Simple Automated Market Maker

Simple version of the Uniswap AMM

Constant Product Market

A group of liquidity providers deposits token $\alpha$and token $\beta$with reserves $R_{\alpha}$and $R_{\beta}$, respectively. The fee for a transaction is $(1-\gamma)$ and the conserved quantity is $k=R_{\alpha}R_{\beta}$. A user wants to trade an amount of token $\alpha$, $\Delta_{\alpha}$, for some amount of token $\beta$,$\Delta_{\beta}$ such that k remains conserved.

Each transaction must satisfy the following equation:

$$(R_{\alpha} - \Delta_{\alpha})(R_{\beta} +\gamma\Delta_{\beta}) = k$$

After the transaction the reserves and k are updated as follows:

$$R_{\alpha} = R_{\alpha} - \Delta_{\alpha}$$

$$R_{\beta} = R_{\beta} + \Delta_{\beta}$$

$$k = (R_{\alpha} - \Delta_{\alpha})(R_{\beta} + \Delta_{\beta})$$

Simple Calculation

Alice wants to sell an amount $\Delta_{\alpha}$ for token $\beta$.

What amount, $\Delta_{\beta}$, does Bob pay to Alice given $R_{\alpha}$, $R_{\beta}$, and $\gamma$?

$$\Delta_{\beta} = -\frac{R_{\beta} \Delta_{\alpha}}{\gamma(\Delta_{\alpha} - R_{\alpha})}$$

The ratio of the changes: $\nu = \Delta_{\alpha} / \Delta_{\beta}$.

$$\nu = -\frac{\gamma \Delta_{\alpha}(\Delta_{\alpha} - R_{\alpha})}{R_{\beta}\Delta_{\alpha}}$$

The market price is $m_{\mu} = R_{\alpha} / R_{\beta}$.

Constant Mean Market

A constant mean market is a generalization of a constant product market. The conserved constant becomes a product over all the reserves for n tokens where $w_{i}$is the reserve weight for the ith token.

$$k = \prod_{i=1}^{n} R_{i}^{w_{i}}$$

The conservation equation becomes

$$c(R_{j} + \gamma_{j}\Delta_{j})^{w_{j}}(R_{\ell} -\Delta_{\ell})^{w_{\ell}} = k$$

where $\gamma_{j}$is the fee for the jth coin and c is given by

$$c = \prod_{\substack{i=1 \\ \ell \neq j}}^{n}R_{i}^{w_{i}}$$

An example can be seen at

Example-Code/constant_product.py at main · logicalmechanism/Example-CodeGitHub