Basic Requirements

• Two assets:
  • cash and an asset 
  • deposits in pool of $c$ cash and $a$ asset
• AMMs should be self-contained: pricing rule based on liquidity only on $a,c$
• implied price of the asset: $$p=\frac{c}{a}.$$
• marginal price: an infinitesimal unit should cost $$p^m=p$$
• trades involve changes of the asset in return for change in cash and transform $a,c$
• consider linear changes:
  • a trade of $q\in(-\infty,a),$ costs $\Delta c(q)$
  • liquidity changes to $a-q,c+\Delta c(q)$
  • $\to$ marginal price after trade of $q$ is $$p^m(q)=\frac{c+\Delta c(q)}{a-q}.$$ 

Some Pricing Rules from Traditional Markets: Uniform Price

• assume cost is price $\times$ quantity: $$\Delta c(q)=q\cdot p(q)$$
• plus assume average price $=$ marginal price
  
  $$p(q)=\frac{c+q\cdot p(q)}{a-q}~~\Leftrightarrow~~p(q)=\frac{c}{a-2q}.$$
   
• formalized and analyzed by Canidio and Fritsch (2023) and implemented in COW-Swap
• Concern: a trader can split orders into infinitesimal quantities and earn a higher payoff.

$q=2$

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

Some 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
  
  $$q\in\left(\sum_{i=1}^{m-1} q_i,\sum_{i=1}^{m} q_i\right]~~~\text{pays}~~\Delta c(q)=\sum_{i=1}^mq_i\cdot p_i.$$
   
• For a simpler exposition,  assume that the price-quantity curve can be expressed by a continuous marginal price function $\rho(s)$ so that

$$\Delta c(q)=\int_0^q\rho(s) ds.$$

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 $$\Delta c(q)=\int_0^q \rho(s) ds~\to~\Delta c'(q)=\rho(q)=p(q),~~ \Delta c(0)=0.$$
• Also "first" unit costs nothing $\Delta c(q)=0$
• This brings differential equation
• $$p(q)=\frac{c+\int_0^q \rho(s) ds}{a-q}~~\Leftrightarrow~~\Delta c'(q)=\frac{c+\Delta c(q)}{a-q},$$
  with solution $$\Delta c(q)=q\cdot\frac{c}{a-q}.$$
• Or, we can write $$\Delta c(q)=\int\limits_0^q p(s)ds=\int\limits_0^q\frac{ac}{(a-q)^2}ds=\frac{cq}{a-q}.$$

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 $$p^m(q)=\frac{c+\frac{cq}{a-q}}{a-q}=\frac{ac}{(a-q)^2}.$$