Analyze stochastic paths with drift, volatility, horizon, shocks. Review estimates, confidence ranges, and simulated trajectories. Built for analysts comparing uncertainty across evolving price series.
| Scenario | S0 | μ | σ | T | Z | Approx Terminal Price |
|---|---|---|---|---|---|---|
| Base Case | 100 | 0.08 | 0.20 | 1.00 | 0.50 | 117.351087 |
| Higher Risk | 100 | 0.08 | 0.35 | 1.00 | -0.30 | 94.648514 |
| Longer Horizon | 100 | 0.10 | 0.25 | 2.00 | 1.10 | 167.356879 |
Geometric Brownian Motion models a positive price path with continuous compounding and random shocks.
Stochastic differential form: dS = μS dt + σS dW
Closed form terminal price: S(T) = S0 × exp[(μ − 0.5σ²)T + σ√T × Z]
Expected price: E[S(T)] = S0 × exp(μT)
Median price: Median = S0 × exp[(μ − 0.5σ²)T]
Variance: Var[S(T)] = S0² × exp(2μT) × (exp(σ²T) − 1)
Step simulation: S(t+Δt) = S(t) × exp[(μ − 0.5σ²)Δt + σ√Δt × Z]
Enter the starting price in Initial Price.
Enter the average growth rate in Drift.
Enter uncertainty in Volatility.
Set the total forecast period in Time Horizon.
Choose the number of Simulation Steps for the path table.
Use Scenario Shock to test a chosen standard normal outcome.
Use Confidence Z for the lognormal range estimate.
Add a Random Seed when you want repeatable path results.
Press Calculate. The results appear above the form.
Use the export buttons to save the summary and simulated path.
Geometric Brownian motion is a standard stochastic process in statistics and quantitative finance. It models values that change continuously and stay positive. This calculator helps you test how drift, volatility, time, and shocks affect a projected path. You can estimate an expected future price, a median outcome, a scenario price, and a dispersion range. You can also inspect a simulated step-by-step path. That makes the tool useful for learning, forecasting, and sensitivity analysis.
GBM matters because many real variables grow multiplicatively. Asset prices, indexes, and some business metrics often behave this way over short periods. The model assumes proportional change, continuous compounding, and random movement from a normal shock. In statistics, that creates a lognormal distribution for the future level. This is helpful because the distribution remains positive. It also separates trend from uncertainty. Drift controls the average direction. Volatility controls the spread. Time expands the effect of both.
Analysts use GBM for scenario planning, Monte Carlo simulation, option inputs, and risk communication. Students use it to understand stochastic calculus and log returns. Traders use it to compare optimistic, neutral, and stressed assumptions. Researchers use it as a baseline model before applying more advanced dynamics. It is simple, fast, and easy to explain. It also connects neatly to expected return, variance, confidence ranges, and path dependence across time steps.
The expected price is the mean of the terminal distribution. The median price is the middle lognormal outcome. The scenario price applies your chosen shock directly. The variance and standard deviation show dispersion. The lower and upper bounds show a practical range based on your selected Z value. The simulated path table shows one possible route through time. That path is not the only answer. It is one random realization under your assumptions.
It is a stochastic model for positive values that evolve continuously. It combines average growth with random shocks. It is widely used for asset price modeling and simulation.
Drift is the average continuous growth rate in the model. A higher drift raises the expected future level over time, assuming all other inputs stay unchanged.
Volatility measures uncertainty. Higher volatility spreads outcomes farther apart. It increases dispersion, widens ranges, and makes simulated paths move more sharply.
Z is a chosen standard normal shock. It lets you test a specific outcome, such as a favorable or adverse move, for the terminal price formula.
GBM creates a lognormal terminal distribution. In a lognormal distribution, the mean is usually above the median. That gap becomes larger as volatility rises.
Steps control the path resolution. More steps give a more detailed route through time. They do not change the basic model assumptions by themselves.
No. It is a simplified baseline model. Real markets can include jumps, regime changes, clustering, and changing volatility that GBM does not capture well.
Use one consistent unit system. If drift and volatility are annualized, time should be in years. Consistent units keep the formulas meaningful.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.