Unified Approach for Hedging Impermanent Loss of Liquidity Provision

arXiv:2407.05146 · q-fin.MF, q-fin.TR · Submitted 2024-07-06 · Read on arXiv

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Introduction to the show: ident: Quantitative Finance. The day in quantitative finance: models of markets, risk and prices.

Ruth: We're looking at the paper: "Unified Approach for Hedging Impermanent Loss of Liquidity Provision".

Warren: Here's the gist of the paper. We develop static and dynamic approaches for hedging of the impermanent loss (IL) of liquidity provision (LP) staked at Decentralised Exchanges (DEXes) which employ Uniswap V2 and V3 protocols.

Ruth: This paper provides detailed definitions and formulas for computing the IL to unify different definitions occurring in the existing literature.

Warren: Conceptual Framework of Impermanent Loss The mechanism behind the impermanent loss of LPs is similar to being short a portfolio of call and put options.

Ruth: The paper defines the IL in three ways: 1.

Title and authors: Ruth: Moving into the core of what this paper does, we see that "Unified Approach for Hedging Impermanent Loss of Liquidity Provision" isn't just defining terms; it’s proposing actual methods to manage the risk associated with liquidity provision. It shows how you can treat impermanent loss as a contingent claim with a non-linear payoff at a fixed maturity date >

Warren: A contingent claim means there's an outcome that depends on future price movements, and the payoff isn't just something you calculate once; it evolves >

Iris: That ties into the idea of introducing an "IL protection claim" which delivers the negative of that impermanent loss payoff at a fixed time T >

Ruth: Precisely. So instead of just accepting the loss, you can sell this claim to hedge against it, and they develop arbitrage-based methods for valuing and managing this specific claim >

Warren: I'm interested in those valuation methods. How do they handle the complexity when dealing with protocols like Uniswap V3 that have concentrated liquidity ranges >

Iris: That’s where the decomposition comes in; they derive a formula for impermanent loss in Uniswap V3 into three distinct parts: vanilla options, digital options, and a square root contract payoff >

Ruth: So they're breaking down the loss itself into these recognizable financial instruments that can be priced separately >

Warren: That makes sense. It moves us from just looking at a total loss number to understanding the individual pieces of what's causing the loss on a protocol like Uniswap V3 >

The paper's summary: Ruth: So, to summarize this paper, it’s about developing both static and dynamic approaches for hedging impermanent loss for liquidity provision staked at DEXes using both Uniswap V2 and V3 protocols >

Warren: It covers the definitions they introduced—the nominal, borrowed, and relative IL definitions—and then shows how these lead into the protection claim idea >

Iris: And then they show that if you have a liquid options market, you can hedge this protection claim perfectly using those traded options through a static replication method >

Ruth: But for when there's no liquid options market, they develop a model-based approach where they decompose the IL function into vanilla options, digital options, and that exotic square root payoff >

Warren: That decomposition is what allows them to derive analytic valuation formulas under models like the Black-Scholes-Merton model and even the log-normal stochastic volatility model >

Iris: And for dynamic hedging when there's no liquid market, they use Fourier transform techniques applied to the moment generating function to get those analytic solutions for IL protection claims >

Ruth: So, in essence, it gives you tools whether you have a perfect options market or not to calculate the value of protecting your liquidity provision from impermanent loss >

Warren: It’s a comprehensive toolkit that covers both static replication and model-dependent valuation under different market conditions >

The paper's improvements: Ruth: One significant improvement they suggest is using the IL protection claim to deliver the negative of the IL payoff at maturity, which allows for precise hedging at a fixed time T >

Warren: So, instead of trying to hedge continuously, you can define a specific point in time where you lock in that hedge using options >

Iris: That gives us a concrete target for risk management; it turns the abstract loss into something we can directly price and manage through option delta hedging under exponential price dynamics >

Ruth: And dynamically, they show that by balancing the premiums you receive from selling these claims against the variance realized through delta-hedging, you optimize your total profit and loss position >

Warren: That sounds like a way to control the overall P andL of a trading desk by managing both the upfront premium income and the ongoing hedging costs >

Iris: They also mention that hedging costs can be estimated either by looking at the cost of replicating portfolio or by looking at what you'd pay a trading desk for an IL protection claim >

Ruth: Which means liquidity providers can actually start making better decisions about which DEX pools they choose based on whether the expected fees they earn outweigh those calculated hedging costs >

Conclusion: Warren: So, to wrap up this paper on the "Unified Approach for Hedging Impermanent Loss of Liquidity Provision," we see that the main implication is that there's a structured way to approach IL risk management across different DEX protocols >

Ruth: The paper gives us a comprehensive framework: static replication for liquid markets and model-dependent valuation with decomposition for those without options markets >

Iris: It really establishes the IL protection claim as a formal contingent claim, which helps move the discussion away from just guesswork about loss towards rigorous mathematical pricing and risk management >

Warren: The key is that they provide tools whether you're looking at static replication or dynamic delta hedging under complex models like jump-diffusions and stochastic volatility >

