Flashloan fee of Uniswap v2

tags: uniswap flashloan fee uniswap-v2

Refer to Uniswap v2 Whitepaper, each swap should satisfy following formula:

$$\begin{array}{}\text{(1)}& (x_1-0.003\cdot x_{in})\cdot (y_1-0.003\cdot y_{in})\ge x_0\cdot y_0\end{array}$$

To simplify the calculation on-chain:

$$\begin{array}{}\text{(2)}& (1000\cdot x_1-3\cdot x_{in})\cdot (1000\cdot y_1-3\cdot y_{in})\ge 1000000\cdot x_0\cdot y_0\end{array}$$

When we flash loan from Uniswap v2 pair, what is the amount including fee we should repay?

Suppose we flash loan amount of token

$y$:

$y_{out}$.

$x_{in}$ and

$y_{in}$ are the amount of token

$x$ and token

$y$ to repay.

$x_0$ and

$x_1$ are the balance of token

$x$ before and after flash loan, respectively,

$y_0$ and

$y_1$ are the balance of token

$y$ before and after flash loan.

The problem is to calculate

$y_{in}$ using

$y_{out}$.

Since we don't flash loan token

$x$, so

$x_{in}=0$,

$x_{out}=0$,

$x_0=x_1$, we can simplify formula (2):

$$\begin{array}{}\text{(3)}& 1000\cdot y_1-3\cdot y_{in}\ge 1000\cdot y_0\end{array}$$

$$\begin{array}{}\text{(4)}& y_1-y_0\ge \frac{3}{1000}\cdot y_{in}\end{array}$$

The relation between

$y_{in}$ and

$y_{out}$ is:

$$\begin{array}{}\text{(5)}& y_{in}=y_1-(y_0-y_{out})\end{array}$$

Using formula (4) and (5):

$$\begin{array}{}\text{(6)}& y_{in}-y_{out}\ge \frac{3}{1000}\cdot y_{in}\end{array}$$

We can calculate

$y_{in}$:

$$\begin{array}{}\text{(7)}& y_{in}\ge \frac{1000}{997}\cdot y_{out}\end{array}$$

In Solidity contract, to avoid rouding down error, we should add 1 to round up:

uint256 repayAmount = amountOut.mul(1000).div(997).add(1)

where repayAmount is

$y_{in}$, amountOut is

$y_{out}$.

So the flashloan fee of Uniswap v2 is:

$$\begin{array}{}\text{(8)}& \frac{1000}{997}-1\approx 0.0030090271\approx 0.30090271\mathrm{\%}\end{array}$$