Exercise: Simulating a GBM Path and Checking Closed-Form Moments

Prerequisites: Geometric Brownian Motion, Log-Normal Distribution

Problem

Fix $S_0 = 100$ , $\mu = 0.10$ , $\sigma = 0.25$ , $T = 2$ years.

Using the exact log-space scheme (Example 2 in the lesson), simulate $N = 50{,}000$ independent GBM paths with $n = 504$ time steps each (two years at 252 trading days per year). Record only the terminal values $S_T$ .
Compute the sample mean, sample median, and sample standard deviation of $S_T$ . Compare each to the closed-form values $\mathbb{E}[S_T]$ , $\text{Median}(S_T)$ , and $\sqrt{\operatorname{Var}(S_T)}$ from the lesson.
Verify empirically that $\ln S_T$ is approximately Gaussian. Compute the sample mean and standard deviation of $\ln S_T$ and compare to $\ln S_0 + (\mu - \tfrac{1}{2}\sigma^2)T$ and $\sigma\sqrt{T}$ .
Estimate $\mathbb{P}(S_T < S_0)$ — the probability that the stock has fallen after two years. Compare to the closed-form value $\Phi\!\left(-\frac{(\mu - \tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}\right)$ .

Use rng = np.random.default_rng(2026) for reproducibility. Since only terminal values are needed, simulate with a single step per path:

S_T = S_0 \exp((\mu - \tfrac{1}{2}\sigma^2)T + \sigma\sqrt{T}\,Z)

with

Z \sim \mathcal{N}(0, 1)

. For part 4, use from scipy.stats import norm and compare (S_T < S0).mean() to norm.cdf(-((mu - 0.5*sigma**2)*T) / (sigma*np.sqrt(T))).

Jump to the solution when you're ready.