Expiring Assets in Automated Market Makers

An automated market maker (AMM) [7] is an automaton (usually implemented as a smart contract) that trades one electronic asset for another at rates set by a fixed mathematical formula. Liquidity providers (LPs) lend assets to the AMM in return for a share of transaction fees and governance rights. Today, AMMs form a multi-billion dollar business [18]. The principal benefit of AMMs over more traditional order-book methods is that an AMM offers its assets at a take-it-or-leave-it price, so trades occur immediately, without need for complex bidding strategies.

AMMs typically trade long-lived assets such as cryptocurrencies or stablecoins. In this paper, we investigate a decentralized exchange (DEX) that can be designed to trade expiring assets, those that have a fixed date after which they cannot be resold. For example, a flight ticket from New York to Paris that departs on Monday cannot be sold on Tuesday. Expiration can have complicated effects on price: a last-minute airline ticket may sell at a premium, but a last-minute ticket for a Broadway show may sell at a discount.

Parties can play three distinct roles. Producers sell (expiring) tokens through the DEX. For ease of exposition, all tokens expire at the same well-known time and date. Consumers use (electronic) cash to purchase those tokens through the DEX. Cash has a stable value, and acts as the numéraire. (Some parties may combine both roles to act as arbitrageurs, buying and selling tokens with the effect of aligning the DEX token price with an exogenous market price.) Finally, liquidity providers (LPs) are market makers, buying tokens from producers and selling them to consumers in return for fees.

For most of the DEX’s lifetime, it acts like a standard automated market maker. Producers and consumers sell and buy tokens at a take-it-or-leave-it price determined by the formula specific to the DEX. All transactions are executed immediately. If there is a known, exogenous “market” where tokens can also be bought and sold, then arbitrageurs keep the DEX price in agreement with the market price. Liquidity providers bootstrap the DEX by loaning assets, and they profit by taking a fee for each transaction. When the the system shuts down, its remaining cash is returned to the LPs.

As the expiration date approaches, standard market mechanisms may fail. For some unknown but brief time before token expiration, token supply may vanish: producers stop selling new tokens on the DEX, no matter the price. (For example, airlines cannot create more seats for an imminent Monday flight to Paris.) Similarly, token demand may vanish: consumers stop buying tokens on the DEX, no matter the price. (Hotel rooms in the Caribbean are not going to be booked by people that cannot get there on time) In addition, airline tickets, hotel rooms, and similar expiring assets have prices that can vary highly in response to exogenous events that may be harder to predict, such as weather, geopolitics, etc. Under these circumstances, the DEX can no longer rely on arbitrage to keep the market efficient.

In particular, conventional AMMs fail under those circumstances. Normally, if an AMM’s price for tokens (in cash) rises above the market price, then an arbitrageur would buy (or create) new tokens and sell them to the AMM, thus collecting a risk-free profit and bringing the AMM’s token price back to the market rate. If supply vanishes, however, each token purchase will drive the AMM price up, and eventually the AMM’s token price will rise just above the market price, and the remaining tokens will expire unsold, even though both the consumers and the LPs would have been better off selling those tokens at a lower price. From this we learn that a DEX for expiring assets cannot rely exclusively on arbitrage to keep prices in line with the market.

In an alternative scenario, if demand vanishes, token producers will dump their expiring tokens on the AMM. Of course, such panicky selling will cause the AMM price to fall, but the token producers will realize at least some cash for their tokens. But these trades come at the expense of the LPs who loaned the cash that is being replaced with soon-to-be worthless tokens. From this we learn that AMMs for expiring assets must provide some way for LPs to intervene to limit their exposure to market distortions due to expiration.

This paper describes the design of a decentralized exchange (DEX) for tokens that expire, with the following design goals.

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  Market clearing: at expiration time, it is never the case that there is a producer willing to sell a token at some price ¹¹1 A producer may have a minimum price even for expiring tokens. For example, a hotel might not rent a room for less than the cost of cleaning that room., and a consumer willing to buy at that price, yet that token remains unsold,

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  Instant gratification: each asset is bought and sold either immediately (during normal operations) or after a known delay (as expiration approaches), and

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  Incentives: each participant has an incentive to participate.

