Every Liquidity Provider is familiar with this infamous graph (for those unfamiliar). Decentralized liquidity is a critical component of the defi ecosystem, but the risk of impermanent loss holds back many market participants from providing more liquidity.
In this article, we are going to explore how we can abstract away that risk from liquidity providers and allow them to inject more liquidity into the ecosystem worry free by using a novel instrument on Solana.
Formalizing Impermanent Loss
For constant product liquidity pools given future price px_1 and current price px_0, the impermanent loss equation can be formalized as follows.
You can actually expand the sqrt(1+r) term as a Taylor series, giving us the following:
The -1 and -r/2 terms actually cancel out, letting us simplify the IL calculation into a quadratic term + other higher order terms.
You can see here how the addition of each higher order term approaches the optimal hedge denoted by the pink line (which is the mirror opposite of first IL graph posted above). This is an important concept for later because it shows the diminishing returns of hedging each subsequent higher order.
Credit to @macrocephalopod for his derivation. Give him a follow!
Introducing Power Perps
Now that we’ve formalized the impermanent loss risk we are looking to hedge as an infinite sum of a quadratic + higher order terms, we now have to find an instrument that lets us mimic that risk profile.
The geniuses at Paradigm introduced an instrument that gives us this risk exposure exactly called Power Perps. Simply put, the price of a SOL² power perp is simply a perpetual where the index price is the price of SOL squared.
If the price of SOL doubles, the SOL² power perp 4Xs, the SOL³ power perp 8Xs, and the SOL⁵ power perp 32Xs.
Power Perps allow for us to get asymmetric upside, similar to the risk profile of an option. To exemplify this consider the following:
If the price of SOL goes up 10%, the SOL² power perp goes up 21%
(1 + 0.1)² = 1.21 = 21%
If the price of SOL goes down 10%, the SOL² power perp goes down only 19%
(1–0.1)² = 0.81, or 19%.
The sickest thing about this instrument is that we can decompose our impermanent loss risk into different order power perps! The SOL² power perp can hedge the quadratic term in the IL risk decomposition, the SOL³ power per can hedge the tertiary term, etc.
If we assume a SOL starting price of 136, the percent return profile looks like this:
Composing the IL Hedge using Power Perps
Let’s tie it all together now. Let’s review the impermanent loss risk we are trying to hedge (raw is the IL(pct) from earlier* price of instrument).
We saw earlier that there was a diminishing return in risk reduction when hedging additional higher order terms (it’s also more expensive). For now, let’s focus on only hedging the quadratic term using a Power² Perp.
After simplifying, we can hedge out the r² term in the desired hedge by buying a fraction of a power perp. Buying the power perp has a delta component that we need to hedge out by selling some regular perpetual futures.
The resultant hedged outcome minimizes impermanent loss spectacularly, reversing most of the impermanent loss. Hedging with the Power² Perp actually gives you impermanent gain if the price of Solana rips upwards.
Historical Backtest
Let’s visualize how this type of hedge would have done in the past 3 months. Let’s assume we provided liquidity in the SOL/USDC pair starting from the end of September until now.
We would have experienced maximum IL of 14% of when SOL peaked at 260.
With the Power² Perp hedge on, we would have actually made an additional 10% on our initial position during the SOL rip in November. Since the value of Solana didn’t go below our starting value of 136, we didn’t experience a single bp of impermanent loss, and actually experienced impermanent gain during the entire period.
Note: In a real life trading situation these hedges will look worse, due to various reasons, including the Power² Perp may not always track SOL² precisely for funding reasons, spreads, and trading fees.
Conclusion [Draft]
Power Perps are a great natural hedge against the risks of impermanent loss and abstracts many of the risks of future volatility for liquidity providers. Enabling this would probably lead to an injection of liquidity into the defi ecosystem, giving founders and VCs the exit liquidity they need.
In the next article, we’ll do a deep dive on historical backtests including trading fees and emissions to see the final yield improvement.
Credit to the following articles/teams for the inspiration:
