• Blockchain: borderless general purpose value and resource management tool

Preliminaries

• DeFi: financial applications that run on blockchains

• \(\Rightarrow\) brought new ideas and tools

• one new market institution: automated market makers

Quick Recap: Automated Market Makers (AMM) 

Liquidity Pool

Key Components

Modelling Calibrate-able Liquidity Supply and Demand in an Automated Market Maker

Facts about modelling liquidity provision

• You provide liquidity when the marginal price at the AMM and elsewhere is \(p(t_0)=p.\)
   
• You make a decision what to do next at \(t_1>t_0\)
   
• Many things can happen, for instance:
  • AMM trades
  • price moves in the broader markets

Two broad approaches for modelling liquidity provision

• Aoyagi and Ito (2021), Cartea, Drissi, Monga (2023), Milionis, Moallemi, Roughgarden, and Zhang (2022), Milionis, Moallemi, Roughgarden (2024)

• Lehar and Parlour (2021), Hasbrouck, Riviera, Saleh (2023), Park (2021), Malinova and Park (2023)[this paper], Cao, Falk, Kogan, Tsoukalas (2025)

Provide-and-forget liquidity provision

• LPs deposit asset & cash when the asset price is \(p\)
• While in the pool: balanced volume \(V\) + order imbalance \(p\to p'\)
• Withdraw at price \(p'\)

Buy and hold

Provided liquidity

in the pool

• \(\Rightarrow\)  positional loss relative to a "buy-and-hold" when \(p'\ne p\)
  • adverse selection

• Arbitrageurs monitor the pool 
• \(\Rightarrow\) LPs deposit/writhdraw at efficient prices!

Modelling provide-and-forget liquidity provision

\[\int_0^\infty(\underbrace{\Delta c(q^*)-q^*Rp )}_{\text{adverse selection loss when the return is \(R\)}}+ \underbrace{F \cdot \Delta c(q^*))}_{\text{fees earned on informed}}~\phi(R)dR + \underbrace{F\cdot p\cdot V}_{\text{fees earned on balanced flow}}\ge \text{cost of capital}.\]

\(q^* \)= function of \(R\) = what (informed) arbitrageurs trade (net) to move the price to reflect \(R=p'/p\)

Basic idea: LP must earn more fees on balanced flow (in AMM: fees) than what they lose on price movement:

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

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

Expected returns to liquidity providers over the "deposit period"

• \(R\) = 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 \(L\) when the asset return is \(R=p'/p\)

Liquidity Demander's Decision & (optimal) AMM Fees

• This calibration: better off with AMM relative to traditional market if
  \[\text{AMM price impact} +\text{AMM fee } F \le \text{bid-ask spread}.\]

Sidebar: modelling adverse selection in traditional microstructure vs. AMM

Traditional microstructure models:

• tick-by-tick analysis
• market makers earn bid-ask spreads 
  • able to liquidate positions at (favourable to them) inefficient prices
• key modelling variable: informed vs. uninformed order flow (PIN)

AMM (provide-and-forget version):

• liquidity providers are passive, long(er) term (daily horizons)
• liquidity deposits and withdrawals occur at efficient prices 
  • no positional gains 
• key modelling variables
  • end-of-day prices (assumed efficient) \(\to\) open-to-close returns
  • balanced volume

What's next?

• Calibrate to stock markets
• AMM Feasible? 
  • Are the AMM costs at the optimal fee \(F\) \(<\) bid-ask spread?

Approach: daily AMM deposits

Background on Data

(Most of) the slides coming up: AMMs based on historical returns 

• 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

AMMs are better on about 85% of trading days

AMMs with"ad hoc" one-day backward look

relative savings: what fraction of transactions costs would an AMM save? \(\to\) about 30%

AMMs with"ad hoc" one-day backward look

average per-day savings: 16 bps

average daily: $9.5K

average annual saving: $2.4 million

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

Sidebar 1: constant product AMM  vs Traditional Pricing Models

Basic Requirements for "unconstrained" two-asset liquidity pool

• Two assets:
  • cash and an asset 
  • deposits in pool of \(c\) cash and \(a\) asset
• AMMs should be self-contained: pricing rule based on
  • deposits \(a,c\) and
  • liquidity demanded \(q\)
• implied value of the asset at deposit: \[p=\frac{c}{a}.\]
• \(\to\) marginal price: an infinitesimal unit should cost \[p^m=p\]

What does an AMM need?

