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

Katya Malinova and Andreas Park

 

Women in Microstructure 

San Francisco, June 25 2023

 

Preliminaries & 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

Decentralized trading with automated market makers (AMM)

AMMs price: mechanical, based on the amount of  liquidity deposits.

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

\[a\]

\[a-q\]

\[c+\Delta c\]

\[c\]

 Given the initial deposits (a, c), the total cost \(\Delta c\)  for \(q\) units is such that:

Remove (buy) \(q\) units of the asset from the pool

Add (pay) \(\Delta c\) cash to the pool

\[a\]

asset (stock)

cash

liquidity after trade \(=\) liquidity before trade

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 (fee) income
  • Liquidity demanders:
    • predictable price
    • continuous trading
    • ample liquidity

Liquidity Supply and Demand in an Automated Market Maker

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

\[a\]

\[a-q\]

\[c+\Delta c\]

\[c\]

  • \(\Rightarrow\) Cost of buying \(q\) \[\Delta c=\frac{c}{a-q}\times q\]
  • Average price per unit \[p(q)=\frac{c}{a-q}\]

Initial marginal price = initial asset value: \[p(0) =\frac{c}{a}\] 

  • Average "spread" paid: \[\frac{p(q)}{p(0)}-1=\frac{q}{a-q}.\]
  • larger liquidity deposits \(a\) \(\Rightarrow\) 
    • lower costs for liquidity demanders

Most Common Pricing: Constant Product: \(L(a,c)=a\cdot c\)

Per unit \(p(q)\) as liquidity deposits increase

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

\[a\]

\[a-q\]

\[c+\Delta c\]

\[c\]

  • \(\Rightarrow\) Total cost of trading \(q\) \[\Delta c=\frac{c}{a-q}\times q\]
  • Average price per unit \[p(q)=\frac{c}{a-q}\]

Initial marginal price = initial asset value: \[p(0) =\frac{c}{a}\] 

Liquidity providers: positional losses

If trade is informed \(\Rightarrow\) new asset value  \[=\frac{c+\Delta c}{a-q}>p(q)\] 

  • larger liquidity deposits \(\Rightarrow\) 
    • larger positional losses for liquidity providers
  • Liquidity providers "undercharge" the informed

Basics of Liquidity Provision in an AMM

\[\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 \(\nearrow\)
    • \(\Rightarrow\) price \(\nearrow\)
      • \(\Rightarrow\) sold the asset for less than its worth
  • fundamental value \(\searrow\)
    • \(\Rightarrow\) price \(\searrow\)
      • \(\Rightarrow\) bought the asset for more than its worth

LP compensation in AMM: liquidity providers earn fees

  • must earn more fees on balanced flow than what they lose on price movement


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

in AMMs:
protocol fee

takes the role of the TradFi bid-ask spread

  • a liquidity trade
    • arbitrageurs 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

\[\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)\]

same as in Barbon & Ranaldo (2022)

Liquidity Provision Decision

  • Express liquidity provision as a "collective" deposit \(\alpha\) of firm's market cap
    • initial deposit  \(\alpha \nearrow \) \(\Rightarrow\) LP payoff \(\searrow\)

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

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

  • Competitive liquidity provision
  • \(\Rightarrow\) Find \(\alpha\) such that LPs break-even, as a function of
    • balanced volume
    • asset volatility/return distribution
    • fee

Liquidity Demander's Decision & (optimal) AMM Fees

  • Wants to trade some quantity \(q\).
     
  • Better off with AMM relative to traditional market if

    \[\text{AMM price impact} +\text{AMM fee} \le \text{bid-ask spread}.\]
  • Two opposing forces when \(F\nearrow\) for liquidity demand
    •  more liquidity provision
      \(\to\) lower price impact
    • more fees to pay
  • Finding: there exists an optimal (for the liquidity demander) fee \(>0\)

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

What's next?

  • Have:
    • equilibrium choices for competitive liquidity provision
    • fee that minimizes liquidity demander AMM costs (\(>0\))
  • Next:
    • Calibrate to stock markets
    • Optimal fees?
    • AMM Feasible? 
      • AMM costs at the optimal fee < bid-ask spread?

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 \(p_0=c/a\) 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
  • \(\to\) intermediated volume could disappear 
  • \(\to\) use volume/2
  • Some caveats, e.g.
    • arbitrageur volumes
    • larger volume if AMM has lower trading costs

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%.

Special Consideration 3:

Where to get 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

Average of the market cap to be deposited for competitive liquidity provision: \(\bar{\alpha}\approx 2\%\)

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

average: 94% of days AMM is cheaper than LOB for liq demanders

average savings: 16 bps

average daily: $9.5K

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

average annual saving: $2.4 million

implied "excess depth" on AMM relative to the traditional market

Sidebar: Capital Requirement

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: AMM requires an off-setting cash amount: \(c =a\cdot p(0)\).
     
  • 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

\(\Rightarrow \) Need about 5% of the value of the shares deposited -- not 100% --  to cover up to a 10% return decline

An alternative to -10% circuit breaker:

max cash needed based on long-run  past average R \(-\) 2 std

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:
    • predictable price
    • continuous trading
    • ample liquidity

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.

@katyamalinova

malinovk@mcmaster.ca

slides.com/kmalinova

https://sites.google.com/site/katyamalinova/

@financeUTM

andreas.park@rotman.utoronto.ca

slides.com/ap248

sites.google.com/site/parkandreas/

youtube.com/user/andreaspark2812/

Learning from DeFi: Are AMMs better? WiMM

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