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

\[\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 \(R\)

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

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

in AMMs:
protocol fee

in tradFi: bid-ask spread

Similar to Lehar and Parlour (2023), Barbon & Ranaldo (2022).

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))

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

  1. AMMs close overnight.
     
  2. Market: opening auction \(\to\) \(p_0\)
     
  3. Determine: optimal fee; submit liquidity \(a,c\)
    at ratio \(p_0=a/c\) until break even \(\alpha=\overline{\alpha}\)
     
  4. Liquidity locked for day
     
  5. At EOD release deposits and fees
     
  6. 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

Return distribution example: Microsoft

Return distribution example: Tesla

  • average \(F^\pi=11\)bps

\(\bar{\alpha}\approx 2\%\)

almost break even on average (average loss 0.2bps \(\approx0\))

average: 94% of days AMM is better than LOB

average savings: 16 bps

average daily: $9.5K

saves around 45% of transaction costs (measured in bid-ask spread)

average annual saving: $2.4 million

Optimally Designed AMMs with
"ad hoc" one-day backward look

Optimal fee \(F^\pi\)

average benefits liquidity provider in bps (average=0)

Insight: Theory is OK - LP's about break even

\(\overline{\alpha}\) for \(F=F^\pi\)

Need about 10% of market cap in liquidity deposits to make this work

actually needed cash as fraction of "headline" amount

Only need about 5% of the 10% marketcap amount in cash

AMMs are better on about 85% of trading days

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

relative savings: what fraction of transactions costs would an AMM save? \(\to\) about 30%

theoretical annual savings in transactions costs is about $15B

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 and Last Words

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

@financeUTM

andreas.park@rotman.utoronto.ca

slides.com/ap248

sites.google.com/site/parkandreas/

youtube.com/user/andreaspark2812/

Learning from DeFi: Are AMMs better? Fed NY FinTech/AI Conference

By Andreas Park

Learning from DeFi: Are AMMs better? Fed NY FinTech/AI Conference

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