Andreas Park PRO
Professor of Finance at UofT
Katya Malinova and Andreas Park
Preliminaries & Some Motivation
Big Picture
Liquidity providers
Liquidity demander
Liquidity Pool
AMM pricing is mechanical:
No effect on the marginal price
Key Components
where do I find these plots? theblock.co/data/
limit order book | periodic auctions | AMM | |
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continuous trading |
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price discovery with orders | |||
risk sharing |
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passive liquidity provision | |||
price continuity |
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continuous liquidity | |||
sniping prevented |
Modelling Calibrate-able Liquidity Supply and Demand in an Automated Market Maker
Liquidity providers: positional losses
Buy and hold
Provided liquidity
in the pool
Basics of Liquidity Provision
Basic idea of liquidity provision: earn more on balanced flow than what you lose on price movement
\[\text{fee income} +\underbrace{\text{what I sold it for}-\text{value of net position}}_{\text{adverse selection loss}} \ge \text{cost of capital} \]
in AMMs:
protocol fee
in tradFi: bid-ask spread
Basics of Liquidity Provision
\[\underbrace{F \int DV \mu(DV) }_{\text{fees earned on}\atop \text{balanced flow}}+\int_0^\infty\underbrace{(\Delta c(q^*)-q^*p_t(R)}_{\text{adverse selection loss} \atop \text{when the return is {\it R}}} +\underbrace{F \cdot \Delta c(q^*))}_{\text{fees earned}\atop \text{from arbitrageurs}}~\phi(R)dR \ge 0.\]
\(q^* \) 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
\[\text{fee income} +\underbrace{\text{what I sold it for}-\text{value of net position}}_{\text{adverse selection loss}} \ge \text{cost of capital} \]
in AMMs:
protocol fee
in tradFi: bid-ask spread
Returns to liquidity providers
Similar to Lehar and Parlour (2023), Barbon & Ranaldo (2022).
adverse selection/positional loss when the return is \(R\)
fees earned
on informed
fees earned
on balanced flow
for reference:
For fixed balanced volume \(V\) & fee \(F\):
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)
Liquidity Demander's Decision & (optimal) AMM Fees
Result:
competitive liq provision\(\to\) there exists an optimal (min trading costs) fee \(>0\)
Similar to Lehar&Parlour (2023) and Hasbrouck, Riviera, Saleh (2023)
What's next?
How we think of the Implementation of an AMM for our Empirical Analysis
Approach: daily AMM deposits
you may say 24/7/365 is great -- but:
could be done with AMM rule with price \[\Delta c(q)=\frac{c}{a-2q}\]
uiii!
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
some volume may be intermediated
AMMs based on historical returns
Return distribution example: Tesla
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
\(\Rightarrow \) Need about 5% of the value of the shares deposited -- not 100% -- to cover up to a 10% return decline
Summary
@katyamalinova
malinovk@mcmaster.ca
slides.com/kmalinova
https://sites.google.com/site/katyamalinova/
Deposit Requirements
\(\Rightarrow \) 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
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/
By Andreas Park
Presentation at the Inaugural Innovations in Financial Services Conference (IFIS) 2024