This paper’s contribution is the design and analysis of a novel DEX for trading assets that expire. Our analysis points out ways in which conventional AMM design fails to address issues raised by expiring tokens, and we propose the following mechanisms to augment conventional AMMs to handle expiring assets.

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  LPs can respond to unforeseen fluctuations in token supply and demand – in particular, in cases resulting in arbitrage failure – by directly intervening to change the shape of the AMM’s curve.

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  LPs can limit its exposure to demand failure by setting a token price below which their cash cannot be used to buy tokens (similar to Uniswap V3’s concentrated liquidity).

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  The AMM is augmented with a pair of order books to ensure market clearing, even in situations of supply and demand failure. For example, just before the tokens expire, any tokens remaining in the DEX are auctioned off, prioritizing LPs in order of cumulative participation.

LPs are assumed to be rational. They may disagree on market predictions, but they do not act maliciously against one another.

Today, the most popular automated market maker is Uniswap [1, 4, 13, 19], a family of constant-product AMMs. Originally trading between ERC-20 tokens and ether cryptocurrency, later versions added direct trading between pairs of ERC-20 tokens, and allowed liquidity providers to restrict the range of prices in which their assets participate. Bancor [14] AMMs permit more flexible pricing schemes, and later versions [6] include integration with external “price oracles” to keep prices in line with market conditions. Balancer [15] AMMs trade across more than two assets, based on a constant mean formula that generalizes constant product. Curve [11] uses a custom curve specialized for trading stablecoins, maintaining low slippage and divergence loss as long as the stablecoins trade at near-parity. The formal model for AMMs used here is adapted from Engel and Herlihy [12].

Xu et al. [18] and Bartoletti et al. [7] provide informative overviews on AMM protocols. Angeris and Chitra [2] introduce a constant function market maker model and focus on conditions that ensure that agents who interact with AMMs correctly report asset prices.

Aoyagi [5] analyzes strategies for constant-product AMM liquidity providers in the presence of “noise” trading, which is not intended to move prices, and “informed” trading, intended to move the AMM to the stable point for a new and more accurate valuation. Angeris et al. [3] propose an economic model relating how the curvature of the AMM’s function affects LP profitability in the presence of informed and noise traders. Bichuch and Feinstein [8] propose a general mathematical framework for AMMs, and Capponi et al. [9] analyze AMMs using a game theoretic model.

Our mechanism where liquidity providers can freeze their liquidity under adverse market conditions is similar to Uniswap V3’s [13] notion of concentrated liquidity, though the purpose and operational details are somewhat different.

Ramseyer et al. [17] describe several ways to integrate constant-function AMMs with batch auctions. Like the exchange described in this paper, their exchanges have hybrid structures, but their goals and techniques are substantially different, as they are concerned with broad economic properties, not with expiring assets.

Our DEX is composed of an AMM and two order books that operate under specific rules. We give now an operational overview of our DEX, starting with the AMM component. We point the reader to the appropriate sections containing detailed discussion as we present the concepts below.

An AMM owns and trades two kinds of assets: a stable asset $X$ (informally called cash), and an expiring asset $Y$ (informally called tokens), These assets are loaned to the AMM by liquidity providers (LPs). As explained later, LPs can be active or inactive.

The AMM tracks a pair $(x,y)\in\mathbb{R}_{>0}^{2}$, where $x$ (respectively $y$) is the amount of $X$ (respectively $Y$) that the AMM offers for trading.

Any party $\ell$ can become a liquidity provider by lending assets to the AMM. In return, $\ell$ receives a share $s_{\ell}\in(0,1]$ proportional to its contribution. In departure from “pure” AMM designs, an LP can dynamically influence AMM prices to a degree proportional to its share. The particular protocol and curve design is discussed in Section 5.

