Learning from DeFi: Would Automated
Market Makers Improve Equity Trading?

Katya Malinova and Andreas Park

Some Motivation

Blockchain: borderless general purpose value and resource management tool

Basic Idea

DeFi: financial applications that run on blockchains

$\Rightarrow$ brought new ideas and tools

one new market institution: automated market makers

Application: decentralized trading with automated market makers

Key Components

Our question:
1. Can an economically viable AMM be designed for current equity markets?
2. Would such an AMM improve current markets?

pooling of liquidity

pro-rated
- fee income
- risk

Liquidity providers:
- use existing assets to earn passive income

Liquidity demanders:
- predicatable price
- continuous trading
- ample liquidity

Liquidity Supply and Demand in an Automated Market Maker

Constant Liquidity (Product) AMM

Purchase $q$ of asset
Deposit cash $\Delta c (q)$ into liquidity pool, extract $q$ of shares
Idea of pricing: liquidity before trade $=$ after trade
\[L(a,c)=L(a-q,c+\Delta c)\]

AMMs require liquidity deposits
Deposits:
- $a$ units of an asset (e.g. a stock)
- $c$ units of cash

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

Basics of Liquidity Provision in an AMM for a Model

fundamental value changes
- $\Rightarrow$ informed arbitrageur
  - $\Rightarrow$ LP incur losses (trade at the "wrong price")

a liquidity trade
- arbitrageurs (eventually) trade back to the original price
- $\to$ liquidity deposits return to the original values $(a, c)$

losses on informed
no gains (nor losses) on noise

Basics of Liquidity Provision

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

\[\text{LP payoff}=\text{what I sold it for}-\text{value of net position}+\text{fee income} \ge 0 \]

Characterize liquidity provision for stocks as a
"collective" deposit $\alpha$ of firm's market cap
- initial deposit $\alpha \nearrow $ $\Rightarrow$ LP payoff $\searrow$

Competitive liquidity provision $\Rightarrow$ initial deposit fraction $\bar{\alpha}$ such that LPs break-even, as a function of
- balanced $-volume
- asset volatility/return $R$ distribution
- fee $F$

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%

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

as in Barbon & Ranaldo (2022)

"Incremental Loss from Long-Run Adverse Selection"

Wants to trade some quantity $q$.
Is better off with AMM relative to traditional market if

\[\text{bid-ask spread}\ge\text{AMM price impact} +\text{AMM fee}.\]

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

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

Liquidity Demander's Decision & (optimal) AMM Fees

Model Summary

Have:
- equilibrium choices for competitive liquidity provision
- fee that minimizes liquidity demander AMM costs ($>0$)
Next:
- Calibrate to stock markets
- AMM Feasible?
  - AMM costs at the optimal fee < bid-ask spread?
    - is the max (break-even) supplied liquidity enough?

How we think of the Implementation of an AMM for our Empirical Analysis

Approach: daily AMM deposits

AMMs close overnight.
Market: opening auction $\to$ $p_0$
Determine: optimal fee; submit liquidity $a,c$
at ratio $p_0=a/c$ until break even $\alpha=\overline{\alpha}$
Liquidity locked for day
At EOD release deposits and fees
Back to 1.

Background on Data

Special Consideration 1: What volume?

some volume may be intermediated
with AMMs: no need for intermediation
$\to$ intermediate volume should disappear
$\to$ use volume/2

Special Consideration 2: What's $q$ (the representative order size)?

use average per day
take long-run average + 2 std of daily averages
(also avg $\times 2$,$\times 4$, depth)

All displayed data CRSP $\cap$ TAQ

CRSP for shares outstanding
WRDS computed TAQ statistics for
- quoted spreads (results similar for effective)
- volume
- open-to-close returns
- average trade sizes, VWAP
Time horizon: 2014 - March 2022
Exclude "tick pilot" period (Oct 2016-Oct 2018)
All common stocks (not ETFs) (~7550).
Explicitly not cutting by price or size
All "boundless" numbers are winsorized at 99%.

Special Consideration 3:

Where to get expected returns and volume?

Approach 1: "ad hoc"
- "one-day-back" look
- take yesterday's return and volume when deciding on liquidity provision in AMM
Approach 2: estimate historical return distribution

AMMs based on historical returns