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Automated Market Making of Trading Pool

Abstract

We introduce a new invariant function associated with generalised mean that underpins the ALEX AMM. ALEX builds DeFi primitives targeting developers looking to build ecosystem on Bitcoin, enabled by Stacks. With a suitable parameterisation, the invariant function supports both risky pairs (i.e.
xy=Lx y=L
), stable pairs (i.e.
x+y=Lx +y=L
) and any linear combination in-between (i.e. Curve). We also show that our invariant function maps
LL
to the liquidity distribution of Uniswap V3.

Introduction

At ALEX, we build DeFi primitives targeting developers looking to build ecosystem on Bitcoin, enabled by Stacks. As such, we focus on trading, lending and borrowing of crypto assets with Bitcoin as the settlement layer and Stacks as the smart contract layer. At the core of this focus is the automated market making ("AMM") protocol, which allows users to exchange one crypto asset with another trustlessly. This paper focuses on technical aspects of AMM.

AMM and Invariant Function

ALEX AMM is built on three beliefs: (i) it is mathematically neat and reflect economic demand and supply and (ii) it is a type of mean, like other AMMs.
We will firstly review some desirable features of AMM that ALEX hopes to exhibit.

Properties of AMM

AMM protocol, which provides liquidity algorithmically, is the core engine of DeFi. In the liquidity pool, two or more assets are deposited and subsequently swapped resulting in both reserve and price movement. The protocol follows an invariant function
f(X)=Lf(X)=L
, where
X=(x1,x2,…,xd)X=\left(x_1,x_2,\dots,x_d\right)
is
dd
dimension representing
dd
assets and
LL
is constant. When
d=2d=2
, which is the common practise by a range of protocols, AMM
f(x1,x2)=Lf(x_1,x_2)=L
can be expressed as
x2=g(x1)x_2=g(x_1)
. Although it is not always true,
gg
tends to be twice differentiable and satisfies the following
  • monotonically decreasing, i.e.
    dg(x1)dx1<0\frac{dg(x_1)}{dx_1}<0
    . This is because price is often defined as
    −dg(x1)dx1-\frac{dg(x_1)}{dx_1}
    . A decreasing function ensures price to be positive.
  • convex, i.e.
    d2g(x1)dx12≥0\frac{d^2g(x_1)}{dx_1^2} \geq 0
    . This is equivalent to say that
    −dg(x1)dx1-\frac{dg(x_1)}{dx_1}
    is a non-increasing function of
    x1x_1
    . It is within the expectation of economic theory of demand and supply, as more reserve of
    x1x_1
    means declining price.
Meanwhile,
ff
can usually be interpreted as a form of mean, for example, mStable relates to arithmetic mean, where
x1+x2=Lx_1+x_2=L
(constant sum formula); one of the most popular platforms Uniswap relates to geometric mean, where
x1x2=Lx_1 x_2=L
(constant product formula); Balancer, which our collateral rebalancing pool employs, applies weighted geometric mean. Its AMM is
x1w1x2w2=Lx_1^{w_1} x_2^{w_2}=L
where
w1w_1
and
w2w_2
are fixed weights. ALEX AMM extends these to create a generalised mean.

ALEX AMM

After extensive research, we consider it possible for ALEX AMM to be connected to generalised mean defined as
(1d∑i=1dxip)1p\left( \frac{1}{d} \sum _{i=1}^{d} x_i^{p} \right)^{\frac{1}{p}}
where
0≤p≤10 \leq p \leq 1
. The expression might remind readers of
pp
-norm when
xi≥0x_i \geq 0
. It is however not true when
p<1p<1
as triangle inequality doesn't hold.
When
d=2d=2
and
p=1−tp=1-t
(0≤t<1)(0\leq t<1)
is fixed, the core component of generalised mean is assumed constant as below.
x11−t+x21−t=L−dx2dx1=(x2x1)t\begin{split}x_1^{1-t}+x_2^{1-t}&=L\\ -\frac{dx_2}{dx_1}&=\left(\frac{x_2}{x_1} \right)^{t}\end{split}
This equation is regarded reasonable as AMM, because (i) function
gg
where
x2=g(x1)x_2=g(x_1)
is monotonically decreasing and convex; and (ii) The boundary value of
t=0t=0
and
t=1t=1
corresponds to constant sum and constant product formula respectively. When
tt
decreases from 1 to 0, price
−dg(x1)x1-\frac{dg(x_1)}{x_1}
gradually converges to 1, i.e. the curve converges from constant product to constant sum (see Appendix 1 for the relevant proofs).
Though purely theoretical at this stage, Appendix 2 maps
LL
to the liquidity distribution of Uniswap V3. This is motivated by an independent research from Paradigm.

