[Ethereum] Understand price impact and liquidity in pancakeswap

bscdapp-developmentpancakeswapsolidityuniswap

I am trying to understand how the price impact and liquidity was calculated in uniswap or pancakeswap through the router contract?

UNI Router: https://etherscan.io/address/0x7a250d5630b4cf539739df2c5dacb4c659f2488d

Pancake Router: https://bscscan.com/address/0x05ff2b0db69458a0750badebc4f9e13add608c7f

Is there is any direct method or what calculations need to do in my code? Pl. Guide me.
Thanks in Advance

Best Answer

Disclaimer: I don't think this is the correct method to calculate the price impact but it's almost 100% accurate for price impacts of less than 1%. This calculation method assumes that multi-hop is disabled i.e. swapping is restricted to direct pairs. I've only tested this calculation method on PancakeSwap and it is less accurate for larger trade sizes compared to what is shown on their website. Attempting to calculate trade sizes for a 2% price impact, showed 1.99% on the PancakeSwap UI, 5% showed 4.99%, and 10% showed 9.97%. While these discrepancies could be attributed to the user interface lagging, I'm not confident that's the only reason. If anyone can better explain what's happening, I'd be very eager to know.

Short answer

Given a liquidity pool with token A and token B, let:

reserve_a_initial be the amount of token A in the liquidity pool before the trade
reserve_b_initial be the amount of token B in the liquidity pool before the trade (not required to calculate the price impact)
fee be the trading fee. For PancakeSwap, this is 0.0025
amount_traded be the amount of token A traded

To calculate the price impact:

amountInWithFee = amount_traded * (1 - fee);
price_impact = amountInWithFee / (reserve_a_initial + amountInWithFee);

Explanation

This article explains how to calculate the price impact. However, please note that what the article refers to as the "current market price" or "market rate", I refer to as the mid_price. Now, let's calculate the price_impact:

constant_product = reserve_a_initial * reserve_b_initial;
reserve_b_after_execution = constant_product / (reserve_a_initial + amountInWithFee);
amountOut = reserve_b_initial - reserve_b_after_execution;
market_price = amountInWithFee / amountOut;
mid_price = reserve_a_initial / reserve_b_initial;
price_impact = 1 - (mid_price / market_price);

Before substituting and simplifying, let:

reserve_a_initial be $r_{a0}$
reserve_b_initial be $r_{b0}$
reserve_b_after_execution be $r_{b1}$
amount_traded be $a_{traded}$
fee be $f$
amountInWithFee be $a_{in}$
constant_product be $k$
amountOut be $a_{out}$
market_price be $p_{market}$
mid_price be $p_{mid}$
price_impact be $price_{impact}$

Now for the math:

$a_{in} = a_{traded}(1-f)$

$k = r_{a0} \cdot r_{b0}$

$r_{b1} = \frac{k}{r_{a0} + a_{in}} = \frac{r_{a0} \cdot r_{b0}}{r_{a0} + a_{in}}$

$a_{out} = r_{b0} - r_{b1} = r_{b0} - \frac{r_{a0} \cdot r_{b0}}{r_{a0} + a_{in}} = \frac{r_{b0}(r_{a0} + a_{in}) - r_{a0} \cdot r_{b0}}{r_{a0} + a_{in}} = \frac{r_{b0} \cdot a_{in}}{r_{a0} + a_{in}}$

$p_{market} = \frac{a_{in}}{a_{out}} = a_{in} \div \frac{r_{b0} \cdot a_{in}}{r_{a0} + a_{in}} = \frac{r_{a0} + a_{in}}{r_{b0}}$

$p_{mid} = \frac{r_{a0}}{r_{b0}$

$price_{impact} = 1 - \frac{p_{mid}}{p_{market}} = 1 - \frac{r_{a0}}{r_{b0}} \div \frac{r_{a0} + a_{in}}{r_{b0}} = 1 - \frac{r_{a0}}{r_{a0} + a_{in}} = \frac{a_{in}}{r_{a0} + a_{in}}$

I found this mind-blowing, but the amount of token B in the pool doesn't affect the price_impact. However, reserve_b_initial will affect amount_out.

Calculating `amount_traded` given the `price_impact`

Rearranging the price_impact equation, it is possible to calculate amount_traded for a given price_impact:

$a_{traded} = \frac{r_{a0} \cdot price_{impact}}{(1 - price_{impact})(1-f)}$

Slippage and minimum amount received

To account for slippage, s, and calculate the minimum received, amount_out_min:

amount_out_min = amount_out * (1 - s);

Slippage does not affect price_impact.

Sample code

I don't know if there's a function in a PancakeSwap contract that will directly return the price_impact, but you can get the reserves of a pool and then calculate it. Let's take CAKE-USDT as an example:

import { Contract, utils, providers } from 'ethers';
const poolAddress = '0xa39af17ce4a8eb807e076805da1e2b8ea7d0755b'; // CAKE-USDT
const poolContract = new Contract(
    poolAddress,
    ['function getReserves() public view returns (uint112 _reserve0, uint112 _reserve1, uint32 _blockTimestampLast)'],
    new providers.WebSocketProvider(PROVIDER_URL)
);

poolContract.getReserves()
    .then((reserves) => {
        let reserve_a_initial = parseFloat(utils.formatUnits(reserves._reserve0));
        let reserve_b_initial = parseFloat(utils.formatUnits(reserves._reserve1));
        console.log(`CAKE in pool: ${reserve_a_initial}`);
        console.log(`USDT in pool: ${reserve_b_initial}`);

        const fee = 0.0025;
        let max_price_impact = 0.01;
        let amount_traded_cake = reserve_a_initial * max_price_impact / ((1 - max_price_impact)*(1 - fee));
        let amount_traded_usdt = reserve_b_initial * max_price_impact / ((1 - max_price_impact)*(1 - fee));
        console.log(`Given a max price impact of ${max_price_impact*100}%, the max amount of CAKE tradeable is ${amount_traded_cake}`);
        console.log(`Given a max price impact of ${max_price_impact*100}%, the max amount of USDT tradeable is ${amount_traded_usdt}`);

        let amountInCAKE = amount_traded_cake * (1 - fee);
        let amountInUSDT = amount_traded_usdt * (1 - fee);
        let price_impact_trade_cake = amountInCAKE / (reserve_a_initial + amountInCAKE);
        let price_impact_trade_usdt = amountInUSDT / (reserve_b_initial + amountInUSDT);
        console.log(`Price impact when trading ${amount_traded_cake} CAKE: ${price_impact_trade_cake*100}%`);
        console.log(`Price impact when trading ${amount_traded_usdt} USDT: ${price_impact_trade_usdt*100}%`);
    }).catch(console.error);

Sample results:

CAKE in pool: 1030240.4016832297
USDT in pool: 19974605.474162016
Given a max price impact of 1%, the max amount of CAKE tradeable is 10432.550079068678
Given a max price impact of 1%, the max amount of USDT tradeable is 202269.36507087937
Price impact when trading 10432.550079068678 CAKE: 1%
Price impact when trading 202269.36507087937 USDT: 1%

Please note that the price shown on the PancakeSwap UI is the execution_price:

execution_price = amount_traded / amountOut;

Best Answer

Short answer

Explanation

Calculating amount_traded given the price_impact

Slippage and minimum amount received

Sample code

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Calculating `amount_traded` given the `price_impact`