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Table of Contents

• About The Project
  • Built With
• Getting Started
  • Prerequisites
  • Installation
• Usage
• Constant Product AMM
• Time Weighted Average Price
• Uniswap V2 Price Oracle
• Constant Product AMM Spot Price Examples
• Uniswap V3 Price Oracle
• Geometric Mean
• Uniswap V3 TWAP and Geometric Mean
• Uniswap V3 TWAP Inverse Price
• Roadmap
• Contributing
• License
• Contact
• Acknowledgments

About The Project

This project shows a basic Constant Product Automated Market Maker.

You will also find all the Maths used in Uniswap V2 and Uniswap V3

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Built With

• [![Hardhat][Hardhat]][Hardhat-url]
• [![Ethers][Ethers.js]][Ethers-url]

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Getting Started

To get a local copy up and running follow these simple example steps.

Prerequisites

• npm

npm install npm@latest -g

• hardhat

npm install --save-dev hardhat

npm install @nomiclabs/hardhat-ethers @nomiclabs/hardhat-waffle

run:

npx hardhat

verify:

npx hardhat verify --network goerli "contract address" "pair address"

Installation

• Clone the repo

git clone https://github.com/Aboudoc/Constant-Product-AMM.git

• Install NPM packages

npm install

• Dependencies

npm i @uniswap/v2-core @uniswap/v2-periphery

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Usage

If you need testnet funds, use the Alchemy testnet faucet.

This project shows a basic example of Constant Product AMM

Constant Product AMM

Constant product AMM (automated market maker) is a decentralized exchange where 2 tokens are traded.

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Why is it called constant product?

Swap - How many dy for dx?

Uniswap trading fee = 0.3%

Add liquidity - How many dx, dy to add?

Add liquidity - How many shares to mint?

Motivation:

What is L0 and L1

Simplify the equation:

Conclusion

Remove liquidity - How many tokens to withdraw?

Time Weighted Average Price

How do we compute the Time Weighted Average Price from Tk to Tn?

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Uniswap V2 Price Oracle

We saw the math to calculate TWAP, you can also calculate it in Uniswap V2 using Solidity.

This is the purpose of UniswapV2Twap contract

In Uniswap V2, the numbers are represented in decimals, but Solidity does not have any decimal datatypes, so we'ell be using FixedPoint.sol library

The datatype FixedPoint.uq112x112 represents a decimal number as follows:

range: [0, 2e112 - 1] resolution: 1 / 2e112

The IUniswapV2Pair contract is the pair contract that holds the two tokens and does the swaps

Constructor

Since we're using Solidity 0.6, constructor must be declared as public

• We set the pair first
• We set token0 and token1 from pair
• We record price0CumulativeLast and price1CumulativeLast and the last time that these variables were updated: blockTimestampLast using getReserves(), it returns 3 outputs but we only need the third

Function update

• To get the current price0Cumulative, current price1Cumulative and blockTimestamp, we call currentCumulativePrices() on UniswapV2OracleLibrary by passing in the address of the pair
• Calculate how much time has elapsed since the last time we called update(). This time elapsed must be greather than the period defined as state variable
• Calculate the price averages (price0Average and price1Average) by taking the current price cumulative, substracting it from the last price cumulative, and dividing it over the time elapsed. Cast it into a FixedPoint

=> Note that up112x112 is a struct and the input for this struct must be a uint224 => cast the calcul expression

The expression below, used to calculate the price averages, is the exact same equation that we derived from How do we compute the Time Weighted Average Price from Tk to Tn?

price0Average = FixedPoint.uq112x112(
     uint224((price0Cumulative - price0CumulativeLast) / timeElapsed)
   );
   price1Average = FixedPoint.uq112x112(
     uint224((price1Cumulative - price1CumulativeLast) / timeElapsed)
   );

Keep in mind that we don't care if the numbers overflows This is why we are not using SafeMath

• Finally, update the state variables: price0CumulativeLast, price1CumulativeLast and blockTimestampLast

Function consult

Giving the token and the amount of token put in, this function will calculate the amount out using the price0Average and price1Average

• The token must be token0 or token1
• Compute amountOutusing price0Average and amountIn argument (.mul). We have to put it back to uint using a function called decode144()

Additional Reading

You can find more in the following article Using the new Uniswap v2 as oracle

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Constant Product AMM Spot Price

P = Y / X from geometry

P = Y / X from calculus

Constant Product AMM Spot Price Examples

Uniswap V2

Uniswap V3

Uniswap V3 Twap

However, virtual reserves are not tracked in Uni V3

The Tick that is used to compute the current price is tracked in Uni V3

Example

Pricing difference between Uni V2 and Uni V3

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Uniswap V3 Price Oracle

Constructor

• Set token0 and token1
• get the pool by calling getPool() on, IUniswapV3Factory and set the state variable
• To call getPool() we need to pass in 3 parameters: address of the tokens, and the fee => set factory address and fee (uint24)
• Check that the pool is a valid pool (not address(0)) and finally set it (state variable)

Function estimateAmountOut

• Define tokenIn and tokenOut
• Get the average tick: Call consult() on OracleLibrary like so:

(int24 tick, ) = OracleLibrary.consult(pool, secondsAgo);

We'll copy paste the code to compute tick directly from the library to save some gas

• Call GetQuoteAtTick() on OracleLibrary to set the amountOut

Forking mainnet

hardhat.config.js

networks: {
        hardhat: {
          forking: {
            url: https://eth-mainnet.alchemyapi.io/v2/${process.env.ALCHEMYAPIKEY},
       },
     },
  }

.env

ALCHEMYAPIKEY=...

terminal1

ALCHEMYAPIKEY=...
npx hardhat node --fork https://eth-mainnet.g.alchemy.com/v2/$ALCHEMYAPIKEY

terminal2

npx hardhat test --network localhost

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Geometric Mean

Example

Comapraison With Arithmetic Mean

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Uniswap V3 TWAP and Geometric Mean

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Uniswap V3 TWAP Inverse Price

Not in Uniswap V2

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Note

This contract assumes that token0 and token1 both have same decimals

Consider fees = 0.3%

Further reading

(...soon)

Sources

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Roadmap

• [ ] Uniswap V3 TWAP
• [ ] Further reading
• [ ] Deploy script
• [ ] Unit test

See the open issues for a full list of proposed features (and known issues).

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Contributing

Contributions are what make the open source community such an amazing place to learn, inspire, and create. Any contributions you make are greatly appreciated.

If you have a suggestion that would make this better, please fork the repo and create a pull request. You can also simply open an issue with the tag "enhancement". Don't forget to give the project a star! Thanks again!

• Fork the Project
• Create your Feature Branch (git checkout -b feature/AmazingFeature)
• Commit your Changes (git commit -m 'Add some AmazingFeature')
• Push to the Branch (git push origin feature/AmazingFeature)
• Open a Pull Request

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License

Distributed under the MIT License. See LICENSE.txt for more information.

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Contact

Reda Aboutika - @twitter - reda.aboutika@gmail.com

Project Link: https://github.com/Aboudoc/Constant-Product-AMM.git

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Acknowledgments

• Smart Contract Engineer

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