# The Formulas

## Automated Market Making

In each trade, traders trade certain amount of a particular token for certain amount of another token with the price defined by a formula. There is no orderbook and waiting for fulfillment.

The formula that mimo uses is the famous $x * y = k$ that has been widely adopted by AMM based DEX, such as uniswap.

## The formula $x * y = k$

Assume that$X$ is the source token, and $Y$ is the destination token. In mimo, $X$, $Y$ could be either IOTX or XRC20 tokens. Let $x$, $y$ be X-token, Y-token in current liquidity pool, respectively.

Based on the famous AMM equation

$x * y = k$

where $k$ is a constant.

The product of $x$ and $y$ remains the same before and after trading. For details, please refer to vbuterin's post.

## Pricing based on the inputs

Let's further define $d_x$ , $d_y$ are how many X-tokens you want to pay, and how many Y-tokens you will get, respectively.

We'd like to know, the price based on $d_x$ or$d_y$. IfgetInputPrice denotes how many Y-Tokens (i.e. $d_y$ ) can be bought by selling a given $d_x$,

$getInputPrice(x, y, d_x) = \dfrac{y * 997 * dx}{1000 * x + 997 * d_x}$

or in code,

getInputPrice(x, y, dx) = (y * 997 * dx) / (1000 x + 997 dx)

If getOutputPrice denotes how many X-tokens is needed to buy $d_y$ Y-tokens,

$getOutputPrice(x, y, d_y) = \dfrac{1000 * x * d_y}{(y-d_y)*997} + 1$

or in code,

getOutputPrice(x, y, dy) = 1000 x * dy / ((y - dy) * 997) + 1

where / in above equations denotes divToInteger, which means divide with rounding to floor of the results.

## Price impact

In AMM, the price would change after each trade, $d_x$ , $d_y$ and $x, y$ the liquidity in the pool. The price impact is what traders want to know before the trade.

There are two ways of calculating price impact. It can be based on $x, y, d_x$ , or $x, y ,d_y$ . One is based on input , one is based on output.

$PriceImpact(x, y, d_x) = \frac{ (1000*x)^2} { (1000*x + 997*d_x)^2} -1$

or

$PriceImpact(x, y, d_y) = \dfrac{(y-d_y)^2}{y^2} -1$

in code,

price impact(x, y, dx) = (1000*x)^2 / (1000*x + 997*dx)^2 - 1
price impact(x, y, dy) = (y - dy)^2 / y^2 - 1

Note that the price impact is always between -1 and 0.

If there are no direct trading pairs between two tokens, like in V1 where we only support IOTX/token pairs, traders need to use one token, such as IOTX, as a bridge to trade among two tokens.

In this case, the price impact would be

$PriceImpact_{cross} = PI_1 * PI_2 + PI_1 + PI_2$

or in code,

price impact = PI1 * PI2 + PI1 + PI2

where

PI1 is the price impact of first trading pair, such as x to IOTX
PI2 is the price impact of second trading pair, such as IOTX to y

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