Ruth: It moves us toward a system where liquidity providers can actively select pools based on the comparison between expected fees and the calculated hedging costs derived from this work >

Iris: Ultimately, it gives us a rigorous way to quantify the loss so that we can build better strategies for managing liquidity provision risk moving forward >

Abu Dhabi Investment Authority & ADIA Lab & Khalifa University · Imperial College & Marex · LGT Bank

q-fin.MF, q-fin.TR

Submitted: 2024-07-06

Updated: 2026-10-28

Code: https://github.com/ArturSepp/StochVolModels

Importance score: 71/100

The gist: We develop static and dynamic approaches for hedging of the impermanent loss (IL) of liquidity provision (LP) staked at Decentralised Exchanges (DEXes) which employ Uniswap V2 and V3 protocols.

Key concepts

Impermanent Loss (IL): IL is the difference between the value of assets held in a liquidity pool versus simply holding those individual assets outside the pool. It occurs because asset prices diverge while they are staked, leading to a potential loss compared to not providing liquidity.
Nominal IL Definitions: The paper defines three ways to measure IL for leveraged or borrowed LPs. These include 'funded' and 'borrowed' definitions, which adjust the calculation based on whether the LP is using capital from their own funds or borrowed assets, offering different perspectives on risk.
Static Replication Methodology: This method uses traditional traded options (calls and puts) to create a portfolio that replicates the payoff of an IL protection claim. This allows for a model-independent way to value IL protection using existing financial instruments if a liquid options market exists.
Model-Based Valuation: When liquid options are missing, this approach decomposes the complex IL function into simpler components like vanilla options and exotic square root payoffs. It then uses advanced models like Black-Scholes or stochastic volatility to derive analytic formulas for accurate valuation.

Terminology

Impermanent Loss — 32× in this paper
Automated Market Making — 9× in this paper
Adam — 8× in this paper · explained in 3 episodes
Core — 7× in this paper · explained in 2 episodes
Valuation in Black-Scholes-Merton — 1× in this paper

Summary

We develop static and dynamic approaches for hedging of the impermanent loss (IL) of liquidity provision (LP) staked at Decentralised Exchanges (DEXes) which employ Uniswap V2 and V3 protocols. This paper provides detailed definitions and formulas for computing the IL to unify different definitions occurring in the existing literature.

Conceptual Framework of Impermanent Loss

The mechanism behind the impermanent loss of LPs is similar to being short a portfolio of call and put options. The paper defines the IL in three ways:

Nominal IL for funded LP is defined by P&L funded(y) = (ptxt + yt) − (p0x0 + y0) / (p0x0 + y0).
Nominal IL for borrowed LP is defined by P&L borrowed(y) = (ptxt + yt) − (ptx0 + y0) / (p0x0 + y0).
Relative IL for borrowed LP is defined by Rel IL borrowed(y) = (ptxt + yt) − (ptx0 + y0) / (ptx0 + y0).

Static Replication Methodology

The authors develop a static model-independent replication method for the valuation of IL protection claim using traded European vanilla call and put options. They extend this to show that the IL protection claim can be hedged perfectly with options if there is a liquid options market.

The replication portfolio for the payoff of IL protection claim is given by the sum of put and call option portfolios. The weights for these options are computed using the first-order derivatives of the IL function at discrete strike points.

Model-Dependent Valuation and Dynamics Hedging

For digital assets without a liquid options market, a model-based approach is developed using the decomposition of the IL function into vanilla options, digital options, and an exotic square root payoff. They derive analytic valuation formulas for IL protection claims under the Black-Scholes-Merton model and the log-normal stochastic volatility model.

The valuation of IL protection claims is computed using Fourier transform techniques applied to the moment generating function (MGF). For instance, the BSM value of the IL protection claim under Uniswap V3 is given by Proposition 5.2.

Application to AMM Protocols

The study applies these results to both Uniswap V2 and V3 protocols. For Uniswap V3, the IL of funded LP can be decomposed into three parts: linear part, exotic square root price, and vanilla option part. The payoff of the IL protection claim for funded LP under Uniswap V3 is given by Proposition 3.6.

Conclusion on Hedging Strategies

The total P&L of a trading desk will be driven by the difference between premiums received (from selling IL protection claims) and the variance realised through delta-hedging. Liquidity providers can focus on selecting DEX pools and liquidity ranges where expected fees could exceed hedging costs. The paper concludes by showing that the protection seller must hedge the claim dynamically using option delta under BSM model.

References

Angeris G., Kao H.T., Chiang R., Noyes C., Chitra T., (2019). An analysis of Uniswap markets Cryptoeconomic Systems Journal.

Alexander C., Chen D., Imeraj A. (2023), Crypto quanto and inverse options, Mathematical Finance 33(4), 1005–1043.

Bergault P., Bertucci L., Bouba D. (2023). Automated market makers: mean-variance analysis of LPs payoffs and design of pricing functions. Digital Finance, https://doi.org/10.1007/s42521-023-00101-0.

