Funding

Parameters:

Symbol	Meaning
loi	long open interest of each currency in USD for all users
soi	short open interest of each currency in absolute value in USD for all users. soi >=0
pr	a risk management parameter for each currency. pr $\in (0,1)$
s	net position of each currency in USD for all users. (loi-soi)
r	funding rate of each currency
LP	total LP tokens’ net asset value, in USD, equal to LP’s cash after instantly closing all user positions without fee

Assume the funding rate velocity is positively correlated with $\frac s {LP}$.

When $r \cdot s >=0$,

$$\frac{dr}{dt} = \left \{ \begin{array}{rcl} -V_{max} && s\leq-pr \cdot LP\\ \\ \frac s {pr \cdot LP} \cdot V_{max} && s \in (-pr \cdot LP, pr \cdot LP]\\ \\ V_{max} && s>pr \cdot LP \end{array} \right.$$

When $r \cdot s <0$,

$$\frac{dr}{dt} = \left \{ \begin{array}{rcl} -2\cdot V_{max} && s\leq-pr \cdot LP\\ \\ 2 \cdot \frac s {pr \cdot LP} \cdot V_{max} && s \in (-pr \cdot LP, pr \cdot LP]\\ \\ 2 \cdot V_{max} && s>pr \cdot LP \end{array} \right.$$