Deep Understanding of Swap

Swap Logic Breakdown

The token swap logic in the Dyson Pair Contract is divided into two main functions: _swap0in and _swap1in, corresponding to swapping token0 for token1 and token1 for token0, respectively. Let's focus on the _swap1in function to understand the core calculation logic.

_swap1in Function

MAX_FEE_RATIO is set to 2^64 .

function _swap1in(uint input, uint minOutput) internal returns (uint fee, uint output) {
    // ... (omitting details for brevity)

    // Fee calculation based on fee ratio for token1
    fee = uint(_feeRatio1) * input / MAX_FEE_RATIO;

    // Adjust input by deducting the fee
    uint inputLessFee = input - fee;

    // Calculate the output using the invariant formula
    output = inputLessFee * reserve0 / (reserve1 + inputLessFee);

    // Ensure the minimum output requirement is met
    require(output >= minOutput, "slippage");

    // Update fee ratio for token0 based on the output proportion of the reserve0w
    uint64 feeRatioAdded = uint64(output * MAX_FEE_RATIO / reserve0);
    _updateFeeRatio1(_feeRatio1, feeRatioAdded);
}

1. Calculate Output:

   • Obtain the fee ratios, feeRatio0 and feeRatio1, by using the getFeeRatio() function. This step updatesd the fee ratios based on the half-life mechanism. When a trade happens, the fee will increase immediately, and decrease as time goes on. The change will keep going until the trading fee reduces to zero.

   • Fee1 calculation based on fee ratio for token1: fee1 = uint(_feeRatio1) * input / MAX_FEE_RATIO

   • Adjust input by deducting the fee: inputLessFee = input - fee1

   • Use the constant product formula xy = k to calculate the output: output = inputLessFee * reserve0 / (reserve1 + inputLessFee)

2. Increase fee0:

   • After the swap, to account for changes in the pool ratios and mitigate arbitrage opportunities for liquidity providers, an additional fee (fee0) is introduced for the swap0in operation.

   • Calculate the added fee0 using: feeRatioAdded = uint64(MAX_FEE_RATIO * output / reserve0) This calculation is based on the proportion of the total token0 in the pool represented by the tokens swapped out during the _swap0in operation.

   • Update the fee ratio for token0 using: _updateFeeRatio0(_feeRatio0, feeRatioAdded); This step involves utilizing the formula newFee = 1 - (1 - x%)(1 - y%) to calculate the new fee ratio, and the result is stored for future use. About this formula please refer to Trading Fee Calculation section in our white paper.

Conclusion

The Dyson Pair Contract's swap mechanism introduces a dynamic fee model and premium calculation to incentivize users based on their chosen lock times. The core swap logic, exemplified by the _swap1in function, showcases the intricacies of fee calculation, input adjustment, and output determination. This approach provides users with a transparent and adaptive swapping experience, aligning incentives with market conditions and user preferences.