Hedging Impermanent Loss with Power Perpetuals (2)
In the first article of this series, we explained how to hedge the impermanent loss of CFMM (of Uniswap V2 style) by power perpetuals. The idea is to hedge the Delta and Gamma to zero at the entry point (the price at which an LP deposits liquidity) so that the LP portfolio value stays flat in a reasonably wide range around the entry point. The algorithm was based on calculating the Delta and Gamma of an LP position of Uniswap-V2-style CFMM. Since the function in the CFMM of Uniswap V3 is different from that of V2, the hedging would be a bit different too. This article explains how to hedge the impermanent loss of LP position in Uniswap V3 by power perpetuals. We will skip the basic concepts (for which you can the details in the previous article) and directly start with Uniswap’s different Greek letters.
Let’s still take the ETH-USDC pair as an example: provide
ETH and
SDC to Uniswap V3 when ETHUSDC=
with the range specified as
When the price of ETH
changes within the range, you would have
ETH and
USDC following the formula:
Just like in Uniswap V2,
and
are functions of the ETH price
(in this article we follow the denotation of Uniswap V3 whitepaper to use
in place of
and
:
The value of your LP position is as follows, for
:
Therefore, we have Delta and the Gamma of your LP tokens as follows:
Noticing
is corresponding to
, we can see that LP position in Uniswap V3 has a smaller Delta than V2 (with an extra
term), but exactly the same Gamma. Therefore, only some small changes are needed to make the LP position Delta-neutral and Gamma-neutral.
Assume you need
units of ETH^2 to make the portfolio Gamma-neutral:
And then you would need to take
units of ETHUSD futures to make the portfolio Delta-neutral:
In other words, you need to short
units of ETHUSD futures.
In practice, as a static hedge entered at
, we just need to fix
and
with the value at this point.
To summarize, the following portfolio has 0 Delta and 0 Gamma at the entry point
of the range order:
- ProvidingETH+USDC to the ETH-USDT pair
- Longunits of ETH^2 power perps
- Shortunits of ETHUSD futures
Please note that the
is the measure of the liquidity density provided by this range order and is determined by
. Given the amount of provided liquidity (given
and
), the narrower the range is (the smaller
is), the bigger
would be (meaning the liquidity is more concentrated).
can be calculated by solving the pricing formula, which leads to:
For Uniswap V3, LP does not have to provide liquidity symmetrically (i.e.,
). However, if we do follow this V2-tradition, then for 1 unit of ETH-USDC pair (i.e., 1ETH +
USDC), we have
Then we can rewrite the hedged portfolio as:
- Providing 1 unit (i.e., 1ETH+USDC) to the ETH-USDT pair
- Longunits of ETH^2 power perps
- Shortunits of ETHUSD futures
Assume you provide one unit of ETH-USDC pair to Uniswap V3 when ETHUSDC=2000, i.e.,
, with the range specified as
. With the formula above we have
. Assuming volatility
, we can calculate the values of the portfolio:
- Providing 1ETH+2000USDC to the ETH-USDC pair
- Shortunits of ETHUSD futures
Constructing the portfolio above comes at a cost. Usually, both the futures and powers positions would incur funding fees. However, the expectation of the futures funding fee is 0. Therefore, we only consider the funding fee of the powers. Per the pricing of powers, the theoretical funding fee of 1 unit of powers for one funding period is:
This leads to an annualized cost in terms of a ratio of the LP value:
Let’s put this annualized cost into numerical scenarios:

Every second, the cost
accumulates at the speed proportional to
. To figure out the average cost, we need to estimate the average
. Statistics show that, for the last 12 months,
for ETH. In the table above, we also show how the annualized cost varies when
.
Please note the table above applies for BTC too. Statistics show that, for the last 12 months,
for BTC. We also list the corresponding cost ratio in the table.
Extra Capital for the Derivative Margin
The futures and powers positions require extra capital to be held as margin.
The power position needs:
Currently, Deri required 16% as initial margin for powers on BSC and 8% on Arbitrum, which correspond to 1635U on BSC and 817U on Arbitrum respectively.
Similarly the futures position needs:
Currently, Deri required 8% as initial margin for futures on BSC and 4% on Arbitrum, which correspond to 1754 U on BSC and 877 U on Arbitrum respectively.
Therefore, the total required margin would be 3389 U on BSC and 1678 U on Arbitrum. Please note that this initial margin is sufficient to cover the price change of +/-10%, so you don’t need to worry about liquidation.
Last modified 6mo ago