Correlated Brownian Motion (Grzelak)¶

Background¶

Correlated Brownian motions with various correlation parameters.

Demonstrates the generation of two correlated Brownian motions using standardized normal increments. Illustrates how correlation affects the joint behavior of stochastic processes.

Reference: Oosterlee & Grzelak (2019). Mathematical Modeling and Computation in Finance. World Scientific.

Code¶

```python

-- coding: utf-8 --¶

""" Correlated Brownian motions with various correlation parameters.

Demonstrates the generation of two correlated Brownian motions using standardized normal increments. Illustrates how correlation affects the joint behavior of stochastic processes.

Reference: Oosterlee & Grzelak (2019). Mathematical Modeling and Computation in Finance. World Scientific. """

import numpy as np import matplotlib.pyplot as plt

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1. Core Computation¶

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def generate_paths_correlated_bm(num_paths, num_steps, t, rho): """ Generate correlated Brownian motion paths.

Parameters
----------
num_paths : int
    Number of Monte Carlo paths.
num_steps : int
    Number of time steps.
t : float
    Terminal time.
rho : float
    Correlation coefficient between W1 and W2.

Returns
-------
paths : dict
    Dictionary containing:
    - 'time': time grid of shape (num_steps+1,)
    - 'W1': W1 paths of shape (num_paths, num_steps+1)
    - 'W2': W2 paths of shape (num_paths, num_steps+1)
"""
z1 = np.random.normal(0.0, 1.0, (num_paths, num_steps))
z2 = np.random.normal(0.0, 1.0, (num_paths, num_steps))
w1 = np.zeros((num_paths, num_steps + 1))
w2 = np.zeros((num_paths, num_steps + 1))

dt = t / float(num_steps)
time = np.zeros(num_steps + 1)

for i in range(0, num_steps):
    # Ensure samples from normal have mean 0 and variance 1
    if num_paths > 1:
        z1[:, i] = (z1[:, i] - np.mean(z1[:, i])) / np.std(z1[:, i])
        z2[:, i] = (z2[:, i] - np.mean(z2[:, i])) / np.std(z2[:, i])

    # Correlate noises
    z2[:, i] = rho * z1[:, i] + np.sqrt(1.0 - rho**2) * z2[:, i]

    w1[:, i + 1] = w1[:, i] + np.sqrt(dt) * z1[:, i]
    w2[:, i + 1] = w2[:, i] + np.sqrt(dt) * z2[:, i]

    time[i + 1] = time[i] + dt

# Store the results
paths = {"time": time, "W1": w1, "W2": w2}
return paths

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2. Plotting Functions¶

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def plot_paths(time_grid, w1, w2, title): """ Plot Brownian motion paths.

Parameters
----------
time_grid : ndarray
    Time grid of shape (num_steps+1,).
w1 : ndarray
    W1 paths of shape (num_paths, num_steps+1).
w2 : ndarray
    W2 paths of shape (num_paths, num_steps+1).
title : str
    Title for the figure.

Returns
-------
None
"""
plt.figure()
plt.plot(time_grid, np.transpose(w1))
plt.plot(time_grid, np.transpose(w2))
plt.grid()
plt.xlabel("time")
plt.ylabel("W(t)")
plt.title(title)

=============================================================================¶

3. Main Computation¶

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def main(): """Run demonstration of correlated Brownian motions.""" # Parameters num_paths = 1 num_steps = 500 t = 1.0

# ===== Negative correlation =====
rho = -0.9
paths = generate_paths_correlated_bm(num_paths, num_steps, t, rho)
time_grid = paths["time"]
w1 = paths["W1"]
w2 = paths["W2"]
plot_paths(time_grid, w1, w2, "Correlated BM (rho = -0.9)")

# ===== Positive correlation =====
rho = 0.9
paths = generate_paths_correlated_bm(num_paths, num_steps, t, rho)
time_grid = paths["time"]
w1 = paths["W1"]
w2 = paths["W2"]
plot_paths(time_grid, w1, w2, "Correlated BM (rho = 0.9)")

# ===== Zero correlation =====
rho = 0.0
paths = generate_paths_correlated_bm(num_paths, num_steps, t, rho)
time_grid = paths["time"]
w1 = paths["W1"]
w2 = paths["W2"]
plot_paths(time_grid, w1, w2, "Correlated BM (rho = 0.0)")

if name == "main": main() ```

Exercises¶

Exercise 1. To generate two correlated Brownian motions \(W_1, W_2\) with \(\text{Corr}(W_1, W_2) = \rho\), write the construction using independent normals \(Z_1, Z_2\).

Exercise 2. For \(\rho = -0.9\), describe qualitatively how the two Brownian motion paths relate visually.

Exercise 3. Verify that the Cholesky decomposition of \(\Sigma = \begin{pmatrix}1 & \rho\\\rho & 1\end{pmatrix}\) gives \(L = \begin{pmatrix}1 & 0\\\rho & \sqrt{1-\rho^2}\end{pmatrix}\), and that \(LL^T = \Sigma\).

Exercise 4. For a 3-factor model with correlation matrix \(\Sigma_{3\times3}\), explain how the Cholesky decomposition generalizes.