A General Framework for Impermanent Loss in Automated Market Makers

Impermanent loss for protocols like Uniswap ([Ada18]) and Balancer ([MM19]) have been well studied ([Eva20], [AD21], [AEC20], [Aoy20], [Bou21]). These geometric mean market maker protocols (G3Ms) can give the false impression that certain non-trivial properties regarding impermanent loss should hold for all Automated Market Makers (AMMs). For instance, Figure 1 implicitly leverages the idea that impermanent loss is a one free parameter function in the case of two dimensional G3Ms. Not by design by true nevertheless, Curve’s StableSwap AMM protocol ([Ego19]) does not allow impermanent loss to be analyzed with one free parameter (so Curve could not be compared in Figure 1). At a high level, the goal of this paper is to provide a useful way to analyze impermanent loss as protocols continue to increase in complexity.

Along these lines, we start by observing and codifying some of the properties of G3Ms. The major property that we focus on in this paper is what we call Exchange Rate Level Independence (ERLI). This property states that impermanent loss can be understood entirely in terms of ratios of initial and final exchange rates, as opposed to the actual rates themselves. In other words, moving from an Eth $:$ BTC exchange rate of $15:1$ to an exchange rate of $30:1$ has the same effect on impermanent loss as moving from an exchange rate of $10:1$ to an exchange rate of $20:1$, since in both scenarios the exchange rate has doubled. G3Ms exhibit this property, though to our knowledge, it has not been clearly formalized for higher dimensional AMMs until now. Among other conceptual benefits, exchange rate level independence allows us to analyze impermanent loss for AMMs with fewer parameters.

Showing that G3Ms (up to a simple transformation) have the ERLI property and are the only AMMs to have this property is the main result of this paper. The connection between portfolio value functions and trading functions pointed out in [AEC21] is a crucial first step towards establishing this result. However, a slightly different viewpoint is used in this paper. We focus on individual level surfaces of an AMM as opposed to the function $A$ that generates them. The following example might motivate this shift in focus. Consider

Readers familiar with the space can skip to Section $3$, in which we derive an impermanent loss formula for higher dimensional constant product market makers. In the process of doing so, we note that these constant product market makers have some helpful properties that we work to concretize in the later sections. We then show that the smallest class of market makers with these properties is the class of market makers whose level surfaces match those of a geometric mean market maker.

In what follows, we will use a definition for an automated market maker (AMM) motivated by [EH21].

A price vector $p=(p_{1},...,p_{n})$ assigns a price to each token so that $p_{i}$ is the price of token $i$ in terms of some measuring currency. A price vector can be thought of in state space as the gradient vector (it points in the direction of steepest increase) of the value function

This value function is naturally defined in that it is the sum of the product of each token quantity by the corresponding token price. If we think of $V$ as a function on the state space (given a fixed price vector $p$), we see that $V$ has a constant gradient $p$. We see in Figure 3 the relationship between the price vector $p$ and the level curves of the value function $V$ that it induces.

In Figure 3, the state $(2,6)$ is referred to as a stable point, in that it is the state with the lowest value $V$ on the current trading curve $A=12$. We will define stable points more formally below. Note that the measuring currency is not important in the context of defining stable points. Changing the measuring currency through which $p$ is defined would stretch or contract $p$, but it would not change its direction. Also, changing the measuring currency would leave the family of level curves (surfaces) intact, though it would alter the $V$ values corresponding to these level curves (surfaces). If, for instance, the measuring currency were taken to be the first token, $p_{1}$ would be $1$ and $p_{2}$ would be $\frac{2}{6}$, which would result in the same level curves for the value function as shown in Figure 3, but with the corresponding values divided by $6$.

