## 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: ```javascript x0 - dx0 + 0.997 dx1 >= x0 ``` Where: * **x0** is the amount of token X in the pair contract before the flash swap * **dx0** is the amount of token X borrowed * **dx1** 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 dx0 tokens, execute a trade, and repay dx1 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 dx0 * amount in = repay dx1 = dx0 + fee * amount in - fee = 0.997 dx1 The difference in a flash swap is that the amount of token X that we repay, dx1, is not the same as the amount that we initially borrowed, dx0. Instead, dx1 is equal to dx0 plus the fee. ### Solving for Fee We will now solve for the fee using the two equations we have: ```javascript x0 - dx0 + 0.997 dx1 >= x0 dx1 = dx0 + fee ``` Let's rewrite the first equation, then cancel out the x0: ```javascript x0 - dx0 + 0.997 dx1 >= x0 - dx0 + 0.997 dx1 >= 0 ``` Now, let's replace dx1 in the equation above using the second equation: ```javascript 0.997 dx1 >= dx0 0.997 (dx0 + fee) >= dx0 ``` Next, let's rearrange this equation and solve for fee: ```javascript 0.997 dx0 + 0.997 fee >= dx0 0.997 fee >= dx0 - 0.997 dx0 0.997 fee >= (1 - 0.997) dx0 0.997 fee >= 0.003 dx0 fee >= 0.003/0.997 dx0 ``` 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.