Inefficiency of CFMs: hedging perspective and agent-based simulations

The CFM are gaining popularity in market design as they guarantee market liquidity (proportional to the amount of assets locked) and they have minimal storage and computational requirements, unlike e.g. limit-order-book-based markets. This makes them ideal for environments where computational power and storage are at a premium like Ethereum [20], or Cardano, Polkadot or Solana.

Running on blockchain gives advantages over centralised exchanges: transparency, pseudonymity, censorship resistance and security, see references in e.g. [16]. However blockchain-based CFMs suffer from a number of problems: namely questionable returns for LPs as has been already mentioned as well as traders (liquidity takers) being front-run by miners (known as miner-extractable-value) [8], [10].

Classical market clearing models suggest, see Glosten & Milgrom [11], Kyle [13, 14] and Grossman & Miller [12], that the volatility of the underlying asset plays an important role in how the market makers should set bid-ask spread (which is comparable to the pool fee as it imacts execution costs). As we show in Theorem 2.1 the fee income due to the CFM LPs is bounded above by a product of quadratic variation of the underlying asset (which is basically volatility) and convexity of the position being hedged.

Section 2 includes all the theoretical findings of the paper. Some readers may wish to skim through Section 2.1 which reviews CFMs and how liquidity provision works and fixes some notation used further in the paper. Section 2.2 recalls what a perpetual option is, Section 2.3 proves the main theoretical result of the paper which is Theorem 2.1. Section 3 describes in detail the agent-based model used and discusses the findings. Finally, in Section 4, we discuss the main findings, their limitations and possible further work. A full implementation of the agent-based simulation is at¹¹1The link is deliberately broken to comply with the anonymous submission format. https://github.com/simtopia/cfm-sim.

A constant function market (CFM) is characterised by

1. i)

   The reserves $(x^{1},x^{2})\in\mathbb{R}^{2}_{+}$ describing amounts of assets in the pool.

2. ii)

   A trading function $\Psi:\mathbb{R}_{+}^{2}\rightarrow\mathbb{R}$ which determines the state of the pool after each trade according to the acceptable fund positions:

   $$\left\{(x^{1},x^{2})\in\mathbb{R}^{2}_{+}\,:\,\Psi(x^{1},x^{2})=\text{constant}\right\}.$$

   (1)

3. iii)

   A trading fee $(1-\gamma)$, for $\gamma\in(0,1]$.

For the purposes of this paper, we assume $\Psi:\mathbb{R}_{+}^{2}\rightarrow\mathbb{R}$ to be twice continuously differentiable and convex (see [6], [5]).

To acquire $\Delta x^{1}_{t}$ of asset $x^{1}$ at time $t$ a trader needs to deposit a quantity $\Delta x^{2}=\Delta x^{2}(\Delta x^{1})$ of asset $x^{2}$ into the pool, and pay a fee $(1-\gamma)\Delta x^{2}$.²²2The fee in Uniswap-V3 is not added to the pool reserves [1]. This is in contrast to Uniswap-V2. $\Delta x^{2}$ and $\Delta x$ needs to satisfy the equation

Once the trade is accepted the reserves are updated according to

The relative price of trading $\Delta x^{1}$ for $\Delta x^{2}$ is defined as

Observe that

Hence the relative price of trading an infinitesimal amount of $\Delta x^{1}$ for $\Delta x^{2}$ is given by

Assume that there is an external market where assets $x^{1}$ and $x^{2}$ can be traded (without frictions) at the prices $S_{t}=(S_{t}^{1},S_{t}^{2})$. The no-arbitrage condition in the case of no fees ($\gamma=1)$ implies that

Conversely, if 7 would not hold, then (in a market with no fees) there would be an arbitrage opportunity between CFM and the external market (assuming frictionless trading is possible), as it would be possible to purchase a combination of assets cheaply in one market, and then sell it in the other.