Let’s make a fixed choice of price $p\in\mathbb{R}_{>0}^{n}$, AMM $A:\mathbb{R}_{>0}^{n}\rightarrow\mathbb{R}$, and number $k\in\mathbb{R}$. We say that a point $(x_{1}^{o},...,x_{n}^{o})\in\{(x_{1},...,x_{n})\in\mathbb{R}_{>0}^{n}:A(x_{1},...,x_{n})=k\}$ is stable if

In what follows, we assume that if the price vector changes (one can think of it as fluctuating exogenously), the state vector in the AMM changes to the corresponding stable point (thanks to arbitrageurs). In [EH21], it is shown that for a given price vector $p$, there is a unique stable point $x$ on the $A=k$ surface. In other words, the stable state is a function of price

As is well known, the above formula demonstrates that impermanent loss is least destructive when $t=1$, that is when the exchange rate between tokens does not change (here $IL=0$). The key takeaway that we will try to generalize in what follows is that impermanent loss is best understood in terms of this one parameter $t$, as opposed to the four parameters $p_{1}^{i}$, $p_{2}^{i}$, $p_{1}^{f}$ and $p_{2}^{f}$.

To gain an intuition for the parameters that should be used to conceptualize impermanent loss in higher dimensions, we start by proving a theorem involving an $n$ dimensional constant product market maker.

Similarly to Example 3, $t_{j}$ is the quotient of the final exchange rate and initial exchange rate between assets $j$ and $1$. The fact that the $IL\leq 0$ for constant product market makers is a direct consequence of the arithmetic mean-geometric mean inequality applied to these exchange rate quotients. Again, the key takeaway is that impermanent loss is best understood in terms of these $n-1$ $t_{j}$ parameters, not the $2n$ price parameters. This result has important economic implications in addition to its overall simplifying nature.

The parameter reduction in the impermanent loss formula from $n$ initial prices and $n$ final prices to $n-1$ quotients of exchange rates can be decomposed into two parts. The first reduction consists of using relative pricing of one asset against another as opposed to the raw prices (in our case, we have been pricing assets in terms of the first asset). This is typically referred to as numeraire independence in finance, but we will refer to it as price level independence. The next reduction takes us from separate relative initial and final exchange rates to quotients of final and initial exchange rates. We call this exchange rate level independence, since the absolute levels of the initial and final exchange rates do not matter, only the quotients. In the next section, we will show that impermanent loss for all AMMs exhibits price level independence, but not necessarily exchange rate level independence.

The goal of this section is to generalize the above idea of reducing the number of parameters involved in defining impermanent loss. We will begin by showing that impermanent loss for all AMMs is price level independent. Price level independence is intuitive and not particularly deep in its own right, but we wish to build towards understanding exchange rate level independence systematically. To this end, we need the following lemma, which formalizes the concept introduced in the background section that rescaling price vectors does not affect stable points. In this section, we think of there being a fixed measuring currency through which $p$ and hence $V(x)$ are defined. In other words, $2p$ corresponds to a world in which all prices have doubled, not one in which the measuring currency has changed.

Price level independence for the impermanent loss of an AMM follows immediately.

Indeed, it is clear from the proof that we could express impermanent loss using exchange rates relative to any fixed asset. Along these lines, it will be convenient at times to change the base token from the first token to another of the $n$ tokens. In other words, we might define

so that we use the $k^{\text{th}}$ token as the base token when computing exchange rates. In this case, $m_{k}=1$. To keep the notation uncluttered, though, we will suppress the $k$ and make sure that it is clear from context when important.

Another way of interpreting Theorem 6 is that impermanent loss is agnostic to the measuring currency used for both $p^{i}$ and $p^{f}$. Indeed, we could use a different measuring currency for $p^{i}$ and $p^{f}$ without affecting impermanent loss, though we will not do this. The above figure offers some insight into how we can choose the measuring currency so that $V(x)=p\cdot x$ can be gleaned geometrically. More concretely, if the $x_{2}$ token is the measuring currency, the pool value is the $x_{2}$ intercept¹¹1We could use any of the tokens as the measuring currency, and then the pool value would be the corresponding intercept value. of the line $L$ with the following properties:

• •

  $L$ passes through the current pool state

• •

  $L$ is perpendicular to $p$

The slope of $L$ is the negative of the exchange rate between the two tokens. This line $L$ will be tangent to the $A=k$ curve if the pool state is the current stable state. In the above example, if the individual had just held their assets, the value of their assets would have been equivalent to the value of $V_{\text{Hold}}$ $x_{2}$ tokens. By providing liquidity, the value of their assets is instead $V_{\text{Pool}}$ $x_{2}$ tokens. This diagram makes it clear why the convexity condition for AMMs ensures that impermanent loss will always be negative.