Trading Formulae

Market transaction, which involves exchange of one crypto asset and another, satisfies the invariant function. Please note the formulae do not account for the fee re-investment, which results in a slight increase of
LL
after each transaction, like Uniswap V2.

Out-Given-In

In order to purchase
Δy\Delta y
amount of token Y from the pool, the buyer needs to deposit
Δx\Delta x
amount of token X.
Δx\Delta x
and
Δy\Delta y
satisfy the following
(x+Δx)1−t+(y−Δy)1−t=x1−t+y1−t(x+\Delta x)^{1-t}+(y-\Delta y)^{1-t}=x^{1-t}+y^{1-t}
After each transaction, balance is updated as below:
x→x+Δxx\rightarrow x+\Delta x
and
y→y−Δyy\rightarrow y-\Delta y
. Rearranging the formula results in
Δy=y−[x1−t+y1−t−(x+Δx)1−t]11−t\Delta y=y-\left[x^{1-t}+y^{1-t}-(x+\Delta x)^{1-t}\right]^{\frac{1}{1-t}}
When transaction cost exists, the actual deposit to the pool is less than
Δx\Delta x
. Assuming
λΔx\lambda\Delta x
is the actual amount and
(1−λ)Δx(1-\lambda)\Delta x
is the fee, above can now be expressed as
(x+λΔx)1−t+(y−Δy)1−t=x1−t+y1−tΔy=y−[x1−t+y1−t−(x+λΔx)1−t]11−t\begin{split} &(x+\lambda\Delta x)^{1-t}+(y-\Delta y)^{1-t}=x^{1-t}+y^{1-t}\\ &\Delta y=y-\left[x^{1-t}+y^{1-t}-(x+\lambda\Delta x)^{1-t}\right]^{\frac{1}{1-t}} \end{split}
To keep
LL
constant, the updated balance is:
x→x+λΔxx\rightarrow x+\lambda\Delta x
and
y→y−Δyy\rightarrow y-\Delta y
.

In-Given-Out

This is the opposite case to above. We are deriving
Δx\Delta x
from
Δy\Delta y
.
Δx=1λ[x1−t+y1−t−(y−Δy)1−t]11−t−x\Delta x=\frac{1}{\lambda}{\left[x^{1-t}+y^{1-t}-(y-\Delta y)^{1-t}\right]^{\frac{1}{1-t}}-x}

In-Given-Price

Sometimes, trader would like to adjust the price, perhaps due to deviation of AMM price to the market value. Define
p′p'
the AMM price after rebalancing the token X and token Y in the pool
p′=(y−Δyx+λΔx)tp'=\left(\frac{y-\Delta y}{x+\lambda\Delta x}\right)^{t}
Then, the added amount of
Δx\Delta x
can be calculated from the formula below
(x+λΔx)1−t+(y−Δy)1−t=x1−t+y1−t1+(yx)1−t=(1+λΔxx)1−t+(y−Δyx)1−t1+p1−tt=(1+λΔxx)1−t+p′1−tt(1+λΔxx)1−tΔx=xλ[(1+p1−tt1+p′1−tt)11−t−1]\begin{split} &(x+\lambda\Delta x)^{1-t}+(y-\Delta y)^{1-t}=x^{1-t}+y^{1-t}\\ &1+\left(\frac{y}{x}\right)^{1-t}=\left(1+\lambda\frac{\Delta x}{x}\right)^{1-t}+(\frac{y-\Delta y}{x})^{1-t}\\ &1+p^{\frac{1-t}{t}}=\left(1+\lambda\frac{\Delta x}{x}\right)^{1-t}+p'^{\frac{1-t}{t}}\left(1+\lambda\frac{\Delta x}{x}\right)^{1-t}\\ &\Delta x=\frac{x}{\lambda}\left[\left(\frac{1+p^{\frac{1-t}{t}}}{1+p'^{\frac{1-t}{t}}}\right)^{\frac{1}{1-t}}-1\right]\\ \end{split}

Appendix 1: Generalised Mean when d=2

ALEX's invariant function is
f(x1,x2;p)=x1p+x2p=L.f(x_{1},x_{2};p)=x{_1}^{p}+x_{2}^{p}=L.
It can be rearranged as
x2=g(x1)=(L−x1p)1px{2}=g(x_{1})=(L-x_{1}^{p})^{\frac{1}{p}}
.
x1x_{1}
and
x2x_{2}
should both be positive meaning the liquidity pool contains both tokens.

Theorem

When
0<p<10<p<1
,
g(x1)g\left(x_{1}\right)
is monotonically decreasing and convex.

Proof

This is equivalent to prove
dg(x1)dx1<0\frac{dg(x_{1})}{dx_{1}}<0
and
d2g(x1)dx12≥0\frac{d^{2}g(x_{1})}{dx_{1}^{2}}\geq0
.
dg(x1)dx1=1p(L−x1p)1p−1(−px1p−1)=−(L−x1px1p)1−pp<0d2g(x1)dx12=−1−pp(L−x1px1p)1−2pp[−px1p−1x1p−(L−x1p)pxp−1x12p]=L(1−p)(x2x1)1−2px1−p−1≥0\begin{split} &\frac{dg(x_{1})}{dx_{1}}=\frac{1}{p}(L-x_{1}^{p})^{\frac{1}{p}-1}\left(-px_{1}^{p-1}\right)=-\left(\frac{L-x_{1}^{p}}{x_{1}^{p}}\right)^{\frac{1-p}{p}}<0\\ &\frac{d^{2}g(x_{1})}{dx_{1}^{2}}=-\frac{1-p}{p}\left(\frac{L-x_{1}^{p}}{x_{1}^{p}}\right)^{\frac{1-2p}{p}}\left[\frac{-px_{1}^{p-1}x_{1}^{p}-(L-x_{1}^{p})px^{p-1}}{x_{1}^{2}p}\right]\\ &=L(1-p)\left(\frac{x_{2}}{x_{1}}\right)^{1-2p}x_{1}^{-p-1}\geq0 \end{split}
The last inequality holds because each component is positive.
When
pp
= 1, it is straightforward to see that the invariant function is constant sum. To show that the invariant function converges to constant product when
pp
= 0, we will show and prove an established result in a generalised
dd
dimensional setting.

Theorem

lim⁡p→0(1d∑i=1dxip)1p=(∏i=1dxi)1d\lim_{p\rightarrow0}\left(\frac{1}{d}\sum_{i=1}^{d}x_{i}^{p}\right)^{\frac{1}{p}}=({\prod_{i=1}^{d}x_{i}})^{\frac{1}{d}}

Proof

​
(1d∑i=1dxip)1p=exp[log(1d∑i=1dxip)p]\left(\frac{1}{d}\sum{i=1}^{d}x_{i}^{p}\right)^{\frac{1}{p}}=\text{exp}\left[\frac{\text{log}\left(\frac{1}{d}\sum{i=1}^{d}x_{i}^{p}\right)}{p}\right]
. Applying L'Hospital rule to the exponent, which is 0 in both denominator and nominator when
p→0p\rightarrow0
, we have
lim⁡p→0log(1d∑i=1dxip)p=lim⁡p→0∑i=1dlog(xi)∑j=1d(xjxi)p=∑i=1dlog(xi)d\lim_{p\rightarrow0}\frac{\text{log}\left(\frac{1}{d}\sum_{i=1}^{d}x_{i}^{p}\right)}{p}=\lim_{p\rightarrow0}\sum_{i=1}^{d}\frac{\text{log}(x_{i})}{\sum_{j=1}^{d}\left(\frac{x_{j}}{x_{i}}\right)^{p}}=\frac{\sum_{i=1}^{d}\text{log}(x_{i})}{d}
Therefore
lim⁡p→0(1d∑i=1dxip)1p=lim⁡p→0exp∑i=1dlog(xi)d=(∏i=1dxi)1d\lim_{p\rightarrow0}\left(\frac{1}{d}\sum_{i=1}^{d}x_{i}^{p}\right)^{\frac{1}{p}}=\lim_{p\rightarrow0}\text{exp}\frac{\sum_{i=1}^{d}\text{log}(x_{i})}{d}=({\prod_{i=1}^{d}x_{i}})^{\frac{1}{d}}

