Automated market makers, first popularized by Hanson's logarithmic market scoring rule (or LMSR) for prediction markets, have become important building blocks, called 'primitives,' for decentralized finance. A particularly useful primitive is the ability to measure the price of an asset, a problem often known as the pricing oracle problem. In this paper, we focus on the analysis of a very large class of automated market makers, called constant function market makers (or CFMMs) which includes existing popular market makers such as Uniswap, Balancer, and Curve, whose yearly transaction volume totals to billions of dollars. We give sufficient conditions such that, under fairly general assumptions, agents who interact with these constant function market makers are incentivized to correctly report the price of an asset and that they can do so in a computationally efficient way. We also derive several other useful properties that were previously not known. These include lower bounds on the total value of assets held by CFMMs and lower bounds guaranteeing that no agent can, by any set of trades, drain the reserves of assets held by a given CFMM.
We study the price-setting problem of market makers under risk neutrality and perfect competition in continuous time. Thereby we follow the classic Glosten-Milgrom model that defines bid and ask prices as expectations of a true value of the asset given the market makers' partial information that includes the customers trading decisions. The true value is modeled as a Markov process that can be observed by the customers with some noise at Poisson times. We analyze the price-setting problem in a mathematically rigorous way by solving a filtering problem with an endogenous filtration that depends…
The continuous-time version of Kyle's (1985) model is studied, in which market makers are not fiduciaries. They have some market power which they utilize to set the price to their advantage, resulting in positive expected profits. This has several implications for the equilibrium, the most important being that by setting a modest fee conditional of the order flow, the market maker is able to obtain a profit of the order of magnitude, and even better than, a perfectly informed insider. Our model also indicates why speculative prices are more volatile than predicted by fundamentals.
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…
The Maker Protocol is a decentralized finance application that enables collateralized lending. The application uses open-bid, second-price auctions to complete its loan liquidation process. In this paper, we develop a bidding function for these auctions, focusing on the costs incurred to participate in the auctions. We then optimize these costs using parameters from historical auction data, and compare our optimal bidding prices to the historical auction prices. We find that the majority of auctions end at higher prices than our recommended optimal prices, and we propose several theories for t…
We consider a stochastic game between three types of players: an inside trader, noise traders and a market maker. In a similar fashion to Kyle's model, we assume that the insider first chooses the size of her market-order and then the market maker determines the price by observing the total order-flow resulting from the insider and the noise traders transactions. In addition to the classical framework, a revenue term is added to the market maker's performance function, which is proportional to the order flow and to the size of the bid-ask spread. We derive the maximizer for the insider's reven…
We introduce a class of utility-based market makers that always accept orders at their risk-neutral prices. We derive necessary and sufficient conditions for such market makers to have bounded loss. We prove that hyperbolic absolute risk aversion utility market makers are equivalent to weighted pseudospherical scoring rule market makers. In particular, Hanson's logarithmic scoring rule market maker corresponds to a negative exponential utility market maker in our framework. We describe a third equivalent formulation based on maintaining a cost function that seems most natural for implementatio…
In most OTC markets, a small number of market makers provide liquidity to other market participants. More precisely, for a list of assets, they set prices at which they agree to buy and sell. Market makers face therefore an interesting optimization problem: they need to choose bid and ask prices for making money while mitigating the risk associated with holding inventory in a volatile market. Many market making models have been proposed in the academic literature, most of them dealing with single-asset market making whereas market makers are usually in charge of a long list of assets. The rare…
University of Pennsylvania（宾夕法尼亚大学）；New York University（纽约大学）
Liquidity Providers on Automated Market Makers generate millions of USD in transaction fees daily. However, the net value of a Liquidity Position is vulnerable to price changes in the underlying assets in the pool. The dominant measure of loss in a Liquidity Position is Impermanent Loss. Impermanent Loss for Constant Function Market Makers has been widely studied. We propose a new metric to measure Liquidity Position PNL based on price movement from the underlying assets. We show how this new metric more appropriately measures the change in the net value of a Liquidity Position as a function o…
We study a financial model with a non-trivial price impact effect. In this model we consider the interaction of a large investor trading in an illiquid security, and a market maker who is quoting prices for this security. We assume that the market maker quotes the prices such that by taking the other side of the investor's demand, the market maker will arrive at maturity with maximal expected wealth. Within this model we concentrate on the issue of contingent claims' hedging.
