Geometric Brownian Motion Simulation¶

Background¶

Black Scholes Gbm

Educational script demonstrating black scholes gbm concepts.

Code¶

```python """ Black Scholes Gbm

Educational script demonstrating black scholes gbm concepts. """

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

black_scholes_GEOMETRIC_BROWNIAN_MOTION.py¶

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

import black_scholes as bs

Model parameters¶

if name == "main": S0 = 100 # Current stock price K = 105 # Strike price T = 1 # 1 year to expiration r = 0.05 # 5% risk-free rate sigma = 0.2 # 20% volatility q = 0.02 # 2% dividend yield

# Create the unified Black-Scholes interface
bs_model = bs.BlackScholes(S0=S0, K=K, T=T, r=r, sigma=sigma, q=q)

# risk neutral path simulation and visualization 
bs_model.plot_paths_and_histogram(
    num_paths=1000, 
    max_paths_display=50, 
    seed=42, 
    title=f'Sample GBM Paths with Final Price Distribution\nunder Risk Neutral Measure r = {r}'
    )

# risk neutral vs real world path simulation and visualization 
mu = 0.08  # 8% real-world drift
bs_model.plot_gbm_comparison(
    num_paths=1000, 
    max_paths_display=50, 
    seed=42, 
    mu=mu
    )

```

Exercises¶

Exercise 1. Write the SDE for geometric Brownian motion under the risk-neutral measure and under the real-world measure. What is the key difference?

Solution to Exercise 1

Under the real-world measure \(\mathbb{P}\): \(dS_t = \mu S_t\,dt + \sigma S_t\,dW_t^{\mathbb{P}}\)

Under the risk-neutral measure \(\mathbb{Q}\): \(dS_t = (r - q)S_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}\)

The key difference is the drift: \(\mu\) (expected return) under \(\mathbb{P}\) versus \(r - q\) (risk-free rate minus dividends) under \(\mathbb{Q}\). The volatility \(\sigma\) is the same under both measures. The change from \(\mathbb{P}\) to \(\mathbb{Q}\) is accomplished by the Girsanov theorem: \(dW_t^{\mathbb{Q}} = dW_t^{\mathbb{P}} + \frac{\mu - r + q}{\sigma}\,dt\).

Exercise 2. Simulate 5 GBM paths with \(S_0 = 100\), \(\mu = 0.08\), \(\sigma = 0.20\), \(T = 1\) using 252 daily steps. Describe the expected range of terminal values.

Solution to Exercise 2

The terminal value is \(S_T = 100\exp\bigl((0.08 - 0.02)T + 0.20\sqrt{T}\,Z\bigr) = 100\exp(0.06 + 0.20Z)\) where \(Z \sim \mathcal{N}(0,1)\).

The mean is \(E[S_T] = 100 e^{0.08} \approx 108.33\). For a 95% range: \(Z \in [-1.96, 1.96]\), giving \(S_T \in [100 e^{0.06 - 0.392}, 100 e^{0.06 + 0.392}] = [100 e^{-0.332}, 100 e^{0.452}] \approx [71.7, 157.2]\).

Five paths will show diverse behavior: some trending up, some down, all with the characteristic jagged appearance of random walks.

Exercise 3. Compare risk-neutral and real-world path simulations. Why do risk-neutral paths have a lower expected return, and why is this correct for pricing?

Solution to Exercise 3

Risk-neutral paths use drift \(r - q\) (typically 3--5%) instead of \(\mu\) (typically 8--12%). The expected terminal stock price is \(S_0 e^{(r-q)T}\) under \(\mathbb{Q}\) versus \(S_0 e^{\mu T}\) under \(\mathbb{P}\).

This is correct for pricing because the risk-neutral measure incorporates the market price of risk. Under \(\mathbb{Q}\), the discounted stock price \(e^{-(r-q)t}S_t\) is a martingale, so \(e^{-rT}E^{\mathbb{Q}}[\text{payoff}]\) gives the arbitrage-free price. Using \(\mathbb{P}\) instead would require knowing the market price of risk \(\lambda = (\mu - r)/\sigma\), which is model-dependent and hard to estimate.

Exercise 4. The code plots GBM paths with a final price distribution. Explain why the distribution is right-skewed (log-normal) and how this relates to the Black-Scholes model.

Solution to Exercise 4

The terminal price \(S_T = S_0 e^X\) where \(X \sim \mathcal{N}(\mu_X, \sigma_X^2)\) follows a log-normal distribution. Log-normal distributions are right-skewed because the exponential function maps the symmetric normal onto a positive, asymmetric domain: values can be arbitrarily large but cannot go below zero.

This is a fundamental assumption of Black-Scholes: stock prices are always positive, and log returns are normally distributed. The right skewness means large upward moves are possible (but rare), while the stock cannot lose more than 100%. The BS formula is derived by integrating the payoff against this log-normal distribution.