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

Katya Malinova and Andreas Park

 



 

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
limit order book periodic auctions AMM
continuous
trading
price discovery with orders
risk
sharing
passive liquidity provision
price
continuity
continuous liquidity
sniping
prevented

Liquidity Supply and Demand in an Automated Market Maker

To answer the question of whether an AMM can work in traditional markets we need a model to calibrate against

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"

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

Returns to 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 \(F\)

Basics of Liquidity Provision

\[\text{LP payoff}=\text{what I sold it for}-\text{value of net position}+\text{fee income}\]

  • \(R\) = 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%

positional loss

Returns to Liquidity Provision

For fixed balanced volume \(V\) and fee \(F\)

  • Larger pool \(\to\) smaller share of fees
  • \(\to\) LP expected return \(\searrow\) pool size

Competitive liquidity provision

  • \(\Rightarrow\) upper bound on pool size above which LPs lose money
  • Here: characterize this amount by the asset's market cap to be deposited to the pool (notation: \(\overline{\alpha}\))

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

  • Two opposing forces in equilibrium for 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)

  • For trading of a fixed quantity:

    \[\text{LD cost}=\text{AMM price impact} +\text{AMM fee} .\]

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

What's next?

  • Given return profile and balanced volume:
    • have fee that minimizes liquidity demander AMM costs (\(>0\)) 
  • Given return profile, balanced volume, and fee:
    • zero expected return condition for liquidity provision
      • in competitive market \(\to\) \(=\) liquidity provided
  • Next:
    • Calibrate to stock markets
    • AMM Feasible? 
      • LP threshold \(\overline{\alpha}\le 1\)?
      • LD costs in AMM 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: Tesla

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

Sanity check: do liquidity providers break even on average?

  • average \(F^\pi=11\)bps

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

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

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

average benefits liquidity provider in bps (average=0)

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

Optimal fee \(F^\pi\)

\(\overline{\alpha}\) 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

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

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

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.

  • 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! 
    • \(\to\) 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

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/