Hedging Impermanent Loss with Gamma Swap

Our previous articles explained a theoretical methodology for hedging impermanent loss (IL) using Power Perpetuals. However, as explained in the introductory paper of Gamma Swap, hedging IL with Power Perpetuals has an extremely low capital efficiency, which makes it not practical at all. This article explains how to hedge IL with Gamma Swap.
As mentioned in the introduction, when volatility stays still, Gamma Swap has the following theoretical price:
Pgamma=11−h0T(p−p0)2P_{gamma}=\frac{1}{1-h_0T}(p-p_0)^2Pgamma​=1−h0​T1​(p−p0​)2
where ppp
 and p0p_0p0​
 are the current underlying price and the entry price, while h0=r+σ02/2h_0=r+\sigma_0^2/2h0​=r+σ02​/2
. We will see that impermanent loss has a very similar dynamic.
​
LP Position Value
Let’s take the ETH-USDC pair on Uniswap V3 as example. Suppose an LP position contains xxx
 ETH and yyy
 USDC. Then xxx
 and yyy
 are functions of the ETH price p (in this article we follow the denotation of Uniswap V3 whitepaper):
x=L(1p−1pb)y=L(p−pa)x=L\left(\frac{1}{\sqrt p}-\frac{1}{\sqrt p_b}\right)\\
y=L\left(\sqrt p-\sqrt p_a\right)x=L(p​1​−p​b​1​)y=L(p​−p​a​)
The value of the LP position is as follows, for p∈(a,b)p\in(a,b)p∈(a,b)
 :
V=xp+y=L(2p−ppb−pa)V=xp+y=L\left(2\sqrt p-\frac{p}{\sqrt p_b}-\sqrt p_a\right)V=xp+y=L(2p​−p​b​p​−p​a​)
Suppose the LP position is added at p=p0p=p_0p=p0​
 , with x0x_0x0​
  ETH and y0y_0y0​
  USDC. The following equation should hold with these initial variables:
L=x01p0−1pb=y0p0−paL=\frac{x_0}{\frac{1}{\sqrt p_0}-\frac{1}{\sqrt p_b}}=
\frac{y_0}{\sqrt p_0-\sqrt p_a}L=p​0​1​−p​b​1​x0​​=p​0​−p​a​y0​​
​
Hedge Impermanent Loss
Impermanent Loss refers to the loss of the LP position relative to the value of holding the original portfolio, x0x_0x0​
  ETH and y0y_0y0​
  USDC:
IL=V−(x0p+y0)=(x−x0)p+(y−y0)=L(2p−pp0−p0)\begin{align*}
IL&=V-(x_0p+y_0)\\
&=(x-x_0)p+(y-y_0)\\
&=L\left(2\sqrt p-\frac{p}{\sqrt p_0}-\sqrt p_0\right)

\end{align*}IL​=V−(x0​p+y0​)=(x−x0​)p+(y−y0​)=L(2p​−p​0​p​−p​0​)​
Denoting Δp=p−p0\Delta p = p-p_0Δp=p−p0​
 , with Taylor expansion, we have
p=p0[1+12Δpp0−18Δp2p02+O(Δp3p03)]\sqrt p=\sqrt p_0\left[1+\frac{1}{2}\frac{\Delta p}{p_0}-\frac{1}{8}\frac{\Delta p^2}{p_0^2}+O\left(\frac{\Delta p^3}{p_0^3}\right)\right]p​=p​0​[1+21​p0​Δp​−81​p02​Δp2​+O(p03​Δp3​)]
IL=−14Lp0−32Δp2+O(Δp3p03)≈−14Lp0−32(p−p0)2\begin{align*}
IL&=-\frac{1}{4}Lp_0^{-\frac{3}{2}}\Delta p^2+O\left(\frac{\Delta p^3}{p_0^3}\right)\\