LPs can withdraw liquidity at well-defined times, called epoch boundaries. LPs can make withdrawals denominated in both types of assets, or in only the stable asset $X$. LPs can freeze their liquidity, in a manner comparable to Uniswap V3’s notion of concentrated liquidity [13]. An LP $\ell$ can set a minimum spot price for $Y$ tokens in terms of $X$ cash. If the AMM’s spot price for tokens falls below that LP’s minimum, that LP becomes inactive and its liquidity is not used in trades. Of course, LPs do not accrue fees while they are inactive. An LP can use freezing to protect itself if it fears that demand has collapsed, causing producers to flood the AMM with $Y$ tokens no one will buy. If the price later rises above an LP’s minimum, that LP becomes active again. Details are discussed in Section 6.

To ensure market clearing, there are complementary buyer and seller order books that are primarily used in the following situations: just before the tokens expire, when there are not enough tokens in the system to execute any more cash-for-tokens trades (demand is too high), or when all LPs are inactive, so no more AMM trades can be executed temporarily. Details can be found in Section 7. If a consumer registers a below-spot-market bid (on $Y$ tokens) on the buyer order book and is still present near the expiration time, that bid is executed with a second price auction. There is also a seller order book, (almost) symmetric to the buyer order book, where sellers offer to sell $Y$ tokens that remain after the AMM token pool is exhausted. The combination of the AMM and the two order books acts like an AMM while normal supply-and-demand laws hold, but acts more like an order book (or auction) near the expiration date when those laws may fail.

We also include a mechanism to incentivize LP participation, so that the AMM (instant-gratification) component of the DEX remains live for a longer period of time. Of course, LPs can withdraw or freeze liquidity. However, if an LP $\ell$ chooses to take on more risk by participating more, they will be given priority in clearing their own tokens, and with a higher price, in the described market clearing auction. Thus, even if two LPs $\ell_{1},\ell_{2}$ have the same share and hold the same number of tokens, if $\ell_{1}$ participates in many more (and larger) trades than $\ell_{2}$, then $\ell_{1}$ will receive a greater share of the auction proceeds than $\ell_{2}$. We discuss these details in Section 7.

We now prepare to describe the system in proper detail, first quickly discussing preliminary notation and concepts that are necessary for a formal system presentation in the sections that follow.

For vectors $x=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$ and $y=(y_{1},\dots,y_{n})\in\mathbb{R}^{n}$, we write $x\leq y$ provided $x_{i}\leq y_{i}$ for all $i\in[n]$, where $[n]=\{1,\dots,n\}$; if also $x\neq y$, we write $x\lneq y$. When referencing topological properties in this paper, we assume that any subset of $\mathbb{R}^{n}$ (for any $n\geq 1$) is given the subspace topology, unless otherwise specified. We define $\mathbb{R}_{>0}=\{x\in\mathbb{R}:x>0\}$. A function $A:\mathbb{R}^{n}\to\mathbb{R}$ is twice-differentiable if all of its second partial derivatives exist and are continuous. We also say that $A$ is strictly increasing in each coordinate if for all $x,y\in\mathbb{R}^{n}$ such that $x\lneq y$, we have $A(x)<A(y)$.

A set $\Omega\subseteq\mathbb{R}^{n}$ is convex if for all distinct $x,y\in\Omega$ and $t\in(0,1)$, we have $tx+(1-t)y\in\Omega$; if every such vector $tx+(1-t)y$ is in the interior of $\Omega$, we say $\Omega$ is strictly convex. It follows that that any convex open set is strictly convex. Given a set $S\subseteq\mathbb{R}^{n}$, the convex hull of $S$, denoted $\operatorname{conv}S$, is defined as the intersection of all nonempty convex sets $\Omega\subseteq\mathbb{R}^{n}$ that contain $S$. Suppose $\Omega\subseteq\mathbb{R}^{n}$ is convex. A function $f:\Omega\to\mathbb{R}$ is said to be strictly convex provided that for every distinct $x,y\in\Omega$ and $t\in(0,1)$, it follows that $f(tx+(1-t)y)<tf(x)+(1-t)f(y)$.

