• one new market institution: automated market makers

Liquidity providers

Liquidity demander

No effect on the marginal price

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

Basic idea of liquidity provision: earn more on balanced flow than what you lose on price movement

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

\[\underbrace{F \int DV \mu(DV) }_{\text{fees earned on}\atop \text{balanced flow}}+\int_0^\infty\underbrace{(\Delta c(q^*)-q^*p_t(R)}_{\text{adverse selection loss} \atop \text{when the return is {\it R}}} +\underbrace{F \cdot \Delta c(q^*))}_{\text{fees earned}\atop \text{from arbitrageurs}}~\phi(R)dR \ge 0.\]

\(q^* \) 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

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

adverse selection/positional loss when the return is \(R\)

fees earned

on informed

fees earned

on balanced flow

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

for orientation:

• If the stock price drops by 10% the incremental loss for liquidity providers is 13 basis points on their deposit
  • \(\to\) total loss=-10.13%
• If the stock price rises by 10%, the liquidity provider gains 12 basis points less on the deposit
  • \(\to\) total gain =9.88%

• Two opposing forces when 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)

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.

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

• (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

• 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

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

Need about 10% of market cap in liquidity deposits to make this work

Only need about 5% of the 10% marketcap amount in cash

AMMs are better on about 85% of trading days

theoretical annual savings in transactions costs is about $15B