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

Katya Malinova and Andreas Park

Agenda

• Very brief overview of Automated Market Makers
  • function
  • difference crypto-trading to traditional markets
  • difference DeFi to CeFi trading
• Research question: what if we used AMMs for equities?
  • theory: liquidity provision & demand
  • data: implementation & trading costs

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

Trading Infrastructure

Application: decentralized trading with automated market makers

New institutions!

• passive "shared" liquidity provision
• new pricing function

Key Components

• Liquidity providers:
  • use existing assets to earn passive income

Overview

Roadmap for the rest of the talk

• How does an AMM work?
   
• liquidity provider income: traditional market vs AMM
   
• mechanism to implement an AMM
   
• costs/benefits
   
• capital requirements
   
• obstacles and challenges

Automated Market Makers

Basics of Liquidity Provision

• Basic idea of liquidity provision:
  • price end-of-day is "truth"
  • during the day this happens
    • buy volume: \(B\)
    • sell volume: \(S\)
    • for simplicity \(B>S\)
    • price moves \(p_0\nearrow p_t\)
       
• earn more on balanced flow than what you lose on price movement
  
  \[\underbrace{\text{fee}\times(B+S)}_{\text{fee income}}+\underbrace{\text{net revenue} (B-S)}_{\text{what I sold it for}}- \underbrace{p_t\times (B-S)}_{\text{value of not position}} \ge 0 \]

The Decision of the Liquidity Provider

• earn more on balanced flow than what you lose on price movement
  
  \[\underbrace{\text{fee}\times(B-S)}_{\text{fee income}\atop\text{unbalanced}}+\underbrace{\text{fee}\times(2\times S)}_{\text{fee income}\atop\text{balanced}}> \underbrace{p_t\times (B-S)}_{\text{value of what I sold}} - \underbrace{\text{revenue} (B-S)}_{\text{what I sold it for}}.\]
   
• Assume:
  • balanced volume \(V\) trades at opening price \(p_0\)
  • unbalanced volume is \(q\)
  • fee collected on cash value of trade
     
• Then \[F\times (\Delta c(q)+Vp_0)>\Delta c(q)-p_tq.\]

AMM Pricing

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

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
  \[L(a,c)=L(a-q,c+\Delta c)\]

The Decision of the Liquidity Provider

• 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\[\frac{p(q)}{p(0)}-1=\frac{q}{a-q}.\]

AMM Properties

• Liquidity deposits/withdrawals do not affect the marginal price by construction
  • deposits \(\Delta a,\Delta c\) must be such that \[p_0=\frac{c}{a}=\frac{c+\Delta c}{a+\Delta a}\]
  • contrasts limit order markets where order changes often move marginal price
     
• Prices move only by trades. Trade \(q\) and pay \(\Delta c(q)\), marginal price
  
  \[p_t=\frac{c+\Delta c(q)}{a-q}\]

AMM Properties

• Let \(q^*\) be such that \[p_t=\frac{c_t}{a_t}=\frac{c_t+\Delta c(q^*)}{a-q^*}\]
• Value of post-trade  position: \[p_tq^*\]
• Cash flow from trade \[\Delta c(q^*)\]
• Regret-free pricing would mean \[p_tq^*=\Delta c(q^*)\]

Liquidity Provider gains/losses

Liquidity Provision

Theory Overview

1. Liquidity supply decision
    
2. Liquidity demand decision AMM vs traditional market
    
3. Welfare optimum

• Goal: liquidity demand-supply decision with AMMs
   
• \(\to\) apply to equity data to see if it's available and beneficial

Big Picture for Liquidity Provision

• You supply capital for a while.
   
• During this time the fundamental value of the asset may change.
   
• In an efficient market folks will move the price to the fundamental.
   
• You may trade at a wrong price in the process.
  
  
   
• Almost every model explicitly or implicitly works like this, e.g., Kyle (1985), Glosten and Milgrom (1986), Grossman-Miller (1988), Biais (1993), Foucoult (1999) Budish, Crampton, Shin (2015)

• Asset with value \(V_0\) that may change to value \(V_t\)
   
• Efficient prices so that \(p_0=V_0\)
   
• Arbitrageurs who move price from \(p_0\) to \(p_t=V_t\) by trading a net quantity \(q^*\) at price \(\Delta (q^*)\)
   
• Additional balanced (dumb) volume \(v\)
   
• Collect a fee \(F\) on all trading volume

Big Picture for Liquidity Provision

• Example: 
  • limit order books:
    • Fee=collect bid-ask spread on uninformed trades
    • amount paid = cumulative cost of multiple trades or walking the book
  • Auction market (a la Kyle)
    • Overcharge uninformed to balanced in expectation against possible informed

Liquidity Provision in AMMs

• Asset value \(V_0\) that may change to value \(V_t\)
• Efficient prices \(p_0=V_0\) and  \(p_t=V_t\)
• Arbitrageurs move price from \(p_0\) to \(p_t=V_t\) by trading a net quantity \(q^*\) at price \(\Delta (q^*)\)
• Price gross return \(R\sim \Phi:\mathbb{R}_+\to[0,1]\) with continuous cdf \(\phi\).
• Balanced volume that trades  \(V\) at \(p_0\)
• Exogenous protocol fee \(F\) on all dollar trading volume
   
• Liquidity providers are risk neutral and competitive: earn zero return in expectation
   
• Liquidity providers collectively choose liquidity deposit \(d=p_0a+c.\)

Big Picture for Liquidity Provision

Liquidity Provider Expected Return

Equilibrium Liquidity Supply

• Expressing terms:
  
  \[\int\limits_0^\infty\left(\sqrt{R}-\frac{1}{2}\left(1+R\right)+\frac{F}{2}|\sqrt{R}-1|\right)~\phi(R)dR+F\frac{V}{2a}.=0\]
   

• Gives us an equilibrium deposit \(a^*\)
  
  \[a^*=\frac{F V}{2}\underbrace{\left(-F\times E[|\sqrt{R}-1|/2]-E[\text{ILLRAS}]\right)^{-1}}_{=:C^\mathsf{CP}(\phi,F)^{-1}}.\]

The Decision of the Liquidity Provider

The Decision of the Liquidity Demander

The Optimal Fee for Liquidity Providers is not Zero

• Loose idea (details later): LPs are price takers, enter the day and then
  submit liquidity until they break even in expectation
  • \(\Rightarrow\) submit \(\alpha=\overline{\alpha}\)
  • All benefits go to liquidity demanders in expectation.
     
