Funding Rate, what is β?

$$\begin{align*} &Symbol = ETH-PERP-USDC &\\ &... &\\ &FundingRate = \beta * f( Trade_{amount},AMM_{liquidityStatus})) &\\ & \beta =0.0069 & \\ &Symbol = ETH2-PERP-USDC &\\ & ... &\\ &Varience_{ETH} = (Volatility_{ETH})^2 = \sigma_t^2 = \lambda * \sigma_{t-1}^2+(1-\lambda)u_{t-1}^2&\\ &u_{t} = ln\frac{S_t}{S_{t-1}}\approx1- \frac{S_t}{S_{t-1}}&\\ &\lambda = 0.94 (The \ RiskMetrics\ database\ produced \ by \ JP \ Morgan)&\\ \\ &FundingRate =\sigma_t^2*(1+\beta*f( Trade_{amount},AMM_{liquidityStatus})) &\\ &\beta = 6.9 & \\ \end{align*}$$

$$\begin{align*} f( Trade_{amount},AMM_{liquidityStatus}) &= \frac{\int_L^{L+\Delta L}\int_m^{m+\Delta m}(\frac{x}{y})^3dxdy}{\Delta L \Delta m} &\\ &=\frac{m^3+\frac{3}{2}m^2\Delta m + m\Delta m^2+\frac{\Delta m^3}{4}}{L*L*(L+\Delta L)^2}*(L+\frac{\Delta L}{2}) &\\ &\approx ( \frac{m}{L} )^3 = (UtilizationRate_{AMM})^3&\\ \\ &m = Liquidity_{Locked} \ before \ The \ Trade &\\ &\Delta m = Liquidity_{Locked} \ for \ The \ Trade &\\ &L = Liquidity_{total} \ before \ The \ Trade &\\ &\Delta L = Liquidity_{changed} \ for \ The \ Trade&\\ \end{align*}$$

In order to get that capture the market's view of the likelihood of changes in a given Underlying price as Implied Volatility, Positions against AMM shall be reflected in the Trade Price via Funding Rate.

As for Positions against AMM, the Utilization Rate shall be used. There is no absolute answer to this usage of Utilization Rate, and trial and error will continue. Initially, the following settings were used.

$$\begin{align*} &Symbol = ETH-PERP-USDC &\\ &... &\\ & \beta =0.0022 & \\ &... &\\ &Symbol = ETH2-PERP-USDC &\\ & ... &\\ &\beta = 5.5 & \\ & ... &\\ \end{align*}$$

$$\begin{align*} f( Trade_{amount},AMM_{liquidityStatus}) &= \frac{m+0.5\Delta m}{\Delta L}(\log_e{(L+\Delta L)}-\log_e{L}) &\\ &\approx ( \frac{m}{L} ) = (UtilizationRate_{AMM})&\\ \\ & ... &\\ \end{align*}$$

However, the Utilization Rate has a linear effect so that the Impact cannot be ignored even if the Utilization Rate is low. Ideally, it should increase the impact and provide a corrective effect when the AMM's Utilization Rate interferes with AMM operations, and serve as some IV when the Utilization Rate is low.

So Predy V2 decided to take advantage of the three-square Utilization Rate.

Funding rates for 8 hours can be obtained from such sites. Currently, Predy V2 set β based on Utilization Rate is around 30% (actually it could be considered as 60% since there are two products and AMM funds are used for two of them). Then Funding Rate shall be set to be similar to or lower than the average of others.

To addition, the number 69 was used within the predecessor Product named "Squeeth" as "because it is a nice round number". It has been very considered various numbers and was decided that 69 is really a nice number and works.