The Microstructure of
Crypto-Asset Trading

Andreas Park

What does an AMM need?

trades create linear changes:
- a trade of $q\in(-\infty,a),$ will cost $\Delta c(q)\in(-c,\infty)$
- liquidity changes
  - $a$ $\to$ $a-q$
  - $c$ $\to$ $c+\Delta c(q)$
- $\to$ marginal price after trade of $q$ is
  
  $p^m(0)=\frac{c}{a}~~\to~~p^m(q)=\frac{c+\Delta c(q)}{a-q}.$

Some Pricing Rules from Traditional Markets: Uniform Price

assume cost is price $\times$ quantity: $\Delta c(q)=q\cdot p(q)$
plus assume average price $=$ marginal price after trade

$p(q)=\frac{c+q\cdot p(q)}{a-q}~~\Leftrightarrow~~p(q)=\frac{c}{a-2q}.$
formalized and analyzed by Canidio and Fritsch (2023) and implemented in COW-Swap
theoretical/implementation problem: splitting an order into smaller units is profitable!

$q=2$

$\Delta c(q)= q\times p^m(q)=2\times 15.5$

Some Pricing Rules from Traditional Markets: Limit Order Book

For AMM: assume continuity $\to$ price of "final" marginal unit coincides with price of "first" marginal unit of next trade $\Delta c(q)=\int_0^q \rho(s) ds~\to~\Delta c'(q)=\rho(q)=p(q).$
Also "first" unit costs nothing $\Delta c(0)=0$
This brings differential equation
$p(q)=\frac{c+\int_0^q \rho(s) ds}{a-q}~~\Leftrightarrow~~\Delta c'(q)=\frac{c+\Delta c(q)}{a-q},$
with solution $\Delta c(q)=q\cdot\frac{c}{a-q}.$
Or, we can write $\Delta c(q)=\int\limits_0^q p(s)ds=\int\limits_0^q\frac{ac}{(a-q)^2}ds=\frac{cq}{a-q}.$

Most Common Pricing Rule in DeFi: Constant Product

With $a\cdot c= (a-q) (c+\Delta c(q))$
we compute cost $\Delta c(q)=q\cdot \frac{c}{a-q},$
average price $p(q)=\frac{c}{a-q}$
and marginal price $p^m(q)=\frac{c+\frac{cq}{a-q}}{a-q}=\frac{ac}{(a-q)^2}.$

Insight: AMM pricing function is the same as a limit order book when we require

that prices are continuous and
that prices are determined by liquidity deposits only

price $\times$ quantity: $p(q)=\frac{c}{a-2q}$
limit order book: $p(q)=\frac{c}{a-q}.$
constant product: $p(q)=\frac{c}{a-q}.$

Some insights on pricing functions

Price $\times$ quantity
- nice feature:
  - regret-free price: if price move is due to information, you have no positional loss
- poor feature:
  - you can always save by order-splitting
  - limit price for splitting = limit order book price

LOB/CP
- nice feature:
  - order splitting doesn't pay
- poor feature:
  - prices not regret-free: if price move is due to information, you always lose

The Pricing Function

Most common form of AMM liquidity rule is Constant Product Pricing
$L(a,c)=a\cdot c~\Rightarrow~a\cdot c= (a-q)\cdot (c+\Delta c).$
Total cost of trading $q$ $\Delta c=\frac{cq}{a-q}.$
Price per unit $p(q)=\frac{c}{a-q}.$
Average spread paid $\frac{p(q)}{p(0)}-1=\frac{q}{a-q}.$

liquidity provider makes asset and cash deposit
more deposits flatten price curve
- may attract more volume
- but larger "positional" dollar loss when prices move

larger liquidity deposits $a$ $\Rightarrow$
- lower costs (price impact) for liquidity demanders

Sidebar: we can quantify how much a PASSIVE LP loses when the price moves by $R$

for orientation:

If the stock price drops by 10% the incremental loss for liquidity providers is 13 basis points on their deposit
- $\to$ total loss=-10.13%
If the stock price rises by 10%, the liquidity provider gains 12 basis points less on the deposit
- $\to$ total gain =9.88%

$\frac{\text{adverse selection loss when the return is $R$}}{\text{initial deposit}}=\sqrt{R}-\frac{1}{2}(R+1)$

see Barbon & Ranaldo (2022)

Liquidity Demander's Decision & (optimal) AMM Fees

Two opposing forces in equilibrium for fee $F\nearrow$
1. more liquidity provision
  $\to$ lower price impact
2. more fees to pay

Result:

competitive liq provision $\to$ there exists an optimal (min trading costs) fee $>0$

$\to$ derive closed form solution for competitive liquidity provision
depends on return distribution, balanced volume, quantity demanded

Similar to Lehar&Parlour (2023) and Hasbrouck, Riviera, Saleh (2023)

For trading of a fixed quantity:

$\text{LD cost}=\text{AMM price impact} +\text{AMM fee} .$

$F^\pi=\frac{1}{E[|\sqrt{R}-1|/2]+V}\left(-2q\ E[\text{position loss}]+ \sqrt{-2qV\ E[\text{position loss}]}\right).$

assume: liquidity providers add liquidity until they break even in expectation

Empirical Work

Lehar and Parlour (2021)

Barbon & Ranaldo (2023)

Other notables:

