# NDMM Pricing Mechanism

1 When users open, close, or are liquidated from positions, liquidity providers (LPs) always passively become the counterparty, transacting at the same price.

2 The deviation rate (DR) for a trading pair is calculated using the following algorithm:

* The trading pair's LP = Total TLP size \* Trading pair's proportion
* The trading pair's long positions (LONG) = Total value of long positions for that trading pair
* The trading pair's short positions (SHORT) = Total value of short positions for that trading pair
* The trading pair's LP exposure (EXPOSURE) = LONG - SHORT

The formula is as follows:

* Trading Pair Deviation Rate (DR) = LP exposure value / The trading pair's LP= (LONG - SHORT) / The trading pair's LP

For example:

* Trading Pair: BTC/USDT
* LP for BTC/USDT: 10,000,000 USDT
* LONG: 1,000,000 USDT
* SHORT: 400,000 USDT
* EXPOSURE = LONG - SHORT = 600,000 USDT

Then in this case the DR for BTC/USDT would be **600,000/10,000,000=0.06**

OI limit = The trading pair's LP \* MULTIPLIER

ADL：

* EXPOSURE > The trading pair's LP 

3 The perpetual contract's price premium rate (PR) is determined by the deviation rate (DR) and a premium rate function, i.e., PR is a function of DR, which is decided by the liquidity data and system parameters. Notably, when DR = 0, PR = 0, meaning that when the deviation rate of the trading pair is zero, the contract's premium rate is zero. 

4 The premium rate function f(x) is an antisymmetric function and should be positively correlated with the deviation rate. 

Based on the risk-neutral principal, we assumed the Deviation Rate of perpetual contract follows Normal distribution, thus the probability density function has following format:

$$
f(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}
$$

<div align="center" data-full-width="false"><figure><img src="https://lh7-us.googleusercontent.com/FX6xbI9p-zOd6PENgA-aSOfTcpHxzIRofg4-9mx65WzrcJJOtnOwgU_FhnCK4DG_oScwVnk7I3ZXFP-NQ_EJbcapR7vDxpGvhXWb_8neeL34voI2Y6Ubr793095zc5lisbmFpIhbhdDrkh1j3yu7HGU" alt=""><figcaption></figcaption></figure></div>

The Premium Rate Function should represent the accumulation of the contract's Deviation Rate. Thus, we performed an integration on the distribution of the deviation rate. As a result, we obtained the Premium Rate Function, which exhibits the characteristics of a cumulative normal distribution function N(x).

$$
PR(x)=\int\_0^x\frac1{\sqrt{2\pi}}e^{-\frac{u^2}2}\mathrm{d}u=N(x)-0.5
$$

<figure><img src="https://lh7-us.googleusercontent.com/ETuNBQ49BMmOB476rQTmwQhoDeXMQNifIEKlkkv51BpDfWlTpcOUu_UdGwIV0cvzvJ74NAn81rS2U7sozoeTfMZHxW8A-KOQVMVPVbjeVCG6lz74-6J0sgN4pcU7jbiuEui__TyePNmhTDk1R4SebO8" alt=""><figcaption></figcaption></figure>

Exact Premium Rate Function is determined by some inner configurations such as premium\_cap etc.&#x20;

Characteristics of the Premium Rate Function:

* It is an antisymmetric function.
* It is positively correlated with the degree of deviation, exhibiting a monotonically increasing relationship.&#x20;

The average transaction price:&#x20;

$$
P\_{a\nu g}=(1+(PR\_0+PR\_1)/2)P\_{index}
$$

where PR0 is the premium rate before the transaction, and PR1 is the premium rate after the transaction.&#x20;

Perpetual contract price formula:&#x20;

$$
P\_{contract}=(1+PR)P\_{index}
$$

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