Discussion of
Decentralized Exchanges

Paper by Alfred Lehar & Christine Parlour
Discussion by Andreas Park


 

AFA 2022
 

By now everyone knows what Bitcoin is ...

How should one organize DEX trading?

the constant product pricing function

automated market maker

Price mechanism:

• \(X=\) contract balance of asset \(A\)
• \(Y=\) contract balance of asset \(B\)
• \(k=\) invariance factor
• key relation \(k=X\times\ Y\)

Prices

• when you want to sell \(x\le X\) you receive \(y\) that maintains invariance. 
• implied exchange rate: \(e=\frac{x}{y}\)

• maintain constant product post trade: \[k=(X+x)(Y-y)~ \Leftrightarrow~y=\frac{xY}{X+x}.\]

Dark side of DEx trading: Miner extractable value

Theoretical Framework

• asset that may see a value change
   
• liquidity providers (two?)
   
• liquidity trader: trades fixed quantity for reasons outside the model
   
• arbitrageur who takes advantage when
  • the value changes (picks off wrongly priced liquidity)
  • reverse price dislocations caused by liquidity traders

Theoretical Framework: Limit order book

• post discriminatory schedule
   
• benefit from liquidity traders
   
• lose against informed arbitrageur
   
• my understanding:
  • competition \(\to\) zero profits to LP
  • there may or may not be competition
    (depends on monitoring)

Theoretical Framework: Constant product pricing

• pricing rule fixed but level of liquidity provision is a choice
   
• What happens when?
  • Liquidity trade: will get reversed, LP gains 2 \(\times\) fee
  • Informed trade: LP loses from being picked off
     
• If information: Arbitrageur picks quantity that maximizes profits
   
• LP problem: balance picking off loss with fee gain
   
• (as I understand proof): pick level of liquidity such that expected fee gain = expected picking off loss
   
• NB: LPs do not directly compete with one another but more liquidity means flatter price curve 

Theoretical Framework: comparison

• LPs in both cases earn zero profits in expectation
   
• Arbitrageur's gain comes at the expense of liquidity trader
   
• \(\to\) How much does the liquidity trader pay for fixed quantity?
  • can be expressed in closed form 
  • clear cut result cannot be obtained but indicative statements based on monitoring costs (high costs \(\to\) AMM better)

Small observation

• It's expensive to adjust the "midprice" in UniSwap for LPs (must withdraw or add liquidity)
• As per authors: this happens rarely.
• Yet prices track LOB prices remarkably well!
• Although somewhat static, some dynamics must hold!

We can see that price impact on Binance almost always exceeds that on Uniswap.

Theoretical Framework: some suggestions

• Being monopolist LP comes at a cost. Maybe relate to arm's race?
   
• AMM has trade reversal by arbitrageur after liquditiy trade - why not the LOB?
   
• (Simpler) comparison result when LOB is guaranteed perfectly competitive?
   
• How robust is the constant product pricing liquidity provision choice (in terms of adequate risk compensation)?
  (Below: interesting empirical fact)

EMpirics

• very nice data collection for UniSwap
  • clear separations of "normal" trades (~80%) and complex trades
    • Not all token pairs are traded directly by one can trade across pairs
    • segment out "attack" trades (exploiting a protocol's vulnerability)
• Very nice description of UniSwap liquidity pool structure and stylized facts

EMpirics: most important result IMO

• Regression \(\to\) price impact on Binance almost always exceeds that on Uniswap.

between .5 and 1 bps

Small suggestions: maybe compure implicit spread measure for AMM for comparison

Mempool \(\Rightarrow\) Front-Running!

So what's the Problem?

Problem: MEMPOOL Frontrunning is intrinsically profitable

\(X\)

\(Y\)

Problem: MEMPOOL Frontrunning is intrinsically profitable

\(X\)

\(Y\)

\(Y-y'\)

\(X+x\)

\(Y-y'-y''\)

\(X+2x\)

\(y'=y''~\Rightarrow\)

front-running is not intrinsically profitable

What would be desirable?

Hard-Coded Market Making

1. Time consistent: cannot profitably split orders over time.
    
2. Front-running is not intrinsically profitable. 
    
3. Liquidity splitting invariance
    
4. No Multi-venue arbitrage/ping-pong trading

CPAMM

canonical

A simple calibration exercise

What does the data say?

What you will learn if you read my work

• With mempools, front-running is always a possibility
  • \(\Rightarrow\) must avoid design that makes it intrinsically profitable
  • \(\Rightarrow\) most swap exchanges make it so! (fees are only a band-aid)
• There is also
  • ping-pong trading
  • rule is ad hoc and nothing says that it has any relationship to demand and supply
• Empirically, mempool arbitrage and gas auctions push up gas costs to astronomical levels.
• There is a better way:
  • correctly designed "linear" (marginal) pricing makes front-running and ping-pong trading unprofitable 
  • \(\Rightarrow\) there is merit to traditional economic thinking even in DeFi \(\ldots\)