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

Katya Malinova  &   Andreas Park

 

11th Annual Conference on Financial Market Regulation

Washington DC, May 9-10, 2024

McMaster
University		 University of Toronto

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 using automated market makers (AMM)

Liquidity providers

Liquidity demander

Liquidity Pool

AMM pricing is mechanical:

  • determined by the amounts of deposits 
  • most common:
    • constant product
    • #USDC \(\times\) #ETH = const

No effect on the marginal price

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!
  • Liquidity providers:
    • pro-rated 
      • trading fee income
      • risk
    • use assets that they own to earn passive (fee) income
      • retain exposure to the asset
  • Liquidity demanders:
    • predictable price
    • continuous trading
    • ample liquidity

Liquidity Supply and Demand in an Automated Market Maker

Liquidity providers: positional losses

  • Deposit asset & cash when the asset price is \(p\)
  • Withdraw at price \(p'\ne p\) 

 

 

Buy and hold

Provided liquidity

in the pool

  • Why?
    • adverse selection losses
    • arbitrageurs trade to rebalance the pool 
  • \(\to\) always positional loss relative to a "buy-and-hold"

Returns to liquidity providers

  • \(R\) = asset return
  • F = trading fee
  • V = balanced volume
  • a = size of the liquidity pool

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

(incremental) adverse selection loss when the return is \(R\)

fees earned

on informed

fees earned

on balanced flow

for reference:

  • If the asset price drops by 10% the incremental loss for liquidity providers is 13 basis points on their deposit
    • \(\to\) total loss=-10.13%
  • If the asset price rises by 10%, the liquidity provider gains 12 basis points less on the deposit
    • \(\to\) total gain =9.88%

For fixed  balanced volume \(V\) & fee \(F\):

  • Larger pool size \(\to\) smaller shares of the fees
  • \(\to \) LP expected return \(\searrow\) in pool size
  • Competitive liquidty provision:
    • \(\to\) find the upper-bound on pool size above which LPs lose money
    • we characterize this by \(\bar{\alpha}\) - fraction of the asset's market cap to be deposited to the pool

Liquidity Demander's Decision & (optimal) AMM Fees

  • 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 fee \(F\nearrow\) 
    1.  more liquidity provision
      \(\to\) lower price impact
    2. more fees to pay

Result:

competitive liq provision\(\to\) there exists an optimal (min trading costs) fee \(>0\)

  • \(\to\) derive closed form solution for competitive liquidity provision
  • depends on return distribution, balanced volume, quantity demanded

Similar to Lehar&Parlour (2023) and Hasbrouck, Riviera, Saleh (2023)

What's next?

  • 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 this 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 the 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: 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

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 not a problem: 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.

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

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! 
    • \(\to\) no (overnight) inventory costs
    • \(\to\) use idle capital
    • \(\to\) + better risk sharing

@katyamalinova

malinovk@mcmaster.ca

slides.com/kmalinova

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

Deposit Requirements

  • 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

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.

@financeUTM

andreas.park@rotman.utoronto.ca

slides.com/ap248

sites.google.com/site/parkandreas/

youtube.com/user/andreaspark2812/

Learning from DeFi: Are AMMs better?

By Katya Malinova

Learning from DeFi: Are AMMs better?

Presentation at the 11th Annual Conference on Financial Market Regulation (CFMR 2024), May 9-10, 2024. SEC, Washington DC.

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