Geometric Brownian Motion: Measures, Exact Simulation, and Model Risk

Audit GBM through physical and risk-neutral measures, lognormal moments, exact versus Euler simulation, Monte Carlo error, path monitoring, and model limitations.

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

Under the physical measure P, constant-parameter geometric Brownian motion is dS_t / S_t = mu dt + sigma dW_t^P. Its exact solution is S_t = S_0 exp[(mu - 0.5 sigma^2)t + sigma W_t^P], so ln(S_t / S_0) ~ N((mu - 0.5 sigma^2)t, sigma^2 t) and finite-horizon prices are strictly positive.

Here mu is the instantaneous expected price-growth parameter, not mean log-return growth. Under the risk-neutral measure Q, an ex-dividend asset with continuous yield q follows dS_t / S_t = (r - q)dt + sigma dW_t^Q. These measures answer different questions and use different Brownian motions; risk-neutral probabilities are not return forecasts.

Build and validate GBM in seven steps

State whether the task is physical forecasting, risk-neutral valuation, stress testing, numerical testing, or teaching; do not mix P and Q outputs.
Lock S_0, currency, price or total-return convention, mu or r - q, volatility, rates, dividends, horizon, timestamp, day count, compounding, and corporate-action treatment.
Write the measure-specific SDE, exact solution, and units. Under P, E[S_t] = S_0 e^(mu t), Var[S_t] = S_0^2 e^(2 mu t)(e^(sigma^2 t) - 1), and E[S_t / S_0 - 1] = e^(mu t) - 1.
Validate the log-return mean and variance, price mean, variance, median, mode, quantiles, and simple-versus-log-return distinction before using paths or payoffs.
For piecewise-constant GBM, simulate with S_(i+1) = S_i exp[(mu - 0.5 sigma^2) Delta t + sigma sqrt(Delta t) Z_i], independent Z_i ~ N(0,1), a recorded generator, and a reproducible seed. Compare Euler only as a discretization study.
Report path count, estimator, standard error, confidence interval, and convergence. For barriers, extrema, first passage, or discrete monitoring, address between-step crossings with Brownian bridges or an explicit monitoring approximation.
Separate parameter, sampling, discretization, and structural model error; stress jumps, stochastic volatility, heavy tails, leverage, default, liquidity, halts, transaction costs, settlement, and regime changes.

Four worked examples

Physical distribution moments. Let S_0 = $100, mu = 8%, sigma = 20%, and T = 1. Log-return mean is 0.06 and variance is 0.04. The median is $106.1836546545, mean $108.3287067675, mode $102.0201340027, variance $478.9188716836, and standard deviation $21.8842151261. Shocks Z = -1 and Z = +1 give $86.9358235399 and $129.6930086666; they are scenarios, not bounds.
Exact path versus Euler. With S_0 = $100, mu = 6%, sigma = 30%, Delta t = 0.25, and Z = [0.5, -1, 0.25, 1.25], exact prices are $108.1933804806, $93.4727720616, $97.4091536282, and $117.9393118711. Euler ends at $119.3300424062. The exact exponential transition preserves positivity for these constant coefficients; Euler has discretization bias and can become negative.
Risk-neutral option value. Let S_0 = $100, K = $105, r = 4%, q = 1.5%, sigma = 25%, and T = 0.75. The forward is $101.8926885052, d_1 = -0.0304963995, and d_2 = -0.2470027504; the European call value is $7.2291486749. The physical drift mu does not enter this controlled no-arbitrage price.
Monte Carlo sampling error. Under P, take S_0 = $100, mu = 7%, sigma = 25%, and T = 0.5. For S_T > $120, the threshold is z_star = 0.9217649222 and the exact probability is 0.1783256040. With N = 100,000 independent paths, theoretical Bernoulli standard error is 0.0012104775; a 95% half-width is 0.0023725358, giving the truth-centered reference interval [0.1759530681, 0.1806981400]. More paths reduce sampling error, not model error.

Input, numerical, and model risks

Mixing physical and risk-neutral measures or Brownian motions.
Treating mu as mean log-return growth or as a finite simple return.
Confusing ex-dividend price, total return, and reinvested distributions.
Mixing r, r - q, discrete dividends, compounding, and forward conventions.
Mixing calendar, trading-day, intraday, and annualized time units.
Applying volatility in percent rather than decimal or with the wrong annualization.
Using stale parameters, revised data, look-ahead information, or inconsistent timestamps.
Relying on a short, noisy estimate of drift for direction.
Assuming independent stationary Gaussian log increments in every regime.
Treating constant volatility as compatible with skew, term structure, clustering, and leverage effects.
Omitting jumps, overnight gaps, default-to-zero events, and trading halts.
Assuming a lognormal model captures heavy tails and crisis dependence.
Ignoring stochastic rates, dividends, borrow, or multi-asset correlation structure.
Using Euler when an exact GBM transition is available or ignoring its negative-price possibility.
Failing to record the generator, seed, antithetic or variance-reduction method, and reproducibility data.
Reporting paths without estimator standard error, confidence interval, and convergence.
Missing between-step barrier crossings, extrema, and first-passage events.
Confusing more time steps with lower structural model error.
Treating calibration to one price or distribution moment as model validation.
Omitting executable prices, liquidity, costs, market closure, product settlement, governance, and reconciliation.

Common misconceptions

“GBM predicts a stock price.” It defines a conditional distribution under chosen parameters.
“The log-return drift is mu.” It is mu - 0.5 sigma^2 under the physical notation used here.
“Risk-neutral probability is a real-world forecast.” It is a valuation measure under stated market assumptions.
“More steps or paths make the market model realistic.” They address discretization or sampling error, not missing economics.
“Positive lognormal prices cap downside risk.” Prices can approach zero arbitrarily closely, and GBM omits sudden default.