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 X (price denoted as $x$) for the long and short sides to have the following PnL, respectively:

A long position pays the funding fee and has the $PnL\propto(x-x_0)^2$, where $x_0$ is the entry price of the position;
A short position collects the funding fee and has the $PnL\propto-(x-x_0)^2$, where $x_0$ is the entry price of the position;

Mathematically, we can split $(x-x_0)^2$ into the following two parts, which can be tracked by long Power Perps and short Perpetual Futures, respectively.

$$ (x-x_0)^2=[x^2-x_0^2]-[2x_0(x-x_0)] $$

Due to the Gamma premium, the theoretical price of 1 unit of power perp is $\frac{x^2}{1-hT}$, where $T$ is the funding period and $h=r+\sigma^2$ ($r$ is the risk-free interest rate and $\sigma$ is the volatility). Therefore, we define Gamma Swap as follows:

$$ 1\text{GammaSwap}=1\text{PowerPerp}-\frac{2x_0}{1-h_0T}\text{PerpFutures} $$

where $h_0$ 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 $\frac{2x_0}{1-h_0T}$ Perpetual Futures. Per this definition, Gamma swap has the following theoretical value:

$$ P_{gamma}=\frac{1}{1-hT}(x^2-x_0^2)-\frac{2x_0}{1-h_0T}(x-x_0) $$

When volatility stays still at $h=h_0$, we have:

$$ P_{gamma}=\frac{1}{1-hT}(x-x_0)^2 $$

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 $\frac{2x_0}{1-h_0T}$ 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.).