Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Monte Carlo Simulation for Derivative Pricing › Visualizing Simulated Paths

Michael BrenndoerferDecember 2, 20252 min read

Quantitative FinanceAll chapters

Part 6 of 15Numerical Methods for Derivative Pricing

Chapter 34Monte Carlo Simulation for Derivative Pricing › Visualizing Simulated Paths

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Seeing sample paths helps build intuition for what Monte Carlo computes. Each path represents one possible future for the underlying asset under risk-neutral dynamics.

In[26]:

Code

np.random.seed(123)

n_paths_display = 20
n_steps = 252
dt = T / n_steps
times = np.linspace(0, T, n_steps + 1)

# Generate paths
paths = []
for _ in range(n_paths_display):
    Z = np.random.standard_normal(n_steps)
    log_increments = (r - 0.5 * sigma**2) * dt + sigma * np.sqrt(dt) * Z
    log_S = np.log(S0) + np.concatenate([[0], np.cumsum(log_increments)])
    S_path = np.exp(log_S)
    paths.append(S_path)

Out[27]:

Visualization

Twenty blue simulated GBM price paths spreading out over one year, with horizontal reference lines at the 100-dollar initial price and 105-dollar strike. — Twenty simulated price paths under geometric Brownian motion with risk-neutral drift. The trajectories illustrate the stochastic diffusion process, with individual paths diverging from the starting price as uncertainty increases over the one-year horizon.

Each path tells a story. Some drift upward, finishing well above the strike and generating substantial payoffs. Others decline, finishing below the strike with zero payoff. The Monte Carlo price is the average of these discounted payoffs, which is a numerical approximation of the risk-neutral expectation.