Learning from DeFi: Would Automated  
Market Makers  Improve Equity Trading?

• one new market institution: automated market makers

AMMs price: mechanical, based on the amount of  liquidity deposits.

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

• $\Rightarrow$ Cost of buying $q$ $$\Delta c=\frac{c}{a-q}\times q$$
• Average price per unit $$p(q)=\frac{c}{a-q}$$

Initial marginal price = initial asset value: $$p(0) =\frac{c}{a}$$ 

• Average "spread" paid: $$\frac{p(q)}{p(0)}-1=\frac{q}{a-q}.$$

• larger liquidity deposits $a$ $\Rightarrow$ 
  • lower costs for liquidity demanders

If trade is informed $\Rightarrow$ new asset value  $$=\frac{c+\Delta c}{a-q}>p(q)$$ 

• larger liquidity deposits $\Rightarrow$ 
  • larger positional losses for liquidity providers

• Liquidity providers "undercharge" the informed

$$\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 $R$

LP compensation in AMM: liquidity providers earn fees

• must earn more fees on balanced flow than what they lose on price movement

$$\text{fee income} +\underbrace{\text{what LP sold it for}-\text{value of net position}}_{\text{adverse selection loss}} \ge 0$$

$$\frac{1}{\text{initial deposit}}\int_0^\infty(\Delta c(q^*)-q^*p_t(R)+F \cdot \Delta c(q^*))~\phi(R)dR +\frac{F p_0 V}{\text{initial deposit}}\ge 0$$

$$\int_0^\infty\left(\frac{\Delta c(q^*)-q^*p_t(R)}{\text{initial deposit}} +F \cdot \frac{\Delta c(q^*)}{\text{initial deposit}}\right)~\phi(R)dR +\frac{F p_0 V}{\text{initial deposit}}\ge 0$$

closed form functions of $R$ only
(see Barbon & Ranaldo (2022))

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%

• Express liquidity provision as a "collective" deposit $\alpha$ of firm's market cap
  • initial deposit  $\alpha \nearrow$ $\Rightarrow$ LP payoff $\searrow$

$$\text{what LP sold it for}-\text{value of net position}+\text{fee income} \ge 0$$

• Competitive liquidity provision
• $\Rightarrow$ Find $\alpha$ such that LPs break-even, as a function of
  • balanced volume
  • asset volatility/return distribution
  • fee

• Two opposing forces when $F\nearrow$ for liquidity demand
  •  more liquidity provision
    $\to$ lower price impact
  • more fees to pay
• Finding: there exists an optimal (for the liquidity demander) fee $>0$

$$F^\pi=\frac{1}{E[|\sqrt{R}-1|/2]+V}\left(-2q\ E[\text{ILLRAS}]+ \sqrt{-2qV\ E[\text{ILLRAS}]}\right).$$

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

Return distribution example: Microsoft

• 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