&\approx -\frac{1}{4}Lp_0^{-\frac{3}{2}}(p-p_0)^2
\end{align*}IL​=−41​Lp0−23​​Δp2+O(p03​Δp3​)≈−41​Lp0−23​​(p−p0​)2​
It is very straightforward to hedge this ILILIL
  with Gamma Swap. We just need to long ggg
  units of Gamma Swap such that
IL+pPGamma=0IL+pP_{Gamma}=0IL+pPGamma​=0
which leads to
g=14Lp0−32(1−h0T)g=\frac{1}{4}Lp_0^{-\frac{3}{2}}(1-h_0T)g=41​Lp0−23​​(1−h0​T)
where L=x01p0−1pb=y0p0−paL=\frac{x_0}{\frac{1}{\sqrt p_0}-\frac{1}{\sqrt p_b}}= \frac{y_0}{\sqrt p_0-\sqrt p_a}L=p​0​1​−p​b​1​x0​​=p​0​−p​a​y0​​
 .
​
Please note that, since the Taylor-expansion-based approximation has an error ~ O(Δp3p03)O\left(\frac{\Delta p^3}{p_0^3}\right)O(p03​Δp3​)
 , this only ensures the portfolio has almost 0 IL around the entry point. For a typical position in Uniswap V3 with a range about ±10%\pm10\%±10%
  around the current price, the error is negligible as ∣Δp3p03∣<0.001\left|\frac{\Delta p^3}{p_0^3}\right|<0.001∣∣​p03​Δp3​∣∣​<0.001
 .
Obviously, this hedging comes at a cost, the funding fee of the Gamma Swap. The next section analyzes the cost. We will see that such a cost, although not constant, is reasonably predictable. That’s the whole point of this hedging operation.
​
Hedging Cost
The theoretical funding fee of 1 unit of Gamma Swap for 1 funding period is:
F=p2hT1−hTF= p^2\frac{hT}{1-hT}F=p21−hThT​
The annualized cost of ggg
  units of Gamma Swap is
gF⋅1yearT=14Lhp2p032=14Lp2σ2p032\frac{gF\cdot1year}{T}=\frac{1}{4}Lh\frac{p^2}{p_0^\frac{3}{2}}=\frac{1}{4}L\frac{p^2\sigma^2}{p_0^\frac{3}{2}}TgF⋅1year​=41​Lhp023​​p2​=41​Lp023​​p2σ2​
Let’s discuss a specific example: a range order±10%\pm10\%±10%
  around the current price, i.e. pa=0.9p0,pb=1.1p0p_a=0.9p_0, p_b=1.1p_0pa​=0.9p0​,pb​=1.1p0​
 . Then, for x0=1x_0=1x0​=1
  ETH, we need y0=p0−pa1p0−1pb≈1.1p0y_0=\frac{\sqrt p_0-\sqrt p_a}{\frac{1}{\sqrt p_0}-\frac{1}{\sqrt p_b}}\approx 1.1p_0y0​=p​0​1​−p​b​1​p​0​−p​a​​≈1.1p0​
  USDC, and L=11p0L=11\sqrt p_0L=11p​0​
 .
Numerical example: When ETHUSDC = 1000, for every 1 ETH added to a range order (900, 1000), y0y_0y0​
 =1102.7 USDC is needed.
Then the annualized cost ratio of the position value is:
R=5.37p2σ2p0p+1.1p02R=\frac{5.37p^2\sigma^2}{p_0p+1.1p_0^2}R=p0​p+1.1p02​5.37p2σ2​
Let’s put this annualized cost ratio into numerical scenarios:
 
Price Change
 
 
 
 
Volatility
-10% 
-5% 
0% 
5% 
10%
40%
35%
38%
41%
44%
47%
50%
54%
59%
64%
69%
74%
60%(~current BTC&ETH)
78%
85%
92%
99%
106%
70%
107%
116%
125%
135%
145%
80%
139%
151%
164%
176%
189%
Now, this liquidity-providing business turns into a question: would the income of liquidity-providing (i.e. the transaction fee on the pair) cover the cost of hedging? Given that both the income and the cost are reasonably predictable, this is a very simple math problem.
Essentially, hedging the Uniswap LP position with Gamma Swap turns the exchange-risk-with-income game into an exchange-cost-with-income one. The latter is much easier to manage and thus suitable for most investors, including those risk-averse ones that otherwise would never take risks to provide liquidity to Uniswap. As long as the transaction fee income is greater than the hedging cost, the portfolio is profitable, with reasonable certainty.
​
​
​
​
	Price Change
Volatility	-10%	-5%	0%	5%	10%
40%	35%	38%	41%	44%	47%
50%	54%	59%	64%	69%	74%
60%(~current BTC&ETH)	78%	85%	92%	99%	106%
70%	107%	116%	125%	135%	145%
80%	139%	151%	164%	176%	189%
Last modified 13d ago