Wiener Square Expectation Rule | Mathematical Finance and Computational Methods

Wiener.py

import numpy as np
import matplotlib.pyplot as plt

# Parameters
dt = 0.01  # Small time step
T = 1.0    # Total time
N = int(T / dt)  # Number of steps
num_paths = 5  # Number of random paths to simulate

# Simulate Wiener process paths
np.random.seed(42)  # For reproducibility
dW = np.sqrt(dt) * np.random.randn(num_paths, N)  # dW = ξ sqrt(dt)
W = np.cumsum(dW, axis=1)  # Compute Wiener process by summing increments

# Compute dW^2 and its expectation (should be close to dt)
dW_squared = dW**2
expected_dW_squared = np.mean(dW_squared, axis=0)  # Average over paths

# Plot Wiener process realizations
plt.figure(figsize=(12, 5))

# Wiener process paths
plt.subplot(1, 2, 1)
for i in range(num_paths):
    plt.plot(np.linspace(0, T, N), W[i], alpha=0.7)
plt.xlabel("Time")
plt.ylabel("$W(t)$")
plt.title("Sample Paths of Wiener Process")

# Expectation of dW^2
plt.subplot(1, 2, 2)
plt.plot(np.linspace(dt, T, N), expected_dW_squared, label=r"$\mathbb{E}[dW^2]$")
plt.axhline(dt, color="r", linestyle="--", label="Expected: $dt$")
plt.xlabel("Time")
plt.ylabel("$\mathbb{E}[dW^2]$")
plt.title("Verification of $\mathbb{E}[dW^2] = dt$")
plt.legend()

plt.tight_layout()
plt.show()