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.