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

Katya Malinova and Andreas Park

Some Motivation

Liquidity Supply and Demand in an Automated Market Maker

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

Basics of Liquidity Provision

$\underbrace{F p_0 V}_{\text{fees earned on balanced flow}}+\int_0^\infty\underbrace{(\Delta c(q^*)-q^*p_t(R)}_{\text{adverse selection loss when the return is $R$}} +\underbrace{F \cdot \Delta c(q^*))}_{\text{fees earned from arbitrageurs}}~\phi(R)dR \ge 0.$

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

fundamental value up
- $\Rightarrow$ price up
  - $\Rightarrow$ sell the asset for less than its worth
fundamental value down
- $\Rightarrow$ price up
  - $\Rightarrow$ buy the asset for more than its worth

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

Basics of Liquidity Provision

$\int_0^\infty\underbrace{(\Delta c(q^*)-q^*p_t(R)}_{\text{adverse selection loss when the return is $R$}} +\underbrace{F \cdot \Delta c(q^*))}_{\text{fees earned from arbitrageurs}}~\phi(R)dR +\underbrace{F p_0 V}_{\text{fees earned on balanced flow}}\ge 0$

$\frac{1}{\text{initial deposit}}\int_0^\infty(\Delta c(q^*)-q^*p_t(R)+F \cdot \Delta c(q^*))~\phi(R)dR +\frac{F p_0 V}{\text{initial deposit}}\ge 0$

$\int_0^\infty\left(\frac{\Delta c(q^*)-q^*p_t(R)}{\text{initial deposit}} +F \cdot \frac{\Delta c(q^*)}{\text{initial deposit}}\right)~\phi(R)dR +\frac{F p_0 V}{\text{initial deposit}}\ge 0$

closed form functions of $R$ only
(see Barbon & Ranaldo (2022))

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 firm's market cap as function of

balanced volume
asset volatility/return distribution
fee

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

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

The Decision of the Liquidity Demander

Wants to trade some quantity $q$ .
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:

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

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 $\cap$ WRDS

CRSP for shares outstanding
WRDS computed 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%.

Deposit Requirements

Our approach: measure liquidity provision in % of market cap
Share-based liquidity provision is trivial: the shares are just sitting at brokerages.
But: requires an off-setting cash amount
Cash is not free:
- at 6% annual rate, must pay 2bps per day.
- Would need to add to fees
But: do we need "all that cash"?
No.

(hand-waving argument)
2nd gen AMMs have liquidity provision "bands": specify price range for which one supplies liquidity
Here: specify range for $R\in(\underline{R},\overline{R})$
Outside range: don't trade.
Inside range: "full" liquidity with constant product formula.
Implication: only need cash and shares to satisfy in-range liquidity demand.

For return $R$ , the following number of shares change hands: $q=a\cdot(1-\sqrt{R^{-1}}).$
Fraction of share deposit used $\frac{q}{a}=1-\sqrt{R^{-1}}.$
Fraction of cash used $\frac{\Delta c ("R")}{c}=\frac{1-\sqrt{R^{-1}}}{\sqrt{R^{-1}}}.$
Example for $R=.9$ (max allowed price drop $=10\%$ ) $\frac{\Delta c ("R")}{c}=-5\%.$
$\Rightarrow$ "real" cash requirements $\not=$ deposits

AMM Literature: a booming field

Theory
- Lehar and Parlour (2021): for many parametric configurations, investors prefer AMMs over the limit order market.
- Aoyagi and Ito (2021): co-existence of a centralized exchange and an automated market maker; informed traders react non-monotonically to changes in the risky asset’s volatility
- Capponi and Jia (2021): price volatility $\to$ welfare of AMM LPs; conditions for a breakdown of liquidity supply in the automated system; more convex pricing $\to$ lower arbitrage rents & less trading.
- Capponi, Jia, and Wang (2022): decision problems of validators, traders, and MEV bots under the Flashbots protocol.
- Park (2021): properties and conceptual challenges for AMM pricing functions
- Milionis, Moallemi, Roughgarden, and Zhang (2022): dynamic impermanent loss analysis for under constant product pricing.
- Hasbrouck, Rivera, and Saleh (2022): higher fee $\Rightarrow$ higher volume
Empirics:
- Lehar and Parlour (2021): price discovery better on AMMs
- Barbon and Ranaldo (2022): compare the liquidity CEX and DEX; argue that DEX prices are less efficient.
Math Literature: large number of papers that study AMMs in applied maths
- e.g. Cartea, Drissi, Monga (2023): closed form solution to IPL and optimizing strategy

Summary

AMMs do not require a blockchain - just a concept
could be run in the existing world (though there are institutional and regulatory barriers)

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

Answers:
1. Yes.
2. Massively.

Source of Savings:
- Liquidity providers $\not=$ Citadel!
- $\to$ passive liquidity provision
- $\to$ use idle capital
- $\to$ + better risk sharing

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

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

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