Vasicek Interest Rate Model

TL;DR

The Vasicek model simulates interest rates that bounce around a long-term average.It uses a mean-reverting SDE where randomness and pull-back forces combine.We simulate it using a simple time-stepping method like we did with GBM.

Vasicek model is a mathematical model used to describe how interest rates change over time.

Key idea: Mean Reversion - Interest rates don’t just wander aimlessly, they tend to go back toward some long-term average ( like gravity pulling them in).

The SDE (Stochastic Differential Equation):

\[ dr_t = k(\theta - r_t)dt + \sigma dZ_t \]

Where:

• \(r_t\) interest rate at time t
• \(\theta\) long term average interest rate (mean level)
• \(k\): speed of mean reversion (how fast we go back to \(\theta\))
• \(\sigma\): volatility (how much randomness per unit time)
• \(dZ_t\): standard Brownian motion (random noise)

Intuition Behind the formula

• If current rate \(r_t < \theta\), the drift \(k(\theta - r_t) > 0\) → rate increases
• If \(r_t > \theta\), then \(k(\theta - r_t) < 0\) → rate drops
• So the rate always tries to pull back toward \(\theta\) over time

How do we simulate it ?

To simulate this using MonteCarlo:

1. Discretize time into steps of \(\Delta t\)
2. Use this recurrence formula: \(r_{t+1} = r_t + k(\theta - r_t)\Delta t + \sigma \sqrt{\Delta t} \cdot \varepsilon\)
3. where \(\varepsilon \sim \mathcal{N}(0, 1)\)

This is the Euler discretization of the Vasicek process.

Long-Run behavior

• Long term mean = \(\theta\)
• Long term variance = \(\frac{\sigma ^2}{2k}\)

So we get interest rate paths that wiggle, but hover around the mean \(\theta\).

Python

Python

from wqu.dp import Vasicek

# Instantiate model
vas = Vasicek(r0=0.03, k=0.15, theta=0.05, sigma=0.01, seed=42)

# Plot a single path
vas.plot(M=1, color='blue')

# Plot multiple paths
vas.plot(M=10, alpha=0.3)

# Compare how 'theta' affects the mean reversion level
vas.compare_parameters('theta', [0.03, 0.05, 0.07])

Python

from wqu.dp.vasicek import Vasicek
import matplotlib.pyplot as plt
import numpy as np

# Parameters
M = 100
N = 100
T = 1.0
r0 = 0.01875
K = 0.20
theta = 0.01
sigma = 0.012

# Time grid
t = np.linspace(0, T, N + 1)

# Create and simulate
vas = Vasicek(r0=r0, k=K, theta=theta, sigma=sigma, T=T, N=N, seed=42)
paths = vas.simulate(M)  # shape (M, N+1)

# Plot each path
plt.figure(figsize=(10, 5))
for j in range(M):
    plt.plot(t, paths[j])

plt.xlabel("Time $t$", fontsize=14)
plt.ylabel("$r(t)$", fontsize=14)
plt.title("Vasicek Paths", fontsize=14)
axes = plt.gca()
axes.set_xlim([0, T])
plt.xticks(fontsize=14)
plt.yticks(fontsize=14)
plt.tight_layout()
plt.show()