Capponi A., and Jia R., (2024). Liquidity provision on blockchain-based decentralized exchanges. Working paper, https://ssrn.com/abstract=3805095.

Cartea A., Drissi F., and Monga M., (2024). Predictable losses of liquidity provision in constant function markets and concentrated liquidity markets. Applied Mathematical Finance, 30, 69–93.

Cartea A., Drissi F., and Monga M., (2024). Decentralised Finance and Automated Market Making: Predictable Loss and Optimal Liquidity Provision. Working paper, https://arxiv.org/abs/2309.08431.

P Carr and D Madan., (2001). Towards a Theory of Volatility Trading, In: E Jouini, J Cvitanic, M Musiela, eds. Handbooks in Mathematical Finance: Option Pricing, Interest Rates and Risk Management, Cambridge University Press 458–476.

Adams H., Zinsmeister N., and D Robinson., (2021). Uniswap V2 Core, White paper https://uniswap.org/whitepaper.pdf.

Adams H., Zinsmeister N., Salem M., Keefer R., and D Robinson., (2021). Uniswap V3 Core, White paper https://uniswap.org/whitepaper-V3.pdf.

Deng J., Zong H., and Y Wang., (2023). Static replication of impermanent loss for concentrated liquidity provision in decentralised markets, Operations Research Letters 51(3) 206–211.

Echenim M., Gobet E., and A.C. Maurice., (2023). Thorough mathematical modelling and analysis of Uniswap V3, Working paper https://hal.science/hal-04214315V2.

Fukasawa M., Maire B., and M Wunsch., (2023). Weighted variance swaps hedge against impermanent loss, Quantitative Finance 23(6) 901–911.

Heimbach L., Schertenleib E., and R Wattenhofer., (2022). Risks and returns of Uniswap v3 liquidity providers, Working paper https://arxiv.org/abs/2205.08904.

Lehar A., and Parlour C.A., (2024). Decentralized Exchange: The Uniswap Automated Market Maker, Journal of Finance, forthcoming https://ssrn.com/abstract=3905316.

Lewis A. L., (2001). Mathematical Methods for Foreign Exchange: A Financial Engineer’s Approach. World Scientific.

Lipton A., (2018). Financial Engineering: Selected Works of Alexander Lipton. World Scientific.

Lipton A., and Hardjono T., (2022). Blockchain Intra- and Interoperability, in Babich V, Birge J, Hilary G (eds), Springer, New York.

Lipton A., and Sepp A., (2008). Stochastic volatility models and Kelvin waves. J. Phys. A: Math. Theor. 41, 344012 (23pp).

Lipton A., and Sepp A., (2022). Automated market-making for fiat currencies, Risk Magazine, May.

Lucic V. and Sepp A., (2024), Valuation and Hedging of Cryptocurrency Inverse Options: With Backtest Simulations using Deribit Options Data, Quantitative Finance, forthcoming https://ssrn.com/abstract=4606748.

Maire B., and Wunsch M., (2024), Market Neutral Liquidity Provision, Preprint Milionis J., Moallemi C C., Roughgarden T., Lee Zhang A. (2022). Automated Market Making and Loss-Versus-Rebalancing, Financial Innovation 8(20) https://api.semanticscholar.org/.

Mohan V., (2022), Automated market makers and decentralized exchanges: a DeFi primer, Financial Innovation 8(20) https://doi.org/10.1186/s40854-021-00314-5.

Park A., (2023), The conceptual flaws of decentralized automated market making, Management Science 69(11) 6731–6751. https://doi.org/10.1287/mnsc.2021.02802.

Sepp A., and Rakhmonov P., (2023) Log-normal Stochastic Volatility Model with Quadratic Drift, International Journal of Theoretical and Applied Finance 26(08), 2450003, https://doi.org/10.1142/S021902492450031.

Improvements for AI systems

Automated risk management for liquidity provision can be enhanced by calculating an IL protection claim which delivers the negative of IL payoff at the maturity date. This allows for a precise hedging strategy using options to eliminate impermanent loss at a fixed maturity time T.
Model-based valuation systems can be developed for digital assets lacking liquid options markets by decomposing the impermanent loss into vanilla options, digital options, and an exotic square root payoff. This decomposition enables model-dependent valuation of claims under various price dynamics.
Dynamic delta hedging strategies can be applied to the IL protection claim using model-based approaches for pricing under exponential price dynamics, which allows the system to optimize total P&L by balancing premiums received (from selling IL protection claims) and the variance realised through delta-hedging.
The system can perform continuous-time valuation of IL protection claims under complex models like jump-diffusions and stochastic volatility by utilizing moment generating functions and Fourier inversion techniques to derive analytic solutions.
Hedging costs for liquidity providers can be accurately estimated beforehand by using either the cost of static options replicating portfolio or costs of buying IL protection from a trading desk, enabling optimal selection of DEX pools and liquidity ranges where expected fees exceed hedging costs.