CIR Reddit Mentions | ChartExchange

So, for ease and formatting, I made a quant stackexchange post [here](https://quant.stackexchange.com/questions/85789/using-the-hull-white-model-to-find-the-2026-value-in-the-us-bond-plot) but I believe that I was incorrect here?
I wanted to output specifically the 2026 value of $\\sim5.01\\%$ for 2026, as seen in the above plot?
I believe that I have an error with SDE used $r = 4.65, \\theta = 4.2$, and $\\sigma = 0.55$, apparently in percentage points. That's internally possible, but then the reported standard deviation calculation is wrong in its interpretation:
$$\\sqrt{\\frac{(0.55)\^2}{2(0.35)}}\\approx 0.657$$
meaning $0.657$ percentage points if rates are measured in percentage points?
Also, the Hull–White equation that I derived isn't actually the standard Hull–White specification we're subsequently describing, as derivation starts from
$$dr\_t=\\kappa(\\theta−r\_t)dt+\\sigma dW\_t,$$
where $\\theta$ is constant. That's essentially the Vasicek model, but the standard one-factor Hull–White is
$$dr\_t=\[\\theta(t)−ar\_t\]dt+\\sigma dW\_t,$$
with a time-dependent drift chosen to fit today's initial yield curve?
My code:
\`\`\`
import numpy as np
\# Let's run a simulation for CIR (Cox-Ingersoll-Ross) and Hull-White models
np.newaxis
np.random.seed(42)
N = 216 # Monthly steps from 2008 to 2026
dt = 1 / 12
\# 1. CIR Model simulation: dr = theta \* (mu - r\_t)dt + sigma \* sqrt(r\_t) \* dW\_t
theta\_cir = 0.4
mu\_cir = 3.3
sigma\_cir = 0.35 # Must satisfy Feller condition: 2 \* theta \* mu >= sigma\^2 to stay positive
r\_cir = np.zeros(N)
r\_cir\[0\] = 4.65
for i in range(1, N):
\# Ensure non-negative inside sqrt
r\_prev = max(0.0, r\_cir\[i-1\])
dr = theta\_cir \* (mu\_cir - r\_prev) \* dt + sigma\_cir \* np.sqrt(r\_prev) \* np.sqrt(dt) \* np.random.randn()
r\_cir\[i\] = r\_prev + dr
\# 2. Hull-White Model (Time-varying mean theta(t) or drift to fit term structure)
\# Simplest time-varying mean formulation: theta(t) matches a shifting trend
theta\_hw = 0.35
sigma\_hw = 0.55
r\_hw = np.zeros(N)
r\_hw\[0\] = 4.65
for i in range(1, N):
t\_val = 2008 + i \* dt
\# Let the long-term mean drift higher post-2021 to capture the inflation regime shift
mu\_t = 3.0 if t\_val < 2021 else 4.2
dr = theta\_hw \* (mu\_t - r\_hw\[i-1\]) \* dt + sigma\_hw \* np.sqrt(dt) \* np.random.randn()
r\_hw\[i\] = r\_hw\[i-1\] + dr
print(f"CIR Model 2026 Terminal Value: {r\_cir\[-1\]:.2f}%")
print(f"Hull-White (Regime-Shift) 2026 Terminal Value: {r\_hw\[-1\]:.2f}%")
\##CIR Model 2026 Terminal Value: 4.71%
\##Hull-White (Regime-Shift) 2026 Terminal Value: 5.01%
\`\`\`
Thanks! ❤️