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import numpy as np
import matplotlib.pyplot as plt

def simulate_geometric_random_walk(S0, T=2, N=1000, mu=0.1, sigma=0.05):

    dt = T / N
    t  = np.linspace(0, T, N)
    W  = np.random.standard_normal(size=N)
    # N(0,dt) = sqrt(dt)
    W = np.cumsum(W) * np.sqrt(dt)
    X = (mu - 0.5 * sigma**2) * t + sigma * W
    S = S0 * np.exp(X)
    print(W)
    
    return t, S

plt.figure(figsize=(12, 5))
t, S = simulate_geometric_random_walk(100)
plt.plot(t, S, color='steelblue', lw=1.2)
plt.title(f'Geometric Brownian Motion  μ=0.1  σ=0.05')
plt.xlabel('Time (years)')
plt.ylabel('Price')
plt.grid(True, alpha=0.3)
plt.show()


if __name__ == "__main__":

    simulate_geometric_random_walk(10)