Lehar, Parlour, Zoican (2022):
- study and document fragmentation of liquidity across different fee levels
- most intense provision-withdrawal activity for low fees $\to$ UniSwap v3 starts mimicking tradfi
Capponi & Jia (2022) study impact of curvature of pricing function
- Theory: curvature of optimal pricing curve $\nearrow$ volatility
- Empirics: cost volatility is 60% lower for stable-coin pairs
- corr(deposit inflows ,volatility)<0
  corr(deposit inflows ,volume)>0

Malinova & Park (2023): AMM applied to equities would reduce trading costs by 30%

UniSwap v3 has "concentrated liquidity provision"

liquidity provision choice in v2:
- submit $a,c$ at the marginal $p_0=\frac{c}{a}$
- $\to$ for given $p_0$ , there is ONE degree of freedom
- provide liquidity
  - for all prices $p\in(0,\infty)$
  - all asset amounts $q\in(0,a)$
  - all cash amounts $\Delta c\in (0,c)$

liquidity provision choice in v3:
- submit $u,\Delta c (d)$
- choose price range $[p_d,p_u]$
- provide liquidity
  - for all prices $p\in[p_d,p_u]$
  - all asset amounts $q\in(-\infty,u)$
  - all cash amounts $\Delta c\in (-\Delta c(d),\infty)$

Just a little more institutional details

users submit liquidity positions $(u,d,p_u,p_d)$ and receive an ERC-721 (=NFT) token as a receipt
the code segments liquidity into discrete exponential intervals $[p_k,p_{k+1}]$ with $p_k=(1+\delta)^k. ~~~(\text{numerically: }p(k)=1.0001^k)$
it aggregates liquidity over these intervals
the pricing curve for each interval is determined by the constant product rule
intervals may be "empty"

How the price is determined

In v2, for given $p_0=\frac{c}{a}$ , users have ONE degree of freedom.

for instance, when you pick prices $p_u,p_d$ and the quantity $u$ that you are willing to supply, then the amount of cash that you have to contribute is pre-determined.
You can pick $p_d=p_0$ so that $\Delta c(d)=0$

In v3, for given $p_0$ , users have THREE degree of freedom.

How the price is determined

some more facts
- if the marginal price $p_u$ is reached, you will sell $u$ units of the asset
- the price is still using constant product
- but we use the term "virtual" liquidity
- its the v2 liquidity that would pertain if you'd supply liquidity on the entire interval of prices

$p_d$

$p_u$

$p_0$

we know this curve has functional form $p^m(q)=\frac{\tilde{a}c}{(\tilde{a}-c)^2}$

where $\tilde{a}$ is the virtual liquidity

quick disclaimer: what follows is not how UniSwap is explained on its website etc. But the resulting maths are the same

Solutions

virtual liquidity factor: $\tilde{a}=\frac{u}{1-\sqrt{p_0/p_u}}=u\cdot \frac{\sqrt{p_u}}{\sqrt{p_u}-\sqrt{p_0}}.$

required quantity to reach $p_d$ $\rho(-\tilde{d})=\frac{\tilde{a}\tilde{c}}{(\tilde{a}+\tilde{d})^2}=p_d.~~\Leftrightarrow~~d^*=-\tilde{a}(1-\sqrt{p_0/p_d})$

required cash deposit $\Delta c(\tilde{d})=-\tilde{a}(1 - \sqrt{p_0/p_d})\sqrt{p_0p_d}=-u\sqrt{p_0p_d}\frac{1 - \sqrt{p_0/p_d}}{1 - \sqrt{p_0/p_u}}$

marginal pricing function $\tilde{\gamma}(q)=\frac{\tilde{a}^2p_0}{(\tilde{a}-q)^2}=\frac{u^2p_0p_u}{((u-q)\sqrt{p_u}+q\sqrt{p_0})^2}.$

Numerical example

Suppose $p_0=\$10$ .
LP is willing to sell up to $u=100$
v2 liquidity:
- Required cash deposit: $c=100\times 10=1000$
- liquidity $c\cdot a=100\times 1,000$

For v3: suppose willing to supply liquidity in $[\$9,\$11]$
v3 liquidity:
- virtual liquidity factor: $\tilde{a}\approx 2,150$
- required cash $c^*=\$1,100$

consider a buy order of $q=10$
cost per unit
- v2: $\frac{1000}{100-10}=11.11$
- v3: $\frac{21,500}{2,150-10}=10.05$

From Vitalik Buterin's post on the topic:

https://ethresear.ch/t/improving-front-running-resistance-of-x-y-k-market-makers/1281

February 2022: UniSwap (V2), ETH-USDC
- 38,100 ETH
- 118M USDC
- $p=\$3,097$
Buy: 100 ETH
- pay approximately $3,105 per ETH. accoridng to bonding curve
100 ETH MEV sandwich attack
- Initial trade cost: $3,105 per ETH
- reversing yields $3,121 per ETH
- total profit of $1,639 $=$ value extracted from original trader.
1,000 ETH MEV attack
- profit/loss $=$ $17K
10,000 ETH MEV attack
- profit/loss $=$ $261K.

Theorem (Park 2023):

Constant product pricing and linear limit order book pricing allows unlimited MEV losses
Uniform pricing has limited MEV losses but profitable splitting

Hypothetical example