Static Hedging of Impermanent Loss in Constant Product AMMs
Replicating LP positions with options to reduce risk and improve capital efficiency
What Is Impermanent Loss and Why It Matters
In decentralized finance (DeFi), liquidity providers (LPs) are the backbone of automated market makers (AMMs) like Uniswap, Balancer, or Sushi. But being an LP comes with a hidden risk: impermanent loss.
Impermanent loss happens when the prices of the assets you deposited in a liquidity pool change over time. As the AMM automatically rebalances your position, you may end up with fewer tokens of the one that appreciated—and more of the one that lost value. When you withdraw your funds, the result might be worse than simply holding the tokens.
Why is this a big deal?
For LPs, impermanent loss can eat into yield, especially in volatile markets.
For protocols, it can discourage liquidity provision, reducing depth and efficiency.
For DeFi as a whole, it’s a key challenge in designing sustainable tokenomics and vaults that offer real yield.
At POL Finance, we're working on research-backed tools to better understand, quantify, and mitigate these risks.
Case Study: Uniswap v2 and Constant Product AMMs
At the heart of protocols like Uniswap v2 lies a simple yet powerful mechanism: the Constant Product Market Maker (CPMM). It ensures that for every trade, the product of the two token reserves remains constant:
Where x and y are the reserves of two tokens in the pool, and k is constant.
This formula makes it easy for anyone to swap tokens on-chain, without relying on order books or centralized intermediaries. But this simplicity comes with hidden complexity when it comes to LP returns.
Here’s why:
As token prices move, the AMM automatically rebalances the pool.
This rebalancing means LPs end up with a different mix of tokens than they started with.
If prices diverge significantly, the LP would have been better off just holding the tokens outside the pool.
This is the essence of impermanent loss—a cost built into the AMM’s design. But what if we could quantify that loss precisely, and even hedge against it?
That's what our research set out to explore.
A Quant-Inspired Solution Using Options
To address impermanent loss, we turned to a classic toolbox from traditional finance: options.
Options are financial instruments that give you the right (but not the obligation) to buy or sell an asset at a specific price. In particular, we use a well-known strategy called a long strangle—a combination of call and put options with different strike prices.
Here’s the idea:
We treat the value of a liquidity position in an AMM as a payoff function.
Then, we construct a static portfolio of options that replicates this payoff or offsets its risk.
This strategy is passive: it doesn’t require rebalancing, just initial calibration.
By applying this framework, we can protect LPs within a price range around the entry price. If prices swing too far, the options “kick in” and reduce the impact of impermanent loss.
At POL Finance, we implemented this using a simple Python model, then extended it with real options data from Deribit to simulate practical use cases.
Results and Benefits for Protocol Designers
This kind of options-based hedging isn’t just theoretical—it opens up real opportunities for DeFi protocols and builders.
Here’s what we found:
The hedging strategy works across a defined price range, especially in medium-volatility environments.
LPs can reduce downside risk without needing to actively manage their position.
Protocols can use these tools to design smarter vaults, optimize fee structures, or offer built-in IL protection.
For protocols working on risk management, yield optimization, or LP retention, these strategies offer a foundation to:
Create vaults that balance yield and safety.
Quantify how different fee structures affect LP behavior.
Simulate user-level outcomes under various market conditions.
This is especially relevant for newer AMM designs or protocols competing for liquidity in today’s fragmented DeFi landscape.
Conclusion and What’s Next
Impermanent loss has long been one of the most misunderstood—and underestimated—risks in DeFi. But with the right tools, we can make it visible, measurable, and even hedgeable.
At POL Finance, we’re building open-source models and simulations to bridge the gap between theory and practice in DeFi risk management. This post is just one example of how ideas from quantitative finance—like static replication and options pricing—can empower LPs and protocol designers alike.
We’ll continue publishing research-backed content and tools to support the ecosystem.
📊 Want to explore the model?
Try the simulation on Colab
Access the full code on GitHub
Read the original academic paper for the full derivation
🧠 If you're building a protocol and want to collaborate, feel free to reach out.
Want to dive deeper into the math behind this?
I wrote a full breakdown of the derivation, including all formulas and code examples, on my personal blog:
👉 Read the technical deep dive