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Abstract Constant function market makers (CFMMs) are the most popular type of decentralized trading venue for cryptocurrency tokens. In this paper,…

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Public blockchains allow any user to submit transactions which modify the shared state of the network. These transactions are independently…

Public blockchains implement a fee mechanism to allocate scarce computational resources across competing transactions. Most existing fee market designs utilize a joint, fungible unit of account (e.g., gas in Ethereum) to price otherwise non-fungible resources such as bandwidth, computation, and storage, by hardcoding their relative prices. Fixing the relative price of each resource in this way inhibits granular price discovery, limiting scalability and opening up the possibility of denial-of-service attacks.

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This post is an exposition on the landscape of privacy in the context of public blockchains (a.k.a. decentralized ledgers, crypto, and Web3). The first part touches on why privacy is a key hurdle to wide-scale adoption and what different aspects of privacy are. The second part surveys three different approaches to privacy: via zero-knowledge proofs, aiming for anonymity only, and via a new abstraction called MOCCAs.

We consider the problem of optimally executing an order involving multiple cryptoassets, sometimes called tokens, on a network of multiple constant function market makers (CFMMs). When we ignore the fixed cost associated with executing an order on a CFMM, this optimal routing problem can be cast as a convex optimization problem, which is computationally tractable. When we include the fixed costs, the optimal routing problem is a mixed-integer convex problem, which can be solved using (sometimes slow) global optimization methods, or approximately solved using various heuristics based on convex optimization. The optimal routing problem includes as a special case the problem of identifying an arbitrage present in a network of CFMMs, or certifying that none exists.

In this paper, we show that any monotonic payoff can be replicated using only liquidity provider shares in constant function market makers (CFMMs), without the need for additional collateral or oracles. Such payoffs include cash-or-nothing calls and capped calls, among many others, and we give an explicit method for finding a trading function matching these payoffs. For example, this method provides an easy way to show that the trading function for maintaining a portfolio where 50% of the portfolio is allocated in one asset and 50% in the other is exactly the constant product market maker (e.g., Uniswap) from first principles. We additionally provide a simple formula for the total earnings of an arbitrageur who is arbitraging against these CFMMs.

The rise of Ethereum and other blockchains that support smart contracts has led to the creation of decentralized exchanges (DEXs), such as Uniswap, Balancer, Curve, mStable, and SushiSwap, which enable agents to trade cryptocurrencies without trusting a centralized authority. While traditional exchanges use order books to match and execute trades, DEXs are typically organized as constant function market makers (CFMMs). CFMMs accept and reject proposed trades based on the evaluation of a function that depends on the proposed trade and the current reserves of the DEX. For trades that involve only two assets, CFMMs are easy to understand, via two functions that give the quantity of one asset that must be tendered to receive a given quantity of the other, and vice versa. When more than two assets are being exchanged, it is harder to understand the landscape of possible trades. We observe that various problems of choosing a multi-asset trade can be formulated as convex optimization problems, and can therefore be reliably and efficiently solved.

We present a method for constructing Constant Function Market Makers (CFMMs) whose portfolio value functions match a desired payoff. More specifically, we show that the space of concave, nonnegative, nondecreasing, 1-homogeneous payoff functions and the space of convex CFMMs are equivalent; in other words, every CFMM has a concave, nonnegative, nondecreasing, 1-homogeneous payoff function, and every payoff function with these properties has a corresponding convex CFMM. We demonstrate a simple method for recovering a CFMM trading function that produces this desired payoff. This method uses only basic tools from convex analysis and is intimately related to Fenchel conjugacy. We demonstrate our result by constructing trading functions corresponding to basic payoffs, as well as standard financial derivatives such as options and swaps.

Constant function market makers (CFMMs) such as Uniswap, Balancer, Curve, and mStable, among many others, make up some of the largest decentralized exchanges on Ethereum and other blockchains. Because all transactions are public in current implementations, a natural next question is if there exist similar decentralized exchanges which are privacy-preserving; i.e., if a transaction’s quantities are hidden from the public view, then an adversary cannot correctly reconstruct the traded quantities from other public information. In this note, we show that privacy is impossible with the usual implementations of CFMMs under most reasonable models of an adversary and provide some mitigating strategies.

Constant Function Market Makers (CFMMs) are a family of automated market makers that enable censorship-resistant decentralized exchange on public blockchains. Arbitrage trades have been shown to align the prices reported by CFMMs with those of external markets. These trades impose costs on Liquidity Providers (LPs) who supply reserves to CFMMs. Trading fees have been proposed as a mechanism for compensating LPs for arbitrage losses. However, large fees reduce the accuracy of the prices reported by CFMMs and can cause reserves to deviate from desirable asset compositions. CFMM designers are therefore faced with the problem of how to optimally select fees to attract liquidity. We develop a framework for determining the value to LPs of supplying liquidity to a CFMM with fees when the underlying process follows a general diffusion. Focusing on a popular class of CFMMs which we call Geometric Mean Market Makers (G3Ms), our approach also allows one to select optimal fees for maximizing LP value. We illustrate our methodology by showing that an LP with mean-variance utility will prefer a G3M over all alternative trading strategies as fees approach zero.

Geometric mean market makers (G3Ms), such as Uniswap and Balancer, comprise a popular class of automated market makers (AMMs) defined by the following rule: the reserves of the AMM before and after each trade must have the same (weighted) geometric mean. This paper extends several results known for constant-weight G3Ms to the general case of G3Ms with time-varying and potentially stochastic weights. These results include the returns and no-arbitrage prices of liquidity pool (LP) shares that investors receive for supplying liquidity to G3Ms. Using these expressions, we show how to create G3Ms whose LP shares replicate the payoffs of financial derivatives. The resulting hedges are model-independent and exact for derivative contracts whose payoff functions satisfy an elasticity constraint. These strategies allow LP shares to replicate various trading strategies and financial contracts, including standard options. G3Ms are thus shown to be capable of recreating a variety of active trading strategies through passive positions in LP shares.