The above example, also studied in [9], demonstrates two key properties of CFMs:

• •

  A GMM automatically re-balances liquidity pools so that the value of the pools with asset $x^{1}$ and $x^{2}$ is $\theta\cdot\psi(S_{t})$ and $(1-\theta)\psi(S_{t})$, respectively

• •

  Providing liquidity to a CFM $\Psi$ is equivalent to entering a long position on a perpetual derivative on the underlying asset with the payoff dictated by the value (in terms of price) $\psi$.

As observed in [3], [4], for any CFM $\psi$ for any non-negative, non-decreasing, concave, 1-homogenous³³3That is, $\lambda\psi(S)=\psi(\lambda S)$ for all $\lambda>0$ and $S$. Equivalently, we can assume that one of the assets in $S_{t}$ is the numeraire asset, in which case the 1-homogeneity assumption is not needed. payoff function $\psi$, there exists a trading function $\Psi$, such that the value of the liquidity provision in CFM with with function $\Psi$ matches this payoff function. In other words, a CFM with appropriately designed trading function $\Psi$ dynamically adjusts portfolio held by liquidity providers so that the value of this portfolio, at any time, is described by the payoff function $\psi$. This gives us two ways to understand a CFM:

• •

  Through the defining trading function $\Psi$, indicating valid combinations of the two assets.

• •

  Through the valuation function $\psi$, indicating the value of the CFM pool in terms of the prices of the two assets.

Concentrated liquidity, introduced in Uniswap V3, gives individual LPs control over what price ranges their capital is allocated to. This gives a practical way of creating liquidity provision that replicates a given function $\psi$, and has been observed and described in [15].

Given a payoff function $\psi$, there is a simple connection between liquidity provision in a CFM and investment in a perpetual American option.

We begin with a precise definition of a perpetual American contract with streaming premium. As we shall see, liquidity provision in a CFM is equivalent to entering a long perpetual derivative with the payoff dictated by the function $\psi$. This derivative is written on the underlying assets $S=(S_{t})_{t\geq 0}$ traded in the pools. We follow the definition from [2].

In order to highlight the connections between a CFM and a perpetual American option, we make the following observations. In a CFM, a liquidity provider (CFM-LP)

• •

  initially deposits assets with value $\psi(S_{0})$,

• •

  receives fees at the rate $f_{t}$ per unit time (which may vary). These fees may be withdrawn (under the Uniswap v3 protocol) at any time,

• •

  at some future time $\tau$ (of the CFM-LPs choosing), may withdraw their assets, which will have value $\psi(S_{\tau})$.

In a perpetual American option with streaming premium and payoff $\psi$, an investor purchasing the option

• •

  initially purchases the option, for a cost $\psi(S_{0})$,

• •

  receives the streaming premium at a rate $g_{t}$ per unit time (which may vary). This streaming premium may be positive or negative, and is received immediately,

• •

  at some future time $\tau$ (of the investor’s choosing), may exercise the option, which rewards them with value $\psi(S_{\tau})$.

As we can see, assuming fees in the CFM are instantaneously predictable (i.e. the fee to be received from the CFM can be accurately estimated in advance), a no arbitrage argument suggests that we must have $f_{t}=g_{t}$, as otherwise one asset is strictly better than the other in every state of the world (over some short time horizon), which implies that an efficient market will focus all its trading in the better of these alternatives.

We make the following assumptions about the market:

• •

  The agent can borrow and lend any amount of cash at the riskless rate.

• •

  The agent can buy and sell any amount of assets $x^{1}$ and $x^{2}$.

• •

  The above transactions do not incur any transaction costs and the size of a trade does not impact the prices of the traded assets.

We denote the risk free rate (which may be stochastic) by $(r_{t})_{t\geq 0}$ and model the money market as

We assume $r_{t}\geq 0$. We further denote the drift and diffusion coefficients (again, possibly stochastic) of the risky asset shadow prices by $\mu=(\mu_{t}^{1},\mu_{t}^{2})\in\mathbb{R}^{2}$ and

respectively, and model the shadow price of the risky asset $(S^{i}_{t})_{t\geq 0}$ by

where $W=(W_{t}^{1},W_{t}^{2})$ is a $2$-dimensional Brownian motion. We assume that $(\mu,\sigma)$ are sufficiently regular that a (unique strong) solution to (14) exists.

We will proceed to derive an arbitrage-free fee bound in a similar way to the construction of the predictable loss in [17], and to classical arguments for no-arbitrage pricing in financial markets.

Assumption 1 holds in the case when the fee $(1-\gamma)=0$ and arbitrageurs continuously close the gap between $S_{t}$ and the price of the CFM, as demonstrated in [17]. Hence we expect the the arbitrage-free fee derived below to be a good proxy for CFMs that charge small fee (e.g Uniswap). If the CFM charges a fee $(1-\gamma)$ which is significantly larger than $0$ then the value of the pool is a function of the reserves and the price one would use for the conversion (external price $S_{t}$ or pool-implied price $P^{2,CFM}_{t}$), see Remark 2 for more details.

Consider the wealth process $Z$ of an agent who

• •

  begins with zero capital⁴⁴4This is simply to avoid having to account for the interest they should earn on their initial capital.,

• •

  initially borrows a quantity $\psi(S_{0})$ at the risk-free interest rate, which they use to establish a CFM position (which they can do by trading in the liquid market to obtain the desired risky assets for the pool),

• •

  until time $t$, trades in the liquid market hold a quantity $(\Delta_{s}^{i})$ of each asset at time $s$.

The dynamics of $Z_{t}$ are given by

We assumed $\psi$ is smooth, hence we can apply Itô’s lemma to obtain

By setting $\Delta_{s}^{i}=-\partial_{i}\psi(S_{s})$, we eliminate the first term, giving

As the quadratic variation is $d\langle S^{i},S^{j}\rangle_{s}=(\sigma_{s}\sigma_{s}^{\top})^{(i,j)}S_{s}^{i}S_{s}^{j}ds$, this simplifies to

If this quantity is positive, then we have a strategy which earns money with probability one over a short time period, that is, an arbitrage. Therefore, after rearrangement, we know that

As $Z$ is increasing in $f$ (an agent who earns more fees will be wealthier) and we assume $r\geq 0$, there is a unique value of $f$ such that above equation is an equality. With this CFM trading fee income per unit of time, we have $dZ_{s}/ds=0$ and $Z_{0}=0$, and hence $Z_{s}=0$. We denote this critical value⁵⁵5[7] give a similar calculation, to obtain a closely related quantity, which they call the permanent loss of the CFM.

We have thus shown the following.

Notice that the critical CFM fee income rate depends on the trading function via $\partial_{i}\partial_{j}\psi(S)$ (which is negative, as $\psi$ is concave as long as the trading function $\Psi$ is convex) and the quadratic variation of traded asset (which is implicitly related to the level of trading activity). On the other hand the actual CFM fee income rate depends on the trade flow in the CFM and the fee the pool sets i.e. $1-\gamma$. It is not clear that one could bring $f_{t}$ in line with $\hat{f}_{t}$ as increasing $1-\gamma$, the fee charged by the CFM will likely lead to decreased trade flow.

Note that this argument only uses a long-position in the CFM, and only establishes⁶⁶6The presentation above assumes the agent starts at time $0$, and derives the critical fee rate on this basis. To obtain the inequality bound $f_{t}\leq\hat{f}_{t}$ for all times, we formally have to consider starting with zero capital at a time where the inequality is not satisfied, and showing that this gives a short-term arbitrage opportunity. the inequality $f_{t}\leq\hat{f}_{t}$. If it were possible to perfectly short-sell the CFM (or the corresponding perpetual option), then a similar argument would yield the converse inequality. This may explain why the fee rate in Uniswap v3, is systematically below the critical fee rate, which is related to liquidity provision in CFMs yielding persistently poor returns.