Corollary

When d = 2,
x1x2=lim⁡p→0[12(x1p+x2p)]2px_{1}x_{2}=\lim_{p\rightarrow0}\left[\frac{1}{2}(x_{1}^{p}+x_{2}^{p})\right]^{\frac{2}{p}}
Proof of the corollary is trivial, as it is a direct application of the theorem. It shows that generalised mean AMM implies constant product AMM when
p→0p\rightarrow0
.

Appendix 2: Liquidity Mapping to Uniswap v3

As Uniswap v3 is able to simulate liquidity curve of any AMM, we are interested in exploring the connection between ALEX's AMM and that of Uniswap's. Interesting questions include: what is the shape of the liquidity distribution? Which point(s) has the highest liquidity? We acknowledge that the section is more of a theoretical study for now.
Uniswap V3 AMM can be expressed as a function of invariant constant
LL
with respect to price
pp
,
LUniswap=dydpL_{\text{Uniswap}}=\frac{dy}{d\sqrt{p}}
. For us, the difference in the invariant function means we can write price as
p=ertp=e^{rt}
(or, equivalently,
r=1tln⁡pr=\frac{1}{t}\ln{p}
) and we have
LUniswap=dydp=2te−12rtdydrL_{\text{Uniswap}}=\frac{dy}{d\sqrt{p}}=\frac{2}{t}e^{-\frac{1}{2}rt}\frac{dy}{dr}
Based on the previous sections, we can then express
yy
as
y=[L1+e−(1−t)r]11−ty=\left[\frac{L}{1+e^{-(1-t)r}}\right]^{\frac{1}{1-t}}
Therefore
dydr=L11−te−(1−t)r(1+e−(1−t)r)2−t1−tLUniswap=2tL11−t(er(1−t)2+e−r(1−t)2)−2+t1−t=2tL11−t{2cosh⁡[r(1−t)2]}−2+t1−t\begin{split} &\frac{dy}{dr}=L^{\frac{1}{1-t}}\frac{e^{-(1-t)r}}{(1+e^{-(1-t)r})^{\frac{2-t}{1-t}}}\\ &L_{\text{Uniswap}}=\frac{2}{t}L^{\frac{1}{1-t}}\left(e^{\frac{r(1-t)}{2}}+e^{\frac{-r(1-t)}{2}}\right)^{\frac{-2+t}{1-t}}\\ &=\frac{2}{t}L^{\frac{1}{1-t}}\big\{2\cosh\left[\frac{r(1-t)}{2}\right]\big\}^{\frac{-2+t}{1-t}} \end{split}
Figure 1
Figure 1 plots
LUniswapL_{\text{Uniswap}}
against
rr
(which is proportional to
pp
) regarding various levels of
tt
. When
0<t<10<t<1
,
LUniswapL_{\text{Uniswap}}
is symmetric around 0% at which the maximum reaches . This is because
  1. 1.
    ​
    cosh⁡[(r(1−t)2)]\cosh\left[(\frac{r(1-t)}{2})\right]
    is symmetric around
    rr
    = 0% with minimum at 0% and the minimum value 1;
  2. 2.
    ​
    xzx^z
    is a decreasing function of
    xx
    when
    xx
    is positive and power
    zz
    is negative. In our case, we have
    z=−2+t1−t<−1z=\frac{-2+t}{1-t}<-1
    . Therefore, it is the maximum rather than minimum that
    LUniswapL_{\text{Uniswap}}
    achieves at 0.
Furthermore, the higher the
tt
, the flatter the liquidity distribution is. When
tt
approaches 1, i.e. AMM converges to the constant product formula, the liquidity distribution is close to a flat line. When
tt
approaches 0, the distribution concentrates around 0%.