Decentralized
Trading

Andreas Park

Decentralized Trading

The Pricing Function

Liquidity Deposit $\Rightarrow$ slope of the price curve

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 $\frac{p(q)}{p(0)}-1=\frac{q}{a-q}.$

makes asset and cash deposit
more deposits flatter price curve
- attracts more volume
- but larger "positional" loss when prices move

The Pricing Function (just a little more)

A limit order book has a marginal price function $\rho(q)$ so that $\Delta c(q)=\int_0^q\rho(s) ds.$
We can do that here, too: for
$\rho(q)=\frac{ac}{(a - q)^2},$
we have $\int_0^q \frac{ac}{(a - s)^2}ds=\frac{cq}{a-q}.$
$\to$ Constant Product = special case of a limit order book.

More on UniSwap v3

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 $q\in(0,u)$
  - all cash amounts $\Delta c\in (0,\Delta c(d))$
$\to$ for given $p_0$ , there are THREE degrees of freedom

Basics of Liquidity Provision

$\underbrace{F \int DV \mu(DV) }_{\text{fees earned on}\atop \text{balanced flow}}+\int_0^\infty\underbrace{(\Delta c(q^*)-q^*p_t(R)}_{\text{adverse selection loss} \atop \text{when the return is {\it R}}} +\underbrace{F \cdot \Delta c(q^*))}_{\text{fees earned}\atop \text{from arbitrageurs}}~\phi(R)dR \ge 0.$

$q^*$ is what arbitrageurs trade to move the price to reflect $R$

Basic idea of liquidity provision: earn more on balanced flow than what you lose on price movement

$\text{fee income} +\underbrace{\text{what I sold it for}-\text{value of net position}}_{\text{adverse selection loss}} \ge 0$

in AMMs:
protocol fee

in tradFi: bid-ask spread

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)

Basics of Liquidity Provision

Liquidity provision measured as "collective" deposit $\alpha$ of token's market cap as function of

balanced volume
return distribution
(some authors use explicit functions so they can relate it to asset volatility)
fee

$E[\text{DL}(R)]+F\cdot E[\text{another function of }R]+F\cdot \frac{\text{E[dollar volume]}}{\text{initial deposit}}\ge 0.$

$\text{what I sold it for}-\text{value of net position}+\text{fee income} \ge 0$

The Decision of the Liquidity Demander

Wants to trade some quantity $q$ at cost

$\text{AMM price impact} +\text{AMM fee}.$

two opposing forces for $F\nearrow$ for liquidity demand
- more liquidity provision
  $\to$ lower price impact
- more fees to pay
Finding: There is a minimal cost fee:

$F^\pi=\frac{1}{E[|\sqrt{R}-1|/2]+E[DV]}\left(-2q\ E[\text{DL}]+ \sqrt{-2qE[DV]\ E[\text{DL}]}\right).$

this is from Malinova and Park (2023); similar result is in Hasbrouk, Riviera, Saleh (2023)

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