Author: 0xAlpha, Richard Chen

Last updated on 2022.9.3

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:

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: