We can show that in Uniswap v3, the derivative is equal to. Appendices Appendix A: Definition of liquidity Finally, not every useful AMM can be easily represented as a function in liquidity space.įuture work will demonstrate some techniques for getting around these problems using numerical approximation, as well as some smart contract tricks to make custom liquidity provision much more gas-efficient.The gas cost of minting and burning is proportional to the number of ticks updated, so providing liquidity at custom levels across all of tick space would be insurmountably inefficient.Ticks are not infinitely divisible, and liquidity providers have to approximate this curve using the granular ticks available.However, there are several limitations that need to be overcome before Uniswap v3 can be used to simulate most of these AMMs: This paper showed how several popular AMMs could be simulated using Uniswap v3, and showed how graphing curves in liquidity space provides insight into their unique ``liquidity fingerprints". So LMSR concentrates more liquidity to support probabilities around 50% than to support more extreme probabilities.Īppendix B shows how this liquidity fingerprint can be computed. In prediction markets, when YES and NO shares are equal, that implies a probability of 50%. Tick indexes ( ) are logarithmic in price, and specify the lower and upper prices at which that position provides liquidity.Īs shown in the Uniswap v3 whitepaper, this is the trading function that describes the relationship between the reserves of a single Uniswap v3 position while its liquidity is in range:Īs you can see, LMSR concentrates more of its liquidity closer to tick 0 (the price of 1). In Uniswap v3, anyone can create a position to provide some amount of liquidity-within a price range between two ticks. (For more background on thinking about AMMs in terms of derivatives, see the YieldSpace paper). is equivalent to the negated derivative of the reserves curve. We will use to refer to the price of asset in terms of asset. For example, the constant product formula used by Uniswap v2 (ignoring fees) can be described with the formula, with roughly corresponding to the total supply of liquidity tokens. You can loosely think of as the number of ``shares" that liquidity providers have in that pool. We will only consider two-asset pools.įor ease of comparison, let's define the constants in each of these formulas in terms of, where is the quantity of when. This paper assumes you are familiar with automated market makers between two assets and that are defined as a trading function (see paper), meaning a relationship between its reserves and. We can visualize the liquidity provided by any AMM as a curve in "tick space." Applying this method to existing AMMs like Curve, Balancer, and the logarithmic market scoring rule (LMSR) shows how those AMMs concentrate their liquidity across different prices, revealing the unique "liquidity fingerprint" corresponding to each of those AMMs.įollow-up work will explore how strategies like this can be efficiently and accurately approximated in practice on Uniswap v3.
Any static AMM can be approximated by a custom liquidity provision strategy involving multiple positions on Uniswap v3. This unlocks tremendous capital efficiency gains for liquidity providers who can manually adjust their exposure.īut this feature also greatly expands the design space for automated liquidity provision. Uniswap v3 ( paper) allows liquidity providers to provide custom amounts of liquidity in selected price ranges.
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