Andreas Park PRO
Professor of Finance at UofT
Katya Malinova and Andreas Park
Agenda
Why did we write this paper?
Seriously?
0. Cynicism aside - big question: Can we improve liquidity for smaller listings?
Some Motivation
Basic Idea
payments network
Stock Exchange
Clearing House
custodian
custodian
beneficial ownership record
seller
buyer
Broker
Broker
Broker
Exchange
Internalizer
Wholeseller
Darkpool
Venue
Settlement
AMM Pricing
Constant Liquidity (Product) AMM
Key Components
Liquidity Supply and Demand in an Automated Market Maker
The Pricing Function
Liquidity Deposit \(\Rightarrow\) slope of the price curve
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
\[\text{fee income} +\underbrace{\text{what I sold it for}-\text{value of net position}}_{\text{adverse selection loss}} \ge 0 \]
in AMMs:
protocol fee
in tradFi: bid-ask spread
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:
\[\frac{\text{adverse selection loss when the return is \(R\)}}{\text{initial deposit}}=\sqrt{R}-\frac{1}{2}(R+1)\]
see Barbon & Ranaldo (2022)
Basics of Liquidity Provision
Liquidity provision measured as "collective" deposit \(\alpha\) of firm's market cap as function of
\[E[\text{IILRAS}(R)]+F\cdot E[\text{another function of }R]+F\cdot \frac{\text{dollar volume}}{\text{initial deposit}}\ge 0.\]
\[\text{what I sold it for}-\text{value of net position}+\text{fee income} \ge 0 \]
The Decision of the Liquidity Demander
\[F^\pi=\frac{1}{E[|\sqrt{R}-1|/2]+V}\left(-2q\ E[\text{ILLRAS}]+ \sqrt{-2qV\ E[\text{ILLRAS}]}\right).\]
Model Summary
How we think of the Implementation of an AMM for our Empirical Analysis
Approach: daily AMM deposits
Background on Data
some volume may be intermediated
AMMs that's true to the "model"
Return distribution example: Microsoft
Return distribution example: Tesla
\(\bar{\alpha}\approx 2\%\)
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
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
@financeUTM
andreas.park@rotman.utoronto.ca
slides.com/ap248
sites.google.com/site/parkandreas/
youtube.com/user/andreaspark2812/
By Andreas Park
SNB presentation