## Flash Swap Fee We'll learn about flash swap fees, how they're derived, and how they work within the Uniswap V2 protocol. ### Flash Swap Fee Equation Uniswap V2 supports flash swaps, which allow smart contracts to borrow tokens, execute a trade, and then repay the borrowed tokens all in the same transaction. Uniswap V2 requires a minimum fee to be paid for flash swaps, and this fee is a function of the amount of tokens borrowed. The equation used to calculate the minimum fee for a flash swap is as follows: $x_0 - dx_0 + 0.997 dx_1 >= x_0$ Where: * **$x_0$** is the amount of token X in the pair contract before the flash swap * **$dx_0$** is the amount of token X borrowed * **$dx_1$** is the amount of token X repaid ### Deriving the Flash Swap Fee Equation We'll use an analogy to understand how this equation is derived. Let's say we are swapping some amount of token X for token Y. If we put in dx tokens, then we will receive back dy tokens, and the fee taken from the trade will be 0.003 dx: **Swap dx for dy** * amount in = dx * amount in - fee = 0.997 dx * amount out = dy Now, let's consider a flash swap where we borrow $dx_0$ tokens, execute a trade, and repay $dx_1$ tokens. We can think of this flash swap as being like the regular swap scenario above, but with some key differences: **Flash Swap** * amount out = borrow $dx_0$ * amount in = repay $dx_1$ = $dx_0$ + fee * amount in - fee = 0.997 $dx_1$ The difference in a flash swap is that the amount of token X that we repay, $dx_1$, is not the same as the amount that we initially borrowed, $dx_0$. Instead, $dx_1$ is equal to $dx_0$ plus the fee. ### Solving for Fee We will now solve for the fee using the two equations we have: $x_0 - dx_0 + 0.997 dx_1 >= x_0$ $dx_1 = dx_0 + fee$ Let's rewrite the first equation, then cancel out the $x_0$: $x_0 - dx_0 + 0.997 dx_1 >= x_0$ $- dx_0 + 0.997 dx_1 >= 0$ Now, let's replace $dx_1$ in the equation above using the second equation: $0.997 dx_1 >= dx_0$ $0.997 (dx_0 + fee) >= dx_0$ Next, let's rearrange this equation and solve for fee: $0.997 dx_0 + 0.997 fee >= dx_0$ $0.997 fee >= dx_0 - 0.997 dx_0$ $0.997 fee >= (1 - 0.997) dx_0$ $0.997 fee >= 0.003 dx_0$ $fee >= \frac{0.003}{0.997 dx_0}$ This equation is the same as the one we started with at the beginning of the lesson, but we have now derived it using our analogy! ### Conclusion The flash swap fee ensures that the amount of token X in the pair contract is not depleted below the level it was before the flash swap. This helps to prevent attacks where a malicious actor could borrow a large amount of tokens and then drain the liquidity of the pool.