Some Motivation

• Blockchain: borderless general purpose value and resource management tool

Big Picture

• DeFi: financial applications that run on blockchains

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

Trading Infrastructure

Application: decentralized trading with automated market makers

New institutions!

• passive "shared" liquidity provision
• new pricing function

Key Components

where do I find these plots? theblock.co/data/

How do people trade on-chain?

How do people trade on-chain?

AMM Theory: Price Functions

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

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

The Pricing Function

Transaction cost (here: price impact) of buying \(q\) 

\[\frac{p(q)-p(0)}{p_0}=\frac{q}{a-q}\]

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]

Provide-and-Forget Liquidity Provision

• 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"

• Asset with value \(V_0\) that may change to value \(V_t\)
   
• Efficient prices so that \(p_0=V_0\)
   
• Arbitrageurs who move price from \(p_0\) to \(p_t=V_t\) by trading a net quantity \(q^*\) at total cost \(\Delta c(q^*)\)
   
• Additional balanced (dumb) volume \(v\)
   
• Collect a fee \(F\) on all trading volume

Big Picture for Liquidity Provision

• Example: 
  • limit order books:
    • Fee=collect bid-ask spread on uninformed trades
    • amount paid = cumulative cost of multiple trades or walking the book
  • Auction market (a la Kyle)
    • Overcharge uninformed to balanced in expectation against possible informed

Basics of Liquidity Provision

Basics of Liquidity Provision

with expectations (and setting cost of capital to zero) and using \(p_t=V_t=Rp_0\)

Equilibrium Liquidity Supply

• Gives us an equilibrium deposit \(a^*\)
  
  \[a^*=\frac{F E[V]}{2}\underbrace{\left(-F\times E[|\sqrt{R}-1|/2]-E[\text{PL}]\right)^{-1}}_{=:C^\mathsf{CP}(\phi,F)^{-1}}.\]

\[\int\limits_0^\infty\left(\sqrt{R}-\frac{1}{2}\left(1+R\right)+\frac{F}{2}|\sqrt{R}-1|\right)~\phi(R)dR+F\frac{E[V]}{2a}.=0\]

• Gives us an equilibrium deposit \(a^*\)
  
  \[a^*=\frac{F E[V]}{2}\left(-F\times E[|\sqrt{R}-1|/2]-E[\text{PL}]\right)^{-1}.\]

Sidebar: we can quantify how much a PASSIVE LP loses when the price moves by \(R\)

\[\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

• For trading of a fixed quantity:
  
  \[\text{LD cost}=\text{AMM price impact} +\text{AMM fee} .\]

assume: liquidity providers add liquidity until they break even in expectation

Model Summary

• We can express the equilibrium choice for liquidity provision.
   
• We can measure the benefit for liquidity demanders who use the AMM.
   
• We can determine the fee that maximizes the liquidity demander benefit (it's not zero!)
   
• Next question:
  • How would this look like when applied to stock markets?
  • What are the optimal fees?
  • Is it feasible?
  • What are the empirical benefits?

Preliminaries & Some Motivation

Decentralized trading using automated market makers (AMM)

Liquidity Pool

Sidebar: Capital Requirement (or: what about UniSwap v3?)

Deposit Requirements

UniSwap v3

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

How we think of the Implementation of an AMM for our Empirical Analysis

Approach: daily AMM deposits

Background on Data

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

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)

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

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 loss for LPs will occur
  • it is conceptually limited
  • you will receive compensation through fees
     
• based on calibration data, LPs would break even with this specification
   
• \(\to\) a form of lower bar for liquidity cost?
   
• \(\to\) that lower bar is not reached maybe tells us something about the competitiveness/frictions in the present market?

UniSwap v3

UniSwap v3 has "concentrated liquidity provision"

\(p_0\)

How the price is determined

UniSwap v3: An Example

\(p_u=15\)

\(p_d=7\)

\(u=2\)

\(p_0=10\) (that's exogenous, not a choice)

marginal price

\[p^m(s)=\frac{\tilde{a}c}{(\tilde{a}-s)^2}.\]

Finding virtual liquidity factor \(\tilde{a}\)

marginal price

\[p^m(s)=\frac{\tilde{a}c}{(\tilde{a}-s)^2}.\]

\(p_u=15\)

\(p_d=7\)

\(u=2\)

\(p_0=10\) (that's exogenous, not a choice)

Finding the fourth parameter \(\Delta c(d)\)

\(p_u=15\)

\(p_d=7\)

\(u=2\)

marginal price

\[p^m(s)=\frac{\tilde{a}c}{(\tilde{a}-s)^2}.\]

Solutions

•  virtual liquidity factor: \[\tilde{a}=\frac{u}{1-\sqrt{p_0/p_u}}=u\cdot \frac{\sqrt{p_u}}{\sqrt{p_u}-\sqrt{p_0}}.\]

• required quantity to reach \(p_d\) \[p^m(-\tilde{d})=\frac{\tilde{a}\tilde{c}}{(\tilde{a}+\tilde{d})^2}=p_d.~~\Leftrightarrow~~d^*=-\tilde{a}(1-\sqrt{p_0/p_d})\]

• required cash deposit \[\Delta c(\tilde{d})=-\tilde{a}(1 - \sqrt{p_0/p_d})\sqrt{p_0p_d}=-u\sqrt{p_0p_d}\frac{1 - \sqrt{p_0/p_d}}{1 - \sqrt{p_0/p_u}}\]

• marginal pricing function \[p^m(q)=\frac{\tilde{a}^2p_0}{(\tilde{a}-q)^2}=\frac{u^2p_0p_u}{((u-q)\sqrt{p_u}+q\sqrt{p_0})^2}.\]

Numerical example

• Suppose \(p_0=\$10\).
• LP is willing to sell up  to \(u=100\)
• v2 liquidity:
  • Required cash deposit: \(c=100\times 10=1000\)
  • liquidity \(c\cdot a=100\times 1,000\)

Want to read more?

• A Primer on UniSwap v3 Math: As Easy as 1,2, v3 https://blog.uniswap.org/uniswap-v3-math-primer
• Elts 2012: Liquidity Math in UniSwap v3
• Hasbrouck, Riviera, Saleh (2023):
  • UniSwap Liquidity Provision is like a covered call,
  • whereas a standard market (Copeland & Galai JF 1983) is a short strangle (call at the ask, put at the bid)

Deposit Requirements

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.