Brownian Motion Calculator
Example Data Table
| Scenario | Initial Value | Drift | Volatility | Time Horizon | Steps | Paths | Seed |
|---|---|---|---|---|---|---|---|
| Balanced Baseline | 0.00 | 0.25 | 1.10 | 1.00 | 24 | 250 | 42 |
| Low Noise | 2.00 | 0.10 | 0.40 | 2.00 | 48 | 500 | 17 |
| High Noise | -1.00 | 0.05 | 1.80 | 1.50 | 60 | 1000 | 84 |
| Negative Drift | 5.00 | -0.30 | 0.90 | 1.00 | 36 | 300 | 99 |
Formula Used
Standard Brownian motion with drift is modeled as X(t) = X₀ + μt + σW(t), where W(t) is a Wiener process.
For each discrete step, the tool uses:
X₍ᵢ₊₁₎ = Xᵢ + μΔt + σ√Δt · Zᵢ
Here, Δt = T / N and Zᵢ follows a standard normal distribution. Theoretical moments for the final value are:
- Expected final value:
E[X(T)] = X₀ + μT - Variance of final value:
Var[X(T)] = σ²T - Standard deviation:
σ√T
The simulation compares these theoretical values with numerical outcomes from many sampled paths.
How to Use This Calculator
- Enter the initial value for the process.
- Set the drift to represent average directional change.
- Set volatility to control randomness strength.
- Choose a time horizon and total time steps.
- Select how many paths to simulate.
- Optionally enter a random seed for repeatable results.
- Press Run Simulation to view results above the form.
- Use the export buttons to save CSV or PDF copies.
FAQs
1. What does this Brownian motion tool calculate?
It simulates a stochastic path using drift, volatility, time, and random shocks. It also summarizes the final distribution, theoretical moments, and a sample path table.
2. Why are simulated and theoretical results slightly different?
Simulation uses random sampling, so outcomes vary around the theoretical target. Increasing the number of simulated paths usually narrows the gap.
3. What is the role of drift?
Drift is the average directional trend over time. Positive drift shifts the process upward, while negative drift pushes expected values downward.
4. What does volatility change?
Volatility controls how widely paths fluctuate. Higher volatility produces larger random swings and a broader final distribution.
5. Why use a random seed?
A random seed makes the same inputs reproduce the same sampled path and summary outputs. That helps with validation, comparison, and teaching.
6. Can this tool support finance or physics examples?
Yes. Brownian motion appears in finance, diffusion studies, statistical physics, and stochastic modeling. The tool is generic, so you can adapt inputs to many contexts.
7. What improves simulation accuracy?
Using more paths improves distribution estimates, while more steps improve time resolution. Balanced settings depend on your accuracy needs and available computing resources.