
where(i0,h0)(i_0,h_0)(i0​,h0​)
    are the (i,h)(i,h)(i,h)
   as of the position entry point, and
Fpower=Mpower−i2TFperp=Mperp−iT\begin{align*}
F_{power}=&\frac{M_{power}-i^2}{T}\\
F_{perp}=&\frac{M_{perp}-i}{T}
\end{align*}Fpower​=Fperp​=​TMpower​−i2​TMperp​−i​​
Notice that the coefficient of FperpF_{perp}Fperp​
  is a bit complicated, constantly varying with iii
 . Essentially, this can be understood as every unit of Gamma swap has −2i01−h0T-\frac{2i_0}{1-h_0T}−1−h0​T2i0​​
  “real futures” as well as 2i1−hT\frac{2i}{1-hT}1−hT2i​
  “virtual futures”, and hence totally (2i1−hT−2i01−h0T)\left(\frac{2i}{1-hT}-\frac{2i_0}{1-h_0T}\right)(1−hT2i​−1−h0​T2i0​​)
  effective futures.
​
The Greeks
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\GammaΓ
  refers to the Greek.
Since1GammaSwap=1PowerPerp−2x01−h0TPerpFutures 1\text{GammaSwap}=1\text{PowerPerp}-\frac{2x_0}{1-h_0T}\text{PerpFutures}1GammaSwap=1PowerPerp−1−h0​T2x0​​PerpFutures
 , we have:
​
Δgamma=Δpower−2x01−h0TΔperp=2x1−hT−2x01−h0T\begin{align*}
\Delta_{gamma}&=\Delta_{power}-\frac{2x_0}{1-h_0T}\Delta_{perp}\\

&=\frac{2x}{1-hT}-\frac{2x_0}{1-h_0T}
\end{align*}Δgamma​​=Δpower​−1−h0​T2x0​​Δperp​=1−hT2x​−1−h0​T2x0​​​
Especially, at the entry point, we have Δgamma=0\Delta_{gamma}=0Δgamma​=0
 . This is exactly what we want from the Gamma swap: pure Γ\GammaΓ
  without Δ\DeltaΔ
 .
However, when the price changes while hhh
  stays still (h=h0)(h=h_0)(h=h0​)
 , we have:
Δgamma=21−h0T(x−x0)\begin{align*}
\Delta_{gamma}=\frac{2}{1-h_0T}(x-x_0)
\end{align*}Δgamma​=1−h0​T2​(x−x0​)​
That is, when  xxx
 drifts away from  x0x_0x0​
, the Gamma swap will linearly build a non-zero Δ\DeltaΔ
 proportional to (x−x0)(x-x_0)(x−x0​)
. This is the Δ\DeltaΔ
  “built” by the Γ\GammaΓ
  , which is the micro-process of how the Gamma swap works. Note that Γ\GammaΓ
   is the second-order differential, which would only affect PnL by its integral, i.e., the first-order differential, Δ\DeltaΔ
 .
Please note that a Gamma swap traded with a specified entry price xsx_sxs​
  has a non-zero Δ\DeltaΔ
  at the beginning:
Δgamma(ent.price=xs)=21−hT(x0−xs)\begin{align*}
\Delta_{gamma}(ent.price=x_s)=\frac{2}{1-hT}(x_0-x_s)
\end{align*}Δgamma​(ent.price=xs​)=1−hT2​(x0​−xs​)​
So, if wanting to obtain Γ\GammaΓ
 together with a non-zero Δ\DeltaΔ
, one can trade Gamma swap with a properly specified entry price.
The Γ\GammaΓ
 of Gamma swap is as follows:
Γgamma=Γpower−2i01−h0TΓperp=21−hT\begin{align*}
\Gamma_{gamma}&=\Gamma_{power}-\frac{2i_0}{1-h_0T}\Gamma_{perp}\\
&=\frac{2}{1-hT}
\end{align*}Γgamma​​=Γpower​−1−h0​T2i0​​Γperp​=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\GammaΓ
 .
​
The effects of varying volatility
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\Delta_{gamma}Δgamma​
  and Γgamma\Gamma_{gamma}Γgamma​
 .
∂Δgamma∂σ=2xT(1−hT)2∂h∂σ=4xTσ(1−hT)2\frac{\partial \Delta_{gamma}}{\partial \sigma}=\frac{2xT}{(1-hT)^2}\frac{\partial h}{\partial \sigma}=\frac{4xT\sigma}{(1-hT)^2}∂σ∂Δgamma​​=(1−hT)22xT​∂σ∂h​=(1−hT)24xTσ​
A small change in volatility, δσ \delta\sigmaδσ
 , would cause:
δΔgamma=δσσ⋅4xTσ2(1−ht)2≈δσσ4Tσ2x\delta\Delta_{gamma}=\frac{\delta\sigma}{\sigma}\cdot\frac{4xT\sigma^2}{(1-ht)^2}\approx \frac{\delta\sigma}{\sigma}4T\sigma^2xδΔgamma​=σδσ​⋅(1−ht)24xTσ2​≈σδσ​4Tσ2x
For small TTT
, e.g. 1 day or 1 week, this is usually a negligible amount.
Similarly, we have:
∂Γgamma∂σ=4σT(1−hT)2\frac{\partial \Gamma_{gamma}}{\partial \sigma}=\frac{4\sigma T}{(1-hT)^2}∂σ∂Γgamma​​=(1−hT)24σT​
δΓgammaΓgamma=δσσ⋅2σ2T1−hT\frac{\delta\Gamma_{gamma}}{\Gamma_{gamma}}=\frac{\delta\sigma}{\sigma}\cdot\frac{2\sigma^2 T}{1-hT}Γgamma​δΓgamma​​=σδσ​⋅1−hT2σ2T​
​
which is around 4% of (δσ/σ)(\delta\sigma/\sigma)(δσ/σ)
  for T=1weekT=1\text{week}T=1week
 , or around 0.5% for T=1dayT=1\text{day}T=1day
 .
Therefore, we can conclude that varying volatility does not substantially affect the constancy of a Gamma swap’s Γ \GammaΓ
 , 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\GammaΓ
  without Δ\DeltaΔ
 .
Margin
One possible margin solution for Gamma swap could be the “Greek-based margin”. That is, with Δgamma\Delta_{gamma}Δgamma​
  and Γgamma\Gamma_{gamma}Γgamma​
 , we can estimate the change of the Gamma swap value due to the underlying price change from xxx
  to x+δx x+\delta xx+δx
 , with a second-order Taylor expansion:
δPgamma≈Δgammaδx+12Γgammaδx2\delta P_{gamma}\approx\Delta_{gamma}\delta x+\frac{1}{2}\Gamma_{gamma}\delta x^2δPgamma​≈Δgamma​δx+21​Γgamma​δx2
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%\delta x/x =\pm5\%δx/x=±5%
 .
​
Summary
This paper introduces a new type of derivative, Gamma Swap, to efficiently and directly provide almost constant and pure Gamma exposure for traders. However, as a composite derivative, it depends on a composite trading venue to facilitate the trading. We will discuss such a composite trading venue in a subsequent paper.
Application-wise, since Gamma is one of the most primitive elements in the financial world (probably only second to Delta), there would be many potential use cases of Gamma swap. In fact, it would be the go-to solution whenever pure Gamma is needed. We will discuss the applications of Gamma swap in subsequent papers.
Discussions are welcome. You can send emails to [email protected], or DM 0xAlpha on Twitter.
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