• What happens with trades? 
  • a trade of \(q\in(-\infty,a),\) will cost \(\Delta c(q)\)
  • liquidity changes
    • \(a\) \(\to\) \(a-q\)
    • \(c\) \(\to\) \(c+\Delta c(q)\)
  • \(\to\) marginal price after trade of \(q\) is
    
    \[p^m(0)=\frac{c}{a}~~\to~~p^m(q)=\frac{c+\Delta c(q)}{a-q}.\] 

Pricing Rules from Traditional Markets: Uniform Price

• cost is price \(\times\) quantity: \[\Delta c(q)=q\cdot p^m(q)\]
• plus assume average price \(=\) marginal price after trade
  
  \[p^m(q)=\frac{c+q\cdot p^m(q)}{a-q}~~\Leftrightarrow~~p^m(q)=\frac{c}{a-2q}.\]
   
• formalized and analyzed by Canidio and Fritsch (2023) and implemented in COW-Swap

• execute all trades at a marginal price after a batch
• \(\to\) no positional loss for LPs

Pricing Rules from Traditional Markets: Limit Order Book

• Liquidity providers submit sell orders \(s_1(p_1,q_1),\ldots,s(p_n,q_n)\)
• Matching engine sorts order increasing by price
• someone who  wants to buy quantity \(q\)
  
  \[\Delta c(q)=\sum_{i=1}^mq_i\cdot p_i.\]
   
• Assume the price-quantity curve can be expressed by a continuous marginal price function \(p^m(s) \)

\[\to \Delta c(q)=\int_0^qp^m(s) ds.\]

note: if you do uniform pricing and split your order into infinitesimal units, then you pay the limit order price

 LOB-style pricing in AMM \(\to\) constant product pricing

• Continuity \(\to\) price of "final" marginal unit coincides with price of  "first" marginal unit of next trade \[\Delta c(q)=\int_0^q p^m(s) ds~\to~\Delta c'(q)=p^m(q).\]
• \(\to\) a differential equation
• \[p^m(q)=\frac{c+\Delta c(q) }{a-q}~~\Leftrightarrow~~\Delta c'(q)=\frac{c+\Delta c(q)}{a-q},\]
• Also  "first" unit costs nothing \(\Delta c(0)=0\)
• \(\to\) solution \[\Delta c(q)=q\cdot\frac{c}{a-q}.\]

AMM constant product pricing  is the same as a LOB under minimal assumptions:

• prices are continuous 
• the marginal price are determined by the ratio of the liquidity deposits in the pool

Most Common Pricing Rule in DeFi: Constant Product

• For AMM: assume "constant" liquidity
• Liquidity defined as product of deposits
• \(\to\) liquidity before and after trade coincides
  \[a\cdot c= (a-q) (c+\Delta c(q))\]
• This implies cost 
  \[\Delta c(q)=q \frac{c}{a-q},\] average price \[p(q)=\frac{c}{a-q}\]
  and marginal price \[p^m(q)=\frac{c+\frac{cq}{a-q}}{a-q}=\frac{ac}{(a-q)^2}.\]

Insight: AMM pricing function is the same as a limit order book when we require

• that prices are continuous and
• that prices are determined by liquidity deposits only

Sidebar 2: Cash deposit requirements

• Asset provision is not a problem: enough idle liquidity
• But: AMM requires off-setting cash: \(c =a\cdot p\).
  • ​Cash is not free: at 4.5% annual rate,  1.75bps per day.
  • Adds to fees
• Several solutions:
  1. Narrow a range of returns for which to provide liquidity (akin to Uniswap V3) or circuit breakers
  2. Other mechanisms being developed 
     • ​Use the same cash for many assets (Balancer protocol)

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

Sidebar 1: Deposit Requirements/Uniswap V3

Sidebar 1: Deposit Requirements/Uniswap V3

• Uniswap V3 allow LPs to provide liquidity only when price is in interval \([\underline{p},\overline{p}]\)
   
• \(\to\) conceptually similar to circuit breakers
   
• Hasbrouck, Riviera, Saleh ( 2025): concerntrated liquidity provision \(\to\) payoff  same as a covered call.
  • liquidity provision decision requires constant adjustments,
  • indirectly related to the concept of LVR=loss-vs-rebalancing)
• modelling caveat: because price interval is a choice, too many degrees of freedom

Summary

• Our question:
  1. Can an economically viable AMM be designed for current equity markets?
  2. Would such an AMM improve current markets?