In this paper we extend the market-making models with inventory constraints of Avellaneda and Stoikov ("High-frequency trading in a limit-order book", Quantitative Finance Vol.8 No.3 2008) and Gueant, Lehalle and Fernandez-Tapia ("Dealing with inventory risk", Preprint 2011) to the case of a rather general class of mid-price processes, under either exponential or linear PNL utility functions, and we add an inventory-risk-aversion parameter that penalises the marker-maker if she finishes her day with a non-zero inventory. This general, non-martingale framework allows a market-maker to make dire…
Silkswap is an automated market maker model designed for efficient stablecoin trading with minimal price impact. The original purpose of Silkswap is to facilitate the trading of fiat-pegged stablecoins with the stablecoin Silk, but it can be applied to any pair of stablecoins. The Silkswap invariant is a hybrid function that generates an asymmetric price impact curve. We present the derivation of the Silkswap model and its mathematical properties. We also compare different numerical methods used to solve the invariant equation. Finally, we compare our model with the well-known Curve Finance mo…
We develop from basic economic principles a continuous-time model for a large investor who trades with a finite number of market makers at their utility indifference prices. In this model, the market makers compete with their quotes for the investor's orders and trade among themselves to attain Pareto optimal allocations. We first consider the case of simple strategies and then, in analogy to the construction of stochastic integrals, investigate the transition to general continuous dynamics. As a result, we show that the model's evolution can be described by a nonlinear stochastic differential…
JPMorgan AI Research（摩根大通人工智能研究院）；JPMorgan Quantitative Research（摩根大通量化研究部）
Market makers play an important role in providing liquidity to markets by continuously quoting prices at which they are willing to buy and sell, and managing inventory risk. In this paper, we build a multi-agent simulation of a dealer market and demonstrate that it can be used to understand the behavior of a reinforcement learning (RL) based market maker agent. We use the simulator to train an RL-based market maker agent with different competitive scenarios, reward formulations and market price trends (drifts). We show that the reinforcement learning agent is able to learn about its competitor…
We develop a single-period model for a large economic agent who trades with market makers at their utility indifference prices. A key role is played by a pair of conjugate saddle functions associated with the description of Pareto optimal allocations in terms of the utility function of a representative market maker.
This paper studies optimal market making for large-tick assets in the presence of latency. We consider a random walk model for the asset price, and formulate the market maker's optimization problem using Markov Decision Processes (MDP). We characterize the value of an order and show that it plays the role of one-period reward in the MDP model. Based on this characterization, we provide explicit criteria for assessing the profitability of market making when there is latency. Under our model, we show that a market maker can earn a positive expected profit if there are sufficient uninformed marke…
We provide an explicit characterization of the optimal market making strategy in a discrete-time Limit Order Book (LOB). In our model, the number of filled orders during each period depends linearly on the distance between the fundamental price and the market maker's limit order quotes, with random slope and intercept coefficients. The high-frequency market maker (HFM) incurs an end-of-the-day liquidation cost resulting from linear price impact. The optimal placement strategy incorporates in a novel and parsimonious way forecasts about future changes in the asset's fundamental price. We show t…
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 the…
We consider a simple model for the evolution of a limit order book in which limit orders of unit size arrive according to independent Poisson processes. The frequencies of buy limit orders below a given price level, respectively sell limit orders above a given level are described by fixed demand and supply functions. Buy (resp. sell) limit orders that arrive above (resp. below) the current ask (resp. bid) price are converted into market orders. There is no cancellation of limit orders. This model has independently been reinvented by several authors, including Stigler in 1964 and Luckock in 200…
In a semi-realistic market simulator, independent reinforcement learning algorithms may facilitate market makers to maintain wide spreads even without communication. This unexpected outcome challenges the current antitrust law framework. We study the effectiveness of maker-taker fee models in preventing cooperation via algorithms. After modeling market making as a repeated general-sum game, we experimentally show that the relation between net transaction costs and maker rebates is not necessarily monotone. Besides an upper bound on taker fees, we may also need a lower bound on maker rebates to…
In this paper we explore optimal liquidation in a market populated by a number of heterogeneous market makers that have limited inventory-carrying and risk-bearing capacity. We derive a reduced form model for the dynamic of their aggregated inventory considering a proper scaling limit. The resulting price impact profile is shown to depend on the characteristics and relative importance of their inventories. The model is flexible enough to reproduce the empirically documented power law behavior of the price impact function. For any choice of the market makers characteristics, optimal execution w…
Università degli Studi di Milano and Politecnico di Milano, Milan, Italy（米兰大学和米兰理工学院，意大利）；University of Ottawa, Ottawa, Canada（渥太华大学，加拿大）；Politecnico di Milano and Università degli Studi di Milano, Milan, Italy（米兰理工学院和米兰大学，意大利）；Università degli Studi di Milano, Milan, Italy（米兰大学，意大利）；Independent researcher（独立研究者）
