trading Infrastructure

Blockchain Infrastructure

Application: decentralized trading with automated market makers

New institutions!

• passive "shared" liquidity provision
• new pricing function

automated market makers Pricing Function

• remove $Q$ of one token
• must add $P(Q)$ of other token 
• so that liquidity stays invariant $$L(X,Y)=L(X-Q,Y+P(Q))$$
• Common rule: constant product $$X\cdot Y=(X-Q)\cdot (Y+P(Q))$$

$X-Q'$

$X$

$X-Q$

$X-Q'-Q$

$Y$

$Y+P(Q)$

$Y+P(Q')$

$Y+P(Q)+P(Q'|X-Q)$

$P(Q')$

Problem: MEMPOOL Frontrunning is intrinsically profitable

Mempool $\Rightarrow$ Front-Running!

So what's the Problem?

How lucrative is liquidity provision?

• Math alert!
• AMMs are built for passive liquidity provision.
• $\to$ compare to passive holding
• Have $X_0,Y_0$ tokens at price $P_0=\frac{Y_0}{X_0}$.
• Initial investment in $: $X_0P_0+Y_0$

• Price moves to $P_1$
• Pool contains $X_1,Y_1$
• Dollar Profit/loss relative to passive holding: $$DL(P_0,P_1)=P_1X_1+Y_1-(P_1X_0+Y_0)$$

How lucrative is liquidity provision?

• When price moves, the contract loses $Q$ of the first tokens and gains $P(Q)$ of the second token so that $$P_1=\frac{Y_1}{X_1}=\frac{Y_0+P(Q)}{X_0-Q}$$
• Dollar Profit/loss relative to passive holding:

$$\begin{array}{rcl} DL(P_0,P_1)&=&P_1(X_0-Q)+Y_0+P(Q)-(P_1X_0+Y_0)\\ &=& P(Q)-P_1\cdot Q \end{array}$$​

Impermanent Loss

• If there are fees, then these are collected on the dollar amount traded.
• If volume is $v$ and the fee rate $f$, total fees are $f\cdot v$.
• The ratio $\nu=\frac{v}{P_0X_0+Y_0}$ is the turnover.
• Therefore IPL with fees is $$IPL_f(P_0,P_1)=\frac{DL}{2Y_0}+f\nu.$$

• The Impermanent Loss is the dollar loss per unit of capital
  
  $$IPL(P_0,P_1)=\frac{DL(P_0,P_1)}{P_0X_0+Y_0}.$$

Impermanent Loss

• If we use $P_1=Y_1/X_1$ and $R\equiv P_1/P_0$, we can solve
  $$P_1=\frac{Y+QY/(X-Q)}{X-Q}$$
  for $Q$ to get
  $$Q=X\left(1-\sqrt{R^{-1}}\right).$$

• In AMMs with constant product pricing $$XY=(X-Q)(Y+P(Q)).$$
• Therefore $$P(Q)=\frac{QY}{X-Q}.$$

• Subbing and simplifying gives $$IPL(R)=\sqrt{R}-\frac{1}{2}(1+R)$$

Impermanent Loss for Limit Order Books

• One can also approximate limit order book pricing

quantity

$Q$

$p(Q)$

• Here: $P(Q)=P_0Q+Q\times \frac{P_1-P_0}{2}.$
• Sparing you the derivation: $$IPL(R)=-\frac{1}{2}\frac{(R-1)^2}{3R+1}$$

Impermanent Loss in numbers

• Jan 2022 for ETH-USDT:
  • $R=0.7$ (30% price drop)
  • $X_0P_0=Y_0=\$118M$
  • Volume $1.6B $\nu=6.8$
  • Fee \(f=30bps)
• Impermanent Loss
  • Constant product: $-1.3\%$
  • Hypothetical Limit order book: $-1.45\%$
  • Fee income: $0.003\cdot 6.8=2\%$

Comparing Impermanent Loss for Return Distributions

Gamma Distribution

Constant
product
worse

Constant
product
better

Comparing Impermanent Loss for Return Distributions

Log-Normal Distribution of Returns

Constant
 product
is always worse

Canadian View

• Heavily influenced by QuadrigaCX debacle
• Custody questions paramount (how stored, how accessible, what risks)
• Custody = crypto trading platform  (CTP) offers a security (irrespective of the underlying)
• Requirement on due diligence when listing
• Current $50K limit for new purchases
• CTP must register with IIROC (which has no process)
• Only BitBuy is compliant, some are in "Sandbox"