Cox-Ingersoll-Ross (CIR) Model

Mathematical Formulation

The CIR model (1985) introduces a square-root term in the diffusion component to prevent negative interest rates:

\[ dr_t = \kappa(\theta - r_t)dt + \sigma \sqrt{r_t} dW_t \]

The Feller condition, \(2\kappa\theta > \sigma^2\), ensures that the rate remains strictly positive.

Python Simulation

Code

import numpy as np
import matplotlib.pyplot as plt

# 1. Set Parameters
r0 = 0.05
theta = 0.08
kappa = 0.6
sigma = 0.15 # Higher sigma to test boundaries
T = 10.0
dt = 0.005
N = int(T / dt)
num_paths = 5

# 2. Simulation Loop
t = np.linspace(0, T, N+1)
rates = np.zeros((N+1, num_paths))
rates[0, :] = r0

np.random.seed(101)

for i in range(1, N+1):
    # Ensure the term under sqrt is non-negative
    r_prev = np.maximum(rates[i-1, :], 0)
    
    dr = kappa * (theta - r_prev) * dt + \
         sigma * np.sqrt(r_prev) * np.sqrt(dt) * np.random.normal(size=num_paths)
    
    rates[i, :] = r_prev + dr

# 3. Visualization
plt.figure(figsize=(10, 6))
plt.plot(t, rates, lw=1.5)
plt.axhline(theta, color='r', linestyle='--', label='Theta')
plt.title('CIR Model Simulation')
plt.xlabel('Time (Years)')
plt.ylabel('Interest Rate')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()