Some questions that cropped up in the past few years have turned out to be really tricky. We at Paradigm are starting a running list of our favorites. We will update that page periodically with new problems – and hopefully with the solutions that are found.

Summary

The properties of constant product markets like Uniswap are extremely interesting. Despite growing to $25B+ in monthly trading volumes, some questions about their fundamental nature remain unanswered:

  1. What is the expected return of a Uniswap LP?
  2. What is the optimal fee to maximize LP wealth?
  3. Can a Uniswap LP’s optimal growth rate exceed a buy-and-hold portfolio’s?

This document contains formal statements of each problem. They’re tough – academics have said of related problems in classical financial mathematics:

The main problem lies in solving such inequalities. They are so difficult that it has been said that ”every explicit solution is a triumph over nature”.

Introduction

Uniswap LPs lose on volatility. Any change in the relative price of assets they provide liquidity for is experienced as “impermanent loss”. In 2019, I presented the first derivation of Uniswap’s zero-fee replicating portfolio, showing that this decay is equivalent to the negative Gamma of a short option straddle and proportional to the square of the pair’s exchange rate volatility1.

Of course, nobody would provide liquidity to Uniswap if it reflected this model in practice; it would be like selling options for zero premium. Uniswap and others implement a transaction fee to offset this decay. The fees are charged whenever someone trades against the pool, in proportion to the size of their trade2.

1. What is the expected return of a Uniswap LP?

We need to incorporate the effect of accrued fees into our model of LP returns (changes in their wealth, \(W(t)\)). We believe that the asymptotic exponential growth rate of LP wealth, \(G\), is uniquely determined by the fee parameter, \(\gamma\), and volatility of the pair’s risky asset (where the other is riskless cash). We want to solve for \(G\) analytically:

\[G = \mathbb{E}\left[\lim_{T \to \infty}\frac{1}{T} \log(W(T))\right]\]
  • Why it matters: theoretical foundation for understanding the economics of LPs in constant product markets. Allows us to answer the question, “what is Uniswap, really?”
  • Status: Solved

2. What is the optimal fee to maximize LP wealth?

The fee should ideally be set such that LP returns are optimized on a per-market basis. We approach this by maximizing the time-average growth rate criterion above, \(G^{\star}\) (the “log-optimal” strategy of its class), which will materialize in the long-time limit almost surely and is known to be game-theoretically optimal (1, 2):

\[G^{\star} = \max_{0 < \gamma < 1}\left(\mathbb{E}\left[\lim_{T \to \infty}\frac{1}{T} \log(W(T))\right]\right)\]
  • Why it matters: allows us to design more efficient constant product markets. This will be especially important for exotic assets (e.g. those with extreme volatility) that express asymptotic behavior on a macro scale.
  • Status: Solved

3. Can a Uniswap LP’s optimal growth rate exceed a buy-and-hold portfolio’s?

Are there conditions on the drift and volatility (\(\mu\) and \(\sigma\)) of pool assets for which the optimal growth rate of LP wealth, \(G^{\star}\) can generate an excess return to an unrebalanced portfolio of those assets:

\[G^{\star} - \max(\mu - \frac{\sigma^2}{2}, 0) \geq 0?\]
  • Why it matters: tells us whether LPs can only outperform holders by collecting non-arbitrage trading fees, or if an optimal fee can beat it without them. Or, whether zero impermanent loss (IL) can be achieved (in expectation).
  • Status: Solved

  1. An alternative derivation that does not rely on Carr-Madan replication was given in a paper I co-authored with some of the folks at Gauntlet

  2. This analysis assumes that Uniswap is a two-party system of arbitrageurs and LPs, and the former will trade against the latter only when it is profitable. In practice, LPs also collect fees from unprofitable (non-arbitrage) trades. They are excluded here, as are some other practical factors (e.g. gas costs), as the goal is to model a theoretical equlibrium.