Uniswap V2 optimal arbitrage amount

Question

Tried out whole week to find an answer somewhere for this question, but, nothing to see, unfortunately,

Let suppose there are 2 USDT/WETH pairs, one on Uniswap and another one on any Uniswap v2 fork ( like Sushi )

For example, keeping it little simpler, let's remove decimals:

UniswapV2 pool has: 900 ETH and 2238300 USDT, which means price per ETH = 2487 K = 900 * 2238300 = 2014470000

Sushiswap pool has: 487 ETH and 1210195 USDT, which means price per ETH = 2485 K = 487 * 1210195 = 589364965

To calculate output amount of a predefined swap amount: Suppose I want to swap 1 eth for USDT

In Uniswap pool:

input = 1 * 997 = 997 numerator = input * 2238300 = 2231585100 denomitor = 900 * 1000 + 997 = 900997 output = numerator / denomitor = 2,476.79 USDT received for 1 ETH // Or use router getAmountsOut

In Sushiswap pool:

input = 1 * 997 = 997 numerator = input * 1210195 = 1206564415 denomitor = 487 * 1000 + 997 = 487997 output = numerator / denomitor = 2472.48 ( actually, getAmountsOut gives me 2465.06 ) - Anyway

How can i calculate the optimal input amount if there is a price discrepancy between pools in order to arb the price of these two pools? Without considering gasFees, just need simple math

Answer 1

Short answer

Plug these numbers in to calculate r, where

• r = optimal amount in
• x_a = reserve out of AMM A
• y_a = reserve in of AMM A
• x_b = reserve in of AMM B
• y_b = reserve out of AMM B
• f = fee (0.03%)

k = (1-f)*x_b + (1-f)**2*x_a
a = k**2
b = 2*k*y_a*x_b
c = (y_a*x_b)**2 - (1-f)**2*x_a*y_b*y_a*x_b

r = (-b + sqrt(b**2 - 4*a*c)) / (2*a)

Long answer

You need to find a value dy_0 for the function f,

f(dy_0) = dy_1 - dy_0

such that f(dy_0) is the maximum of the function

where dy_0 is the input amount of token and dy_1 is the amount of same token you receive after arbitrage.

If we know dy_0, we can calculate amount out (dx)

dx = dy_0 * (1 - f) * x_a / (y_a + dy_0 * (1 - f))

And, if we know dx, we can calculate dy_1 (amount out from second swap)

dy_1 = dx * (1 - f) * y_b / (x_b + dx * (1 - f))

Now working backwards, re-write f using the 2 equations above, we get a complicated function in terms of dy_0.

To find the maximum of f, take the derivative and then find where the derivative = 0. Finally use the quadratic equation, (-b + sqrt(b**2 - 4ac)) / (2a) to find where the derivate = 0.

Here is the graph of f in red, green is derivative of f and blue is the maximum of f, the optimal input amount assuming gas fee is 0.

https://www.desmos.com/calculator/gjujz3ayta