As discussed in Section 3, our DEX has an AMM meant to provide an instant buying price for consumers. In a conventional constant-function AMM, token prices are kept consistent with the market by external arbitrageurs; if the token price rises above the market price, an arbitrageur may sell overpriced tokens to the AMM, making a risk-free profit. Conversely, if the AMM price falls below the market price, an arbitrageur may buy the bargain tokens from the AMM and resell them on the market. Arbitrage works well for a conventional AMM because there are sufficiently many tokens available to be traded, so that arbitrageurs are always able to trade in the direction needed to bring the AMM price back in line with the market. But close to expiration date, token supply or demand may vanish, and the normal price regulation cannot take place. For example, airlines will not schedule new seats to Paris as the flight time becomes too close. Similarly it makes little sense to book a room in the Caribbean starting in one hour if the room is five hours away. In addition, airline tickets, hotel rooms, event seats, and similar expiring assets have prices that may be influenced by hard-to-predict, exogenous events, such as weather, geopolitical events, etc.

In conventional AMMs, LPs can deposit or withdraw liquidity, but are otherwise passive, allowing producers and consumers to set asset prices via trading. As noted, however, arbitrageurs may become unable to intervene to keep prices close to market levels as expiration approaches. So we propose a mechanism to allow LPs to intervene actively to adjust the curve when conventional methods are in danger of failing.

More specifically, in the period while expiration is far off, when the DEX acts like a conventional AMM, arbitrageur profits come at the expense of LPs, a cost known as divergence loss. The LPs effectively gamble that asset prices will be stable enough that the fees collected will outweigh any divergence loss. As expiration approaches, however, prices may become volatile and arbitrage may become ineffective, so LPs require the ability protect their investments by intervening directly to adjust prices (also to actively protect their capital – see Section 6), effectively assuming the role of conventional arbitrageurs, but without divergence loss.

The duration between when the DEX is started and when the token expires is divided into disjoint intervals called epochs, separated by epoch boundaries. There are two special epoch boundaries: $t_{e}$, when the tokens expire, and $t_{r}<t_{e}$, when the AMM halts and the last-minute auctions occur. Let $T$ be the set of all epoch boundaries strictly before $t_{r}$.

We say that a state curve is a strictly decreasing homeomorphism $f:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$. If $x>0$, we say that the pair $(x,f(x))$ is a state of $f$, and we denote the set of all states of $f$ by $\operatorname{state}(f)$. Operationally, we say that the AMM is in state $(x,y)\in\operatorname{state}(f)$ if it contains $x$ units of $X$ and $y$ units of $Y$. Trades change the amounts $x$ and $y$ of $X$ and $Y$, respectively, so that $(x,y)$ is always in $\operatorname{state}(f)$.

We consider only a fixed subfamily of possible state curves. Let $0<a<b$ be fixed positive lower and upper bounds. (We may wish to require that $a<1$ and $b=\frac{1}{a}$ for symmetry, though this is not required.) For every $c\in[a,b]$ and possible state $(x_{0},y_{0})\in\mathbb{R}_{>0}^{2}$, define the state curve

for all $x>0$. Then $(x_{0},y_{0})\in\operatorname{state}(f_{c,x_{0},y_{0}})$. Let $L$ be the current set of active liquidity providers (whose assets are allocated to trading), and $L^{\prime}$ is the set of inactive liquidity providers (whose assets are frozen). The sets $L$ and $L^{\prime}$ and their associated methods are formally defined in Section 6.

At each time in $T$, each liquidity provider $\ell\in L\cup L^{\prime}$ chooses some $c_{\ell}\in[a,b]$ that informally represents the price $\ell$ believes is most economically efficient, with knowledge of the current state of the AMM. Each $c_{\ell}\in[a,b]$ is aggregated into a single $c\in[a,b]$ with some deterministic aggregation algorithm $\mathtt{aggregate}()$ by computing $c\leftarrow\mathtt{aggregate}()$ at each epoch boundary. The $\mathtt{aggregate}()$ algorithm has parameters $(c_{\ell})_{\ell\in L}$ and $(s_{\ell})_{\ell\in L}$²²2We exclude the $c_{\ell}$’s and shares of inactive liquidity providers because their share of liquidity is not well-defined, as it is not necessarily the same over both of the assets. See Section 6 for details. and returns some element of $[a,b]$. For fixed $L$, since $\mathtt{aggregate}()$ is deterministic, it can be viewed as a map

where $A^{B}$ is the set of all tuples indexed by $B$ with values in $A$ for any sets $A,B$. Now we define the soundness properties for an aggregation algorithm.