As mentioned above, by analyzing the data in the Uniswap v3 pool (e.g. [7]) one can see that, in general, the fee rate is below critical fee rate. Nevertheless, since one cannot simply short position at the Uniswap (unless using an over-the-counter bespoke arrangement) it is not clear how one could realise a potential arbitrage opportunity when $f<\hat{f}$.

We consider the CFM with $x^{1}$ being the numeraire. Price discovery occurs in a second reference market, where $S_{t}$ denotes the price of asset $x^{2}$ in terms of the numeraire $x^{1}$ at time $t$. We model the price $S_{t}$ by a Geometric Brownian Motion

Whenever the price of asset $x^{2}$ in terms of the numeraire $x^{1}$ in the CFM is different than $S_{t}$, there might be an arbitrage opportunity. An arbitrageur will automatically make the necessary trades to maximise their profit. Furthermore we model the arrival of liquidity takers by Poisson process $(N_{t})_{t}$ with intensity $\lambda$. We run the simulation of discrete times $\Pi:=\{0,t_{1},t_{2},...,T\}$ and initialise the simulation with reserves $(x^{1}_{0},x^{2}_{0})$ and fee $(1-\gamma)$; initial price $S_{0}$ and volatility $\sigma$; liquidity takers arrival intensity $\lambda$. We run the following simulation.

For every $t\in\Pi$:

• •

  The arbitrageur makes the necessary trades resulting from optimisations (18) (19) in the CFM and in the reference market to make a risk-free profit.

• •

  If $N_{t}-N_{t-}=1$ then a price-sensitive liquidity trader arrives and trades $\Delta_{t}\in\mathbb{R}$ sampled from $\mathcal{N}(0,1)$ of asset $x^{1}$. The noise trader chooses the market with better execution price with some high probability $p$, i.e. we allow for the noise trader to be irrational with probability $(1-p)$⁷⁷7Perfectly rational liquidity trader would consider splitting the order to optimise the execution cost. We omit this extension in our simulations..

We take $\sigma\in\{0.2,0.4,0.6,0.8\},\lambda\in\{50,75,100\}$ and fix $S_{0}=10,x_{0}^{1}=100,x_{0}^{2}=10$ and run the above simulation 100 times for different values of $\gamma\in[0.96,1]$.

Figures 2, 3, 4 and 5 consider two simulated scenarios: high fee value $\gamma=0.96$ ( i.e fee $4\%$); low fee value $\gamma=0.997$ of ( i.e fee $0.3\%$), $\lambda=50$ and $\sigma=0.4$. Figures 2 and 4 provide the evolution of the price process for the two simulated scenarios in the CFM for the same price process $S_{t}$ in the reference market. The smaller the fee (the higher the value of $\gamma$) the closer the CFM price process depicted in blue follows the reference market price $S_{t}$. For these two scenarios, the CFM and arbitrage-free fee rates are shown in Figures 3 and 5, and the area under the fee rates corresponds to accumulated fees in that particular simulation. High values of fees (low values of $\gamma$) indicate less arbitrage opportunities, and therefore the number of trades will be lower than in the low fees scenarios. Hence increasing the fee rate does not necessarily lead to higher fee income. However, in higher fees regimes, the trade sizes will necessarily be higher (there will only be an arbitrage opportunity when the gap between the CFM price and the reference price is big enough to compensate for the fee).

Fixing $\lambda=50$, Figure 1 aggregates the generated results for all the considered $\gamma\in[0.96,1],\sigma\in\{0.2,0.4,0.6,08\}$. Figure 1(a) indicates that higher trades will bring higher fees on average, yet they will not be able to cover the cost of hedging the impermanent loss, which increases with the volatility $\sigma$.

Figure 6 aggregates all the simulations for three considered intensity regimes for the noise trader. Again, in average the CFM fee does not cover the cost of hedging impermanent loss and, furthermore, this is not impacted by the arrival intensity of noise traders.