Withdraw-Swap

To download notebook to this tutorial, see here

import os
import math as mth
import numpy as np
from termcolor import colored

from uniswappy import *

user_nm = 'user0'
eth_amount = 1000
tkn_amount = 100000

Problem Defined

Given the definition of constant product trading (CPT) as:

\begin{equation} (x-\Delta x)(y - \gamma\Delta y) = L^2 \end{equation}

where

$x$ -> reserve0 (r0)
$y$ -> reserve1 (r1)
$\Delta x$ -> swap x (a0)
$\Delta y$ -> swap y (a1)
$L$ -> total supply
$\gamma$ -> fee $\left(ie, \frac{997}{1000} \right)$

We define the indexing problem via the following linear system of equations:

\begin{equation} \Delta x = \frac{\Delta L x}{L} \tag{Eq. 1} \end{equation}

\begin{equation} \Delta y = \frac{\Delta L y}{L} \tag{Eq. 2} \end{equation}

\begin{equation} \Delta y_{(i)} = \Delta y + \frac{\gamma \Delta x(y-\Delta y)}{(x - \Delta x) + \gamma \Delta x} \tag{Eq. 3} \end{equation}

where

$\Delta y_{(i)}$ -> indexed token
$\Delta L$ -> liquidity deposit

tkn = ERC20("TKN", "0x111")
eth = ERC20("ETH", "0x09")
exchg_data = UniswapExchangeData(tkn0 = eth, tkn1 = tkn, symbol="LP", address="0x011")

factory = UniswapFactory("ETH pool factory", "0x2")
lp = factory.deploy(exchg_data)
lp.add_liquidity(user_nm, eth_amount+100, tkn_amount, eth_amount+100, tkn_amount)

print('***\nInitial LP\n***')
lp.summary()

amt_out = 100
token_out = eth
trading_token = tkn
user_nm1 = 'user0'
#rate = 0

# Step 1: withdrawal
p_out = 0.5
removeLiq = RemoveLiquidity()
res = removeLiq.apply(lp, token_out, user_nm1, p_out*amt_out)

print('***\nLP post step 1\n***')
lp.summary()

# Step 2: swap
out = Swap().apply(lp, trading_token, user_nm1, res[trading_token.token_name])

print('***\nLP post step 2\n***')
lp.summary()

withdrawn = res[eth.token_name] + out

print('Total withdrawn is {:.6f} + {:.6f} = {:.6f} ETH'.format(p_out*amt_out, out, withdrawn))
print('Of the requested {} ETH, a total of {:.6f} ETH has been withdrawn when using a 50/50 split'.format(amt_out, withdrawn))

*** Initial LP *** Exchange ETH-TKN (LP) Reserves: ETH = 1100.0, TKN = 100000.0 Liquidity: 10488.088481701516 *** LP post step 1 *** Exchange ETH-TKN (LP) Reserves: ETH = 1050.0, TKN = 95454.54545454546 Liquidity: 10011.35718707872 *** LP post step 2 *** Exchange ETH-TKN (LP) Reserves: ETH = 1002.4094194662908, TKN = 100000.0 Liquidity: 10011.35718707872 Total withdrawn is 50.000000 + 47.590581 = 97.590581 ETH Of the requested 100 ETH, a total of 97.590581 ETH has been withdrawn when using a 50/50 split

Solve Problem

Using the system of equations outlined in the indexing problem, Eq. 3 can be rearranged as:

\begin{equation} (\Delta y_{(i)}x) - (\Delta y_{(i)}\Delta x) + (\gamma \Delta y_{(i)} \Delta x) - (\Delta y x) + (\Delta y\Delta x) - (\gamma y\Delta x) = 0 \end{equation}

Plug Eq. 1 and Eq. 2 into above, and we get:

\begin{equation} (\Delta y_{(i)} x) - (\frac{\Delta y_{(i)} \Delta L x}{L}) + (\frac{\Delta y_{(i)} \gamma \Delta L x}{L}) - (\frac{\Delta L xy}{L}) + (\frac{\Delta L^2 xy}{L^2}) - (\frac{\Delta L \gamma x y}{L}) = 0 \end{equation}

The above equation gets reduced to the following quadratic:

\begin{equation} \Delta L^2 \left( \frac{xy}{L^2} \right) - \Delta L \left(\frac{1000 \Delta y_{(i)} x - 997\Delta y_{(i)} x + 1000xy + 997 xy}{1000L} \right) + \Delta y_{(i)} x = 0 \end{equation}

Now, solve for $\Delta L$ using calc_lp_settlement

def calc_lp_settlement(lp, token_in, itkn_amt):

    if(token_in.token_name == lp.token1):
        x = UniV3Helper().gwei2dec(lp.reserve0)
        y = UniV3Helper().gwei2dec(lp.reserve1)
    else:
        x = UniV3Helper().gwei2dec(lp.reserve1)
        y = UniV3Helper().gwei2dec(lp.reserve0)

    L = UniV3Helper().gwei2dec(lp.total_supply)
    gamma = 997

    a1 = x*y/L
    a2 = L
    a = a1/a2
    b = (1000*itkn_amt*x - itkn_amt*gamma*x + 1000*x*y + x*y*gamma)/(1000*L);
    c = itkn_amt*x;

    dL = (b*a2 - a2*mth.sqrt(b*b - 4*a1*c/a2)) / (2*a1);
    return dL

eth = ERC20("ETH", "0x09")
tkn = ERC20("TKN", "0x111")
exchg_data = UniswapExchangeData(tkn0 = eth, tkn1 = tkn, symbol="LP", address="0x011")

factory = UniswapFactory("ETH pool factory", "0x2")
lp = factory.deploy(exchg_data)
lp.add_liquidity(user_nm, eth_amount+100, tkn_amount, eth_amount+100, tkn_amount)
lp.summary()

eth_amt = 100
dL = calc_lp_settlement(lp, eth, eth_amt)

print('A request of {} ETH requires a settlement of {:.6f} LP token'.format(eth_amt, dL))