For brevity, let $c=\mathtt{aggregate}((c_{\ell})_{\ell\in L},(s_{\ell})_{\ell\in L})$.

The first condition implies that for a small fixed change in $\log c_{\ell}$ from an LP $\ell$ with share $s_{\ell}$, the corresponding change in $c$ is proportional to $s_{\ell}$. We use a logarithmic scale since derivatives are limits of additive changes, and $\log$ is an isomorphism from the group $\mathbb{R}_{>0}$ under multiplication to the group $\mathbb{R}$ under addition. (Note that $c_{\ell}\in\mathbb{R}_{>0}$, so it only makes sense to give them a multiplicative structure.)

The second condition is a simple boundary condition to guarantee that if all LPs are in consensus with a particular value $c$, the aggregated constant is the common value $c$.

In our model, we define $\mathtt{aggregate}((c_{\ell})_{\ell\in L},(s_{\ell})_{\ell\in L})$ to return

the geometric mean of each $c_{\ell}$ weighted by the share $s_{\ell}\in[0,1]$ each liquidity provider holds in the system. Interestingly, we show in the Appendix that this aggregation algorithm is the unique valid aggregation algorithm.

The state curve is implied from the global variables $c,x_{0},y_{0}$ stored in the AMM. Until the next time in $T$ in which liquidity providers may update their value of $c_{\ell}$, we fix $c$ as defined above. Note that when the state $(x,y)$ changes from adding/removing liquidity, freezing/unfreezing liquidity, and liquidating (specified in Section 6), we update $x_{0}\leftarrow x$ and $y_{0}\leftarrow y$, and we must also update the state curve in these cases. This is necessary because the liquidity state $(x,y)$ after completing any of these methods is no longer in the state space of the current state curve $f_{c,x_{0},y_{0}}$. We omit updates to $(x_{0},y_{0})$ in any pseudocode of these methods for simplicity of presentation.

With expiring assets (and exogenous events that affect price), it is important to avoid that LPs have their cash ($X$) drained upon a sudden decrease in demand for tokens ($Y$). In this section, we discuss liquidity freezing, a tool allowing an LP to alter how much of its liquidity the AMM can use for trading under current market conditions.

LPs add liquidity in a way almost identical to AMMs such as Uniswap [1]. If $(x,y)$ is the current liquidity state of the AMM, then any party $\ell$ may deposit $x_{\ell}>0$ of $X$ and $y_{\ell}>0$ of $Y$ in a proportion that reflects the current price: $\frac{x_{\ell}}{y_{\ell}}=\frac{x}{y}$.³³3In an initial state where $x=0$ or $y=0$, there are no restrictions on how liquidity is added. Then the state $(x,y)$ is updated appropriately, we do $x_{0}\leftarrow x$ and $y_{0}\leftarrow y$, and finally, $L\leftarrow L\cup\{\ell\}$. The depositing LP receives shares in the usual way, but in addition it is allowed to specify its contribution $c_{\ell}\in[a,b]$ in the $\mathtt{aggregate}()$ algorithm (Equation 1), changing the state curve at the next epoch boundary. However, an LP can freeze its liquidity when it considers market conditions to be adverse. The LP establishes a bound on the spot price of tokens ($Y$) in terms of cash ($X$) below which its liquidity will not participate. Each liquidity provider $\ell\in L\cup L^{\prime}$ sets a lower bound $d_{\ell}\geq 0$, updated at the beginning of each epoch (after $\mathtt{removeLiquidity}()$ and $\mathtt{liquidate}()$ defined subsequently). A liquidity provider $\ell$ can alter its $d_{\ell}$ at any time, but no change will occur until the next epoch boundary.