• The contemporaneous profit for a liquidity demander is
  
  \[\pi=\sigma-\frac{q}{\overline{\alpha}(F)S-q}-F\]
   
• which has a local maximum for
  
  \[F^\pi=\frac{1}{E[|\sqrt{R}-1|/2]+V}\left(-2q\ E[\text{ILLRAS}]+ \sqrt{-2qV\ E[\text{ILLRAS}]}\right).\]

Model Summary

• We can express the equilibrium choice for liquidity provision.
   
• We can measure the benefit for liquidity demanders who use the AMM.
   
• We can determine the fee that maximizes the liquidity demander benefit (it's not zero!)
   
• Next question:
  • How would this look like when applied to stock markets?
  • What are the optimal fees?
  • Is it feasible?
  • What are the empirical benefits?

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

Approach: daily AMM deposits

1. AMMs are closed overnight.
    
2. Market starts with opening auction to determine \(p_0\)
    
3. For fixed fees, LP submit liquidity \(a,c\) at ratio \(p_0=a/c\) until \(\alpha=\overline{\alpha}\)
    
4. Liquidity is locked for the day.
    
5. At the end of the day, remaining deposits are released back to LPs.
    
6. Back to 1.

• LPs act backward-looking: Submit liquidity today based on yesterday's
  • dollar-volume 
  • fee
  • returns
     
• and provide liquidity iff \(\overline{\alpha}>\underline{\alpha}\) on the previous day.

Assumptions for Empirical Investigation

1. Peaking at the data to understand relations
   • AMMs come with exogenous fees => analyze hypothetical behavior for different fee levels
2. Using optimal fees \(F^\pi\) but 
   • ignore fees for price-moving trade and optimize based on "yesterday's" return
   • (version in the written paper)
3. Using optimal fees based on return distribution 2014-2021
   • specification that is "true" to the model

Presentation of the Results in THREE steps

Data-Peeking

Getting a Sense of the Relationships

Some Cross-Sectional Relationships

Threshold for Feasibility: \(\overline{\alpha}-\underline{\alpha}\)

Days with feasibility

Average Benefit of AMM per day

Optimally Designed AMMs

Business implementation:

• Fee \(F=F^\pi\) is set today based on yesterday's 
  • av trade size + 2 std(trade size) at the VWAP
  • quoted half-spreads
  • return post-open to pre-close
  • volume

Assumptions for Optimality

The optimal fee \(F^\pi\) maximizes

\[\pi(F)=\sigma-\frac{q}{\bar{\alpha}S-q}-F\]
with solution (no expectations)

\[F^\pi=\frac{-2qp\ \text{ILLRAS}}{V}+ \sqrt{\frac{-2qp\ \text{ILLRAS}}{V}}\]

Optimal fee \(F^\pi\)

quoted spread minus AMM price impact minus AMM fee (all measured in bps)

relative savings:
fees paid in AMM/fees paid with spreads

average benefits liquidity provider in bps (average=0)

AMMs with expectations

Assumptions for Optimality

• The optimal fee \(F^\pi\) with the full specification is
  \[F^\pi=\frac{1}{E[|\sqrt{R}-1|/2]+V}\left(-2q\ E[\text{ILLRAS}]+ \sqrt{-2qV\ E[\text{ILLRAS}]}\right).\]
   
• Means we need
  • \(E[|\sqrt{R}-1|/2]\) and
     
  • \(E[\text{ILLRAS}].\)

Two approaches

1. Fit a parametric distribution to returns
   (used Gamma; Log-Normal is similar)
    
2. Fit a non-parametric distribution

Implementation notes:

• restrict \(R\in[0.5,2]\)
• require 100 observations per firm
• use data from Jan 2014 - March 2021
• test on April 2021-March 2022

Return distribution example: Tesla

• average \(F^\pi=13\)bps
• average \(\overline{\alpha}=30\%\)

average savings: 18 bps

average daily: $12K

average annual: $2.95 million

Some Numbers
(based on "yesterday's")

average per stock and day benefit for liquidity takers

aggregate benefit for liquidity takers (in B$)

Sidebar: Capital Requirement

Deposit Requirements

Literature

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.

The Bigger Picture. Obstacles, Solutions, and Last Words

• Background: The S.E.C. suggests that their retail flow auction proposal could save retail traders $1.5B per year
  • average annual transaction costs currently are <$1.85B
  • \(\to\) retail would pay almost no spread.
  • retail would have to perfectly meet a "natural" counterparty.
  • How would that work? \(\to\) strong belief in magic

Where do the savings come from?

Obstacles

Summary

• Idea: Implement and AMM with equity trading
  • use learnings from two centuries of trading (info disclosures, price finding, etc)
  • design it to benefit liquidity providers and demanders
    • deploy assets to achieve more than buy-and-hold
    • trade cheaper than limit order book
• Propose process that is viable based on historical data