As introduced in [AEC21], there is an elegant mathematical relationship between automated market makers and their corresponding portfolio value functions. In that paper, the relationship is formulated between the AMM $A(x)$ and its value function $V(x)$ via the Legendre-Fenchel transform. Example 7 motivates us to proceed a bit differently. The core object that we will work with instead of $A(x)$ is $f(x_{1},x_{2},\ldots x_{n-1})$, which we will define as the function that defines the specific level curve (or surface) $A(x)=k$:

By our AMM assumptions and the implicit function theorem, $x_{n}=f(x_{1},x_{2},\ldots,x_{n-1})$ is a continuously differentiable surface. Henceforth, we will use the $x_{n}$ token as the measuring currency through which $p$ and hence $V(x)$ are defined. In other words, $p_{n}=1$ and $p_{i}$ is the exchange rate between token $i$ and token $n$ for $1\leq i\leq n-1$. To maintain consistency with the previous section, we will use $m_{i}:=p_{i}$ to reference these exchange rates. Define $\bar{V}$ as

In other words, given the exchange rates $m_{1},m_{2},\ldots,m_{n-1}$, a unique stable point $x$ is determined, and $W$ is the value of the corresponding stable portfolio. We will use $x_{i}(m_{1},\ldots,m_{n-1},1)$ interchangeably with $x_{i}(m_{1},\ldots,m_{n-1})$, mainly to simplify notation and to avoid introducing new notation. From Example 7, we see that $-m_{1}=f^{\prime}(x_{1}(m_{1}))$. In other words, at stable points, we can obtain the exchange rate as the negative derivative of $f$ ²²2In Example 7, $x^{f}$ is the stable point at the final exchange rate $m_{1}^{f}$, while $x^{i}$ is not a stable point.. The following lemma proves and generalizes this observation to cases in which $x$ is a function of $n-1$ exchange rates.

The following theorem gives a precise formulation of how $A(x_{1},\ldots,x_{n-1})$ and $W(m_{1},\ldots,m_{n-1})$ are dual functions. For readers unfamiliar with the Legendre transform, the remainder of the paper is self-contained: all properties that we need are readily derived from the formulas that we have established. As such, the following theorem is just a form of mental bookkeeping. We recommend [ZRM09] as a useful resource to learn more about the Legendre transform and its applications.

Again, the relationship between $A$ and $V$ via the Legendre-Fenchel transform was pointed out in [AEC21], and this observation accounts for the heavy lifting. It can be cumbersome in some settings, however, to think in terms of $A$ (a family of level curves or surfaces) as opposed to the single AMM surface given by $x_{n}=f(x_{1},\ldots,x_{n-1})$. For instance, to replicate a payoff function $W(x_{1},\ldots,x_{n-1})$, one can extend $W$ to $V$ via 1-homogeneity, transform $V$ via the Legendre-Fenchel transform into an $\tilde{A}$, and then manipulate $\tilde{A}$ to obtain a more amenable AMM form $A$. Alternatively, one can use Theorem 10 to pass from $W$ to $f$ directly. The graphs in Figure 5 illustrate the relationship between $f$ and $W$ in two dimensions.

In general, price level independence of impermanent loss allows us to reduce the number of parameters from $2n$ to $2(n-1)$. When we derived the impermanent loss formula for the $n$-dimensional constant product market maker, however, we were able to reduce the number of relevant parameters even further to $n-1$. These $n-1$ parameters were quotients of the initial and final exchange rates (relative to a fixed token).

Our first goal will be to write impermanent loss in terms of the value function $W$ defined in the previous section. To this end, we need the following lemma. We warn the reader of the notational simplification

We are now ready to recast impermanent loss in terms of $W$.

We will finish the section by providing a necessary and sufficient condition on $A$ for its impermanent loss to satisfy the ERLI condition. To this end, we need a lemma that connects the homogeneity properties of $f$ to the homogeneity properties of $W$.

1. 1.

   For each $1\leq j\leq n-1$, $f(x_{1},\ldots,x_{n-1})$ is homogeneous of degree $\alpha_{j}$ in $x_{j}$.

2. 2.

   For each $1\leq j\leq n-1$, $W(m_{1},\ldots,m_{n-1})$ is homogeneous of degree $\beta_{j}$ in $m_{j}$.

Furthermore,