For any initial state $(x_{0},y_{0})$ and current state $(x,y)$ of the AMM with aggregate constant $c$, let $p$ denote the (instantaneous) spot price

as defined in Definition 5.4. This value is stored in the AMM and is updated appropriately. As soon as $p<d_{\ell}$ for some $\ell$, the AMM moves $\ell$’s liquidity into a distinct, untraded liquidity pool. If, at a later time, $p\geq d_{\ell}$, the AMM unfreezes $\ell$’s liquidity by moving it back into the trading state. To ensure no consumer or producer pushes $p$ to be much smaller than some $d_{\ell}$, trades are restricted to one unit increments. The assumption is primarily made for simplification of the proof of Proposition 6.3. For completeness, we write the exact trading algorithm for multiple tokens $Y$ and the corresponding system state changes in pseudocode below. A more efficient implementation is a discretized version of Uniswap V3’s concentrated liquidity [13]. We delay discussion of Lines 1-1 and  1-1 to Section 7.

We showed in Proposition 5.7 that the price $p$ is invariant under adding and removing liquidity, so we update $p$ only on epoch boundaries and after every trade. The AMM maintains the set $\{d_{\ell}\}_{\ell\in L\cup L^{\prime}}$ and a map $\mathtt{fr}:L^{\prime}\to\mathbb{R}^{2}$ to keep track of the liquidity that inactive providers have the right to. Immediately after every epoch boundary and every trade, we execute the method $\mathtt{freeze\_unfreeze}()$ defined in Algorithm 2.

The $\mathtt{freeze}()$ function finds all active liquidity providers $\ell$ whose price tolerance $d_{\ell}$ is less than the current price, and moves their liquidity into a non-trading pool. If $L\neq\emptyset$, $\mathtt{unfreeze}()$ finds the inactive liquidity providers $\ell$ whose price tolerance $d_{\ell}$ is at least $p$, and moves as much of their inactive liquidity back into the trading pool as possible, subject to preserving the ratio of $x$ to $y$ in the liquidity pool. If, however, $L=\emptyset$, the instantaneous price $p$ is not well-defined, so we retrieve the values of $x$ and $y$ from the last point in time where some liquidity provider was active, and unfreeze all the liquidity providers to have a liquidity ratio of $x/y$. In both cases $L=\emptyset$ or $L\neq\emptyset$, the $\mathtt{freeze\_unfreeze}()$ method freezes or unfreezes liquidity providers as needed.

Finally, we define $\mathtt{removeLiquidity}()$ and $\mathtt{liquidate}()$. These methods are similar, but with one key difference. When $\mathtt{removeLiquidity}()$ is called by some $\ell\in L\cup L^{\prime}$, the AMM adds $\ell$ to a set $L_{\mathtt{remove}}$ initialized to $\emptyset$; at the next epoch boundary, the AMM simply sends the assets that each $\ell\in L_{\mathtt{remove}}$ has the right to (computed in the obvious way, depending on if $\ell\in L$ or $\ell\in L^{\prime}$) back to $\ell$, while doing the appropriate bookkeeping. As this is the only difference from usual practice (e.g., [1]), so pseudocode is omitted.

Like $\mathtt{removeLiquidity}()$, when $\mathtt{liquidate}()$ is called by an LP $\ell\in L\cup L^{\prime}$, then $\ell$ is added to a set $L_{\mathtt{liquidate}}$. At each epoch boundary, the following is executed in an atomic step: for each liquidity provider in $L_{\mathtt{liquidate}}$, get the amounts of $X$ and $Y$ they have the right to, remove this liquidity, sell the total units of $Y$ that have been traded for $X$, and distribute these funds to each member of $L_{\mathtt{liquidate}}$ in proportion to their shares of asset $X$; then set $L_{\mathtt{liquidate}}\leftarrow\emptyset$. This protocol is listed as Algorithm 6. Both these methods run atomically at each epoch boundary, where $\mathtt{removeLiquidity}()$ is executed first.