Traditional market makers are losing their importance as automated systems have largely assumed the role of liquidity provision in markets. We update the model of Glosten and Milgrom (1985) to analyze this new world: we add multiple securities and introduce an automated market maker who uses the relationships between securities to price order flow. This new automated participant transacts the majority of orders, sets prices that are more efficient, and increases informed and decreases uninformed traders' transaction costs. These results can explain the recent dominance of high frequency tradin…
Automated market makers (AMM) have grown to obtain significant market share within the cryptocurrency ecosystem, resulting in a proliferation of new products pursuing exotic strategies for horizontal differentiation. Yet, their theoretical properties are curiously homogeneous when a set of basic assumptions are met. In this paper, we start by presenting a universal approach to deriving a formula for liquidity provisioning for AMMs. Next, we show that the constant function market maker and token swap market maker models are theoretically equivalent when liquidity reserves are uniform. Proceedin…
University of Liverpool（利物浦大学）；L3S Research Center（L3S研究中心）；Leibniz University Hannover（汉诺威莱布尼茨大学）
This paper introduces a new representation for the actions of a market maker in an order-driven market. This representation uses scaled beta distributions, and generalises three approaches taken in the artificial intelligence for market making literature: single price-level selection, ladder strategies and "market making at the touch". Ladder strategies place uniform volume across an interval of contiguous prices. Scaled beta distribution based policies generalise these, allowing volume to be skewed across the price interval. We demonstrate that this flexibility is useful for inventory managem…
In most illiquid markets, there is no obvious proxy for the market price of an asset. The European corporate bond market is an archetypal example of such an illiquid market where mid-prices can only be estimated with a statistical model. In this OTC market, dealers / market makers only have access, indeed, to partial information about the market. In real time, they know the price associated with their trades on the dealer-to-dealer (D2D) and dealer-to-client (D2C) markets, they know the result of the requests for quotes (RFQ) they answered, and they have access to composite prices (e.g., Bloom…
As an integral part of the decentralized finance (DeFi) ecosystem, decentralized exchanges (DEXs) with automated market maker (AMM) protocols have gained massive traction with the recently revived interest in blockchain and distributed ledger technology (DLT) in general. Instead of matching the buy and sell sides, automated market makers (AMMs) employ a peer-to-pool method and determine asset price algorithmically through a so-called conservation function. To facilitate the improvement and development of automated market maker (AMM)-based decentralized exchanges (DEXs), we create the first sys…
Market makers provide liquidity to other market participants: they propose prices at which they stand ready to buy and sell a wide variety of assets. They face a complex optimization problem with both static and dynamic components. They need indeed to propose bid and offer/ask prices in an optimal way for making money out of the difference between these two prices (their bid-ask spread). Since they seldom buy and sell simultaneously, and therefore hold long and/or short inventories, they also need to mitigate the risk associated with price changes, and subsequently skew their quotes dynamicall…
We consider an auction market in which market makers fill the order book during a given time period while some other investors send market orders. We define the clearing price of the auction as the price maximizing the exchanged volume at the clearing time according to the supply and demand of each market participants. Then we derive in a semi-explicit form the error made between this clearing price and the efficient price as a function of the auction duration. We study the impact of the behavior of market takers on this error. To do so we consider the case of naive market takers and that of r…
When prices reflect all available information, they oscillate around an equilibrium level. This oscillation is the result of the temporary market impact caused by waves of buyers and sellers. This price behavior can be approximated through an Ornstein-Uhlenbeck (O-U) process. Market makers provide liquidity in an attempt to monetize this oscillation. They enter a long position when a security is priced below its estimated equilibrium level, and they enter a short position when a security is priced above its estimated equilibrium level. They hold that position until one of three outcomes occur:…
Market makers continuously set bid and ask quotes for the stocks they have under consideration. Hence they face a complex optimization problem in which their return, based on the bid-ask spread they quote and the frequency at which they indeed provide liquidity, is challenged by the price risk they bear due to their inventory. In this paper, we consider a stochastic control problem similar to the one introduced by Ho and Stoll and formalized mathematically by Avellaneda and Stoikov. The market is modeled using a reference price $S_t$ following a Brownian motion with standard deviation $σ$, arr…
In corporate bond markets, which are mainly OTC markets, market makers play a central role by providing bid and ask prices for a large number of bonds to asset managers from all around the globe. Determining the optimal bid and ask quotes that a market maker should set for a given universe of bonds is a complex task. Useful models exist, most of them inspired by that of Avellaneda and Stoikov. These models describe the complex optimization problem faced by market makers: proposing bid and ask prices in an optimal way for making money out of the difference between bid and ask prices while mitig…