Summary: alternative view

• AMMs force LPs to optimize against a prescribed, fully contingent limit order book function
   
• the positional loss and the break-even condition that we derive does not require strong assumptions such as behavior or number of informed traders
  • a positonal loss for LPs will occur
  • LPs will receive compensation through fees
     
• based on calibration data, LPs would break even with this specification
   
• \(\to\) a form of lower bar for trading costs? Compensating for adv selection only
   
• \(\to\) that lower bar is not reached maybe tells us something about the competitiveness/frictions in the present market?

Basic Requirements for "unconstrained" two-asset liquidity pool

• Two assets:
  • cash and an asset 
  • deposits in pool of \(c\) cash and \(a\) asset
• AMMs should be self-contained: pricing rule based on
  • deposits \(a,c\) and
  • liquidity demanded \(q\)
• implied value of the asset at deposit: \[p=\frac{c}{a}.\]
• marginal price: an infinitesimal unit should cost \[p^m=p\]

What does an AMM need?

• trades create linear changes:
  • a trade of \(q\in(-\infty,a),\) will cost \(\Delta c(q)\)
  • liquidity changes
    • \(a\) \(\to\) \(a-q\)
    • \(c\) \(\to\) \(c+\Delta c(q)\)
  • \(\to\) marginal price after trade of \(q\) is
    
    \[p^m(0)=\frac{c}{a}~~\to~~p^m(q)=\frac{c+\Delta c(q)}{a-q}.\] 

Some Pricing Rules from Traditional Markets: Uniform Price

• assume cost is price \(\times\) quantity: \[\Delta c(q)=q\cdot p^m(q)\]
• plus assume average price \(=\) marginal price after trade
  
  \[p^m(q)=\frac{c+q\cdot p^m(q)}{a-q}~~\Leftrightarrow~~p^m(q)=\frac{c}{a-2q}.\]
   
• formalized and analyzed by Canidio and Fritsch (2023) and implemented in COW-Swap

\(q=2\)

Example: \(\Delta c(q)= q\times p^m(q)=2\times 15.5\)

Main pricing rule in stock exchanges: limit order book

Some Pricing Rules from Traditional Markets: Limit Order Book

• For AMM: assume continuity \(\to\) price of "final" marginal unit coincides with price of  "first" marginal unit of next trade \[\Delta c(q)=\int_0^q p^m(s) ds~\to~\Delta c'(q)=p^m(q).\]
• Also "first" unit costs nothing \(\Delta c(0)=0\)
• This gives us a differential equation
  \[p^m(q)=\frac{c+\Delta c(q) }{a-q}~~\Leftrightarrow~~\Delta c'(q)=\frac{c+\Delta c(q)}{a-q},\]
• with solution \[\Delta c(q)=q\cdot\frac{c}{a-q}.\]

Most Common Pricing Rule in DeFi: Constant Product

• With \[a\cdot c= (a-q) (c+\Delta c(q))\]
   
• we compute cost \[\Delta c(q)=q\cdot  \frac{c}{a-q},\]
   
• and marginal price \[p^m(q)=\frac{c+\frac{cq}{a-q}}{a-q}=\frac{ac}{(a-q)^2}.\]

Some insights on pricing functions

• Price \(\times\) quantity
  • nice feature:
    • regret-free price: if price move is due to information, you have no positional loss
  • poor feature:
    • you can always save by order-splitting (formally shown by Canidio and Fritsch (2023)) 
    • limit price for splitting = limit order book price

• LOB/CP
  • nice feature:
    • order splitting doesn't pay
  • poor feature:
    • prices not regret-free: if price move is due to information, you always lose

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

Optimal fee \(F^\pi\)

average benefits liquidity provider in bps (average=0)

with circuit breakers: what fraction of \(\alpha\) needs to be deposited as cash

quoted spread minus AMM price impact minus AMM fee (all measured in bps)

Optimal fee \(F^\pi\)

average benefits liquidity provider in bps (average=0)

actually needed cash as fraction of "headline" amount

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%