Assuming the yield volatility scales with the current yield level with zero drift, we have:
$$ dy=\sigma_0ydW\\d\ln y=-\frac{1}{2}\sigma_0^2dt+\sigma_0dW $$
where $y$ is the yield, $\sigma_0$ is the proportional yield volatility, $t$ is the time, and $W$ is the Weiner process.
This is called the Constant Proportional Yield Volatility Model, or the CP Model.
Therefore starting with yield $y_0$ at time $0$, the distribution of yield $\ln y$ at time $t$ is:
$$ f(\ln y)=\frac{1}{\sqrt{2\pi\sigma_0^2t}}e^{-\frac{(\ln y -\ln y_0+\frac{1}{2}\sigma_0^2t)^2}{2\sigma_0^2t}} $$
There are several ways to determine $y_0$ and $\sigma_0$:
Looking at how the yield range is set up for existing assets
Assuming the existing yield range with maturity $t_0$ is from $y_\text{min}$ to $y_\text{max}$, $y_0$ can be determined as:
$$ y_0e^{-\frac{1}{2}\sigma_0^2t_0}=e^{\frac{\ln y_\text{min}+\ln y_\text{max}}{2}}=\sqrt{y_\text{min}y_\text{max}}\\y_0=e^{\frac{1}{2}\sigma_0^2t_0}\sqrt{y_\text{min}y_\text{max}} $$
$\sigma_0$ can be determined as:
$$ \sigma_0=\frac{\ln(y_\text{max}/y_\text{min})}{2\sqrt{t_0}} $$
Therefore the yield range with maturity $t$ is $(y_0e^{-\frac{1}{2}\sigma_0^2t-\sigma_0\sqrt{t}},y_0e^{-\frac{1}{2}\sigma_0^2t+\sigma_0\sqrt{t}})$.
Looking at historical data
$y_0$ can be determined as the arithmetic mean of the most recent data, for example the past 7 days:
$$ y_0=(y_1+...+y_n)/n $$
$\sigma_0$ can be determined as the arithmetic mean of the yield percentage changes over, for example the past 365 days, or the entire maturity:
$$ \sigma_0=\sqrt{365\cdot\frac{(y_2/y_1-1)^2+...+(y_n/y_{n-1}-1)^2}{n-1}} $$
The yield range with maturity $t$ is still $(y_0e^{-\frac{1}{2}\sigma_0^2t-\sigma_0\sqrt{t}},y_0e^{-\frac{1}{2}\sigma_0^2t+\sigma_0\sqrt{t}})$.
Note that yield is implied rate - 1.
Looking at relation between maximum allowed Loan-To-Value (LTV) $\theta$ and yield volatility $\sigma_0$
$$ \ln\theta=\ln\left[\frac{D(1+\mathbb{E}[y]t_0)}{C_0}\right]+z_\delta\sqrt{\sigma_0^2t_0+\sigma_c^2t_0} $$
where $D$ is the initial debt, $C_0$ is the initial collateral value, $\mathbb{E}[y]$ is the expectation of yield at maturity $t_0$, $z_\delta$ is the zeta score with the probability of default $\delta$, $\sigma_0$ is the yield volatility, and $\sigma_c$ is the collateral’s price volatility.
$$ \mathbb{E}[y]=y_0 $$
$$ \ln\theta=\ln\left[\frac{D(1+y_0t_0)}{C_0}\right]+z_\delta\sqrt{\sigma_0^2t_0+\sigma_c^2t_0} $$
Knowing $y_0$, $z_\delta$, and $\sigma_c$, we can solve for $\sigma_0$.
$y_0$ can be determined using the first two methods. $z_\delta$ can be determined by setting a default probability. $\sigma_c$ can be determined using the second method.
As we can see, since $z_\delta$ is a negative, the bigger the yield volatility, the smaller the maximum allowed LTV.
Note that this formula is obtained from using a joint lognormal approximation, I personally don’t recommend this method since it introduces new variables to characterize such as $z_\delta$and $\sigma_c$.