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Let's say the instantaneous price of ETH/USDT pair is 2000.

If I trade a lot of ETH for USDT, there could be a lot of slippage, and my trade might fulfill at an average execution price of 1950 USDT per ETH.

Let's say I'm ok with trading at 1950 USDT. How can I calculate the max amount of ETH I can trade at this average execution price?

How can I calculate this in either direction, either from ETH to USDT or vice versa, in Solidity, using the Uniswap V2 interface?

David Callanan
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1 Answers1

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TL;DR - Final formula is at the end. It would be great if someone can verify that I didn't make any mistakes.

If we exchange token x for token y, as per the constant product formula:

x * y = k

Let a be the amount of x we are exchanging to get b amount of y. Therefore:

(x + a) * (y - b) = k

The execution price of the trade, by definition, is just b / a.

If our target execution price is e, then b / a = e => b = ea

Therefore:

(x + a) * (y - ea) = k

But x * y is also equal to k, therefore:

x * y = (x + a) * (y - ea)

Now we can just rewrite the equation to get a in terms of the other variables.

x * y = x * (y - ea) + a * (y - ea)
xy = xy - eax + ay - ea^2
- eax + ay - ea^2 = 0
-ea^2 + a(y - ex) = 0

We know a is not zero, else the price would be undefined which is not possible. Because a is not zero, we can safely divide across by a:

-ea + y - ex = 0

Now a few more slight adjustments:

ea = y - ex
a = (y - ex) / e
a = (y / e) - x

So this is the final formula:

a = (y / e) - x

where a is the maximum amount we can trade to get an execution price of e or better, and x and y are the number of input and output tokens in the pool before the trade respectively.

TEST CASE - where 1 ETH is worth 2,000 USD

x = 100 ETH
y = 200,000 USD
e = 1,950

a = (y / e) - x
  = (200,000 / 1,950) - 100
  = 2.564 ETH
David Callanan
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