Exchange ETH-TKN (LP) Reserves: ETH = 1100.0, TKN = 100000.0 Liquidity: 10488.088481701516 A request of 100 ETH requires a settlement of 488.787567 LP token

y = UniV3Helper().gwei2dec(lp.reserve0)
x = UniV3Helper().gwei2dec(lp.reserve1)
L = UniV3Helper().gwei2dec(lp.total_supply)
gamma = 997

(dL**2)*x*y/(L*L) - dL*((1000*eth_amt*x - eth_amt*gamma*x + 1000*x*y + x*y*gamma)/(1000*L)) + eth_amt*x

-5.587935447692871e-09

Token Distribution

Using $\Delta L$ , we can determine the splitting distribution for withdrawal. Reconsidering Eq. 3, we redefine $\Delta y$ and $\Delta y_{swap}$ by portion $\alpha$ , thus:

\begin{equation} \Delta y_{(i)} = \Delta y + \Delta y_{swap} \end{equation}

\begin{equation} \Delta y_{(i)} = \alpha \Delta y_{(i)} + (1- \alpha) \Delta y_{(i)} \end{equation}

Therefore, using Eq. 2 we calculate our distribution as:

\begin{equation} \alpha =\frac{\Delta y}{\Delta y_{(i)}}= \frac{\Delta L y}{\Delta y_{(i)} L} \end{equation}

Hence, using the above equation and $L$ from our solver, we can calculate the withdraw distribution $\alpha$ via calc_portion

def calc_withdraw_portion(lp, token_in, amt):

    if(token_in.token_name == lp.token1):
        x = UniV3Helper().gwei2dec(lp.reserve0)
        y = UniV3Helper().gwei2dec(lp.reserve1)
    else:
        x = UniV3Helper().gwei2dec(lp.reserve1)
        y = UniV3Helper().gwei2dec(lp.reserve0)

    L = UniV3Helper().gwei2dec(lp.total_supply)
    gamma = 997/1000

    dL = calc_lp_settlement(lp, token_in, amt)
    dx = dL*x/L
    dy = dL*y/L
    aswap = (gamma*dx)*(y-dy)/(x-dx+gamma*dx)

    return dy/amt

alpha = calc_withdraw_portion(lp, eth, eth_amt)
print('The correct portion (for step 1) is {:.6f}'.format(alpha))

The correct portion (for step 1) is 0.512645

WithdrawSwap Steps

Finally, lets run through the steps to a WithdrawSwap and compare above

tkn = ERC20("TKN", "0x111")
eth = ERC20("ETH", "0x09")
exchg_data = UniswapExchangeData(tkn0 = eth, tkn1 = tkn, symbol="LP", address="0x011")

factory = UniswapFactory("ETH pool factory", "0x2")
lp = factory.deploy(exchg_data)
lp.add_liquidity(user_nm, eth_amount+100, tkn_amount, eth_amount+100, tkn_amount)

print('***\nInitial LP\n***')
lp.summary()

amt_out = 100
token_out = eth
user_nm = 'user0'

# Step 1: withdrawal
p_out = calc_withdraw_portion(lp, token_out, amt_out)
removeLiq = RemoveLiquidity()
res = removeLiq.apply(lp, token_out, user_nm, p_out*amt_out)

print('***\nLP post step 1\n***')
lp.summary()

# Step 2: swap
out = Swap().apply(lp, trading_token, user_nm, res[trading_token.token_name])

print('***\nLP post step 2\n***')
lp.summary()

withdrawn = res[eth.token_name] + out

print('Total withdrawn is {:.6f} + {:.6f} = {:.6f} \
ETH'.format(p_out*amt_out, out, withdrawn))
print('Of the requested {} ETH, a total of {:.6f} ETH \
has been withdrawn'.format(amt_out, withdrawn))

*** Initial LP *** Exchange ETH-TKN (LP) Reserves: ETH = 1100.0, TKN = 100000.0 Liquidity: 10488.088481701516 *** LP post step 1 *** Exchange ETH-TKN (LP) Reserves: ETH = 1048.735527472211, TKN = 95339.59340656463 Liquidity: 9999.300914574962 *** LP post step 2 *** Exchange ETH-TKN (LP) Reserves: ETH = 999.9999999999999, TKN = 100000.0 Liquidity: 9999.300914574962 Total withdrawn is 51.264473 + 48.735527 = 100.000000 ETH Of the requested 100 ETH, a total of 100.000000 ETH has been withdrawn

Final Solution

Finally, let’s check when our solution is integrated into WithdrawSwap

tkn = ERC20("TKN", "0x111")
eth = ERC20("ETH", "0x09")
exchg_data = UniswapExchangeData(tkn0 = eth, tkn1 = tkn, symbol="LP", address="0x011")

factory = UniswapFactory("ETH pool factory", "0x2")
lp = factory.deploy(exchg_data)
lp.add_liquidity(user_nm, eth_amount+100, tkn_amount, eth_amount+100, tkn_amount)
lp.summary()

amt_out = 100
out = WithdrawSwap().apply(lp, eth, user_nm, 100)
lp.summary()

print('Total withdrawn is {:.6f} ETH, as per request'.format(out))

Exchange ETH-TKN (LP) Reserves: ETH = 1100.0, TKN = 100000.0 Liquidity: 10488.088481701516 Exchange ETH-TKN (LP) Reserves: ETH = 1000.0, TKN = 100000.0 Liquidity: 9999.300914574964 Total withdrawn is 100.000000 ETH, as per request