Blockchain 2: Crypto Trading

Workflow Comparison:
Traditional Equity Trading vs Crypto

Some Stylized Facts on Crypto Trading

And then there is FTX

Core problem with QuadrigaCX:

Gerry Cotton sold people crypto he did not have.

Core problem FTX

used customer funds to prop up their market maker Alameida
=> they gave Alameida customer assets

In finance terms

call it unhedged "leverage"
hedge means that you have an off-setting position

Possible solution

prove your aggregate positions:
prove your assets & liabilities
if asset = liabilities $\Rightarrow$ solvency

Source: https://vitalik.ca/general/2022/11/19/proof_of_solvency.html

How do you organize DEX trading? EXAMPLE

automated market maker

sell 4 DAI for USDC

what price will therefore be quoted?

\begin{array}{rcl} k&=&\#\text{DAI}\times\#\text{USDC}\\ 16&=&(4+4)\times(4-y)\\ y&=&2 \end{array}

\begin{array}{rcl} k&=&\#\text{DAI}\times\#\text{USDC}\\ 16&=&(4+4)\times(4-y)\\ y&=&2 \end{array}

how many USDC?

e=x/y~~\to~~e=2

e=x/y~~\to~~e=2

How do you organize DEX trading? EXAMPLE

automated market maker

Problem: large "slippage" (or price impact)

imagine: deposit is 100 DAI & USDC:
- $k=100\times100=10,000~\to$ for $x=4$ need $y=100-10000/104=3.85$
imagine: deposit is 10,000 DAI & USDC:
- $k=10,000\times10,000=100,000,000~\to$ for $x=4$ need $y=10,000-100,000,000/10,004=3.998$
$\to$ the more money is in the contracts, the lower the price impact

$X-Q$

$X$

$Y+P(Q))$

$Y$

$c=X\cdot Y$

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 $X\cdot Y=(X-Q)\cdot (Y+P(Q))$

$X-Q'$

$X$

$X-Q$

$X-Q'-Q$

receive $-P(-Q|X-Q'-Q)$

pay $P(Q)$

$P(Q'|X-Q)$

$Y$

$Y+P(Q)$

$Y+P(Q')$

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

$P(Q')$

Problem: MEMPOOL Frontrunning is intrinsically profitable

Problem: MEMPOOL "Sandwich Attacks" intrinsically profitable

Example: Feb 2022, ETH-USDC trade

Trade:
- 100 ETH
- av price: $\$3,105$
Sandwich attack:
- trade 100 ETH, wait for original, sell 100 ETH:
  - cost: $-100\times \$3,105$
  - revenue: $100 \times \$3,121$
  - profit: $\$1,639$
- trade $\pm 1,000$ : profit $\approx \$17,000$
- trade $\pm 10,000$ : profit $\approx \$261,000$

original trade: 1.5 ETH for APE tokens
attacker:
- $\pm 1,668$ APE
- bought for 4.14 ETH, sold for 4.46 ETH

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

price

$Q$

$p(Q)$

$P^{\mathsf{dp}}(Q)=\int_0^Qp(q)~dq$

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

Option	Exchange-side risk	User-side risk
Custodial exchange (eg.Coinbase today)	User funds may be lost if there is a problem on the exchange side	Exchange can help recover account
Non-custodial exchange (e.g.Uniswap or dydx today)	User can withdraw even if exchange acts maliciously	User funds may be lost if userscrews up