Model stochastic differential equations with controllable numerical steps. Estimate paths, moments, and terminal distributions quickly. Track random motion accurately across time using intuitive controls.
Calculated values appear here after submission and before the form.
| Time | Representative Path | Mean Across Paths | Min Path | Max Path |
|---|
For geometric Brownian motion, drift is applied as μX and diffusion as σX. Other models adjust these coefficients automatically.
| Model | X₀ | μ | σ | θ | m | T | Δt | Paths |
|---|---|---|---|---|---|---|---|---|
| Geometric Brownian Motion | 100 | 0.08 | 0.20 | 1.50 | 90 | 1.00 | 0.01 | 150 |
| Ornstein-Uhlenbeck | 75 | 0.00 | 0.35 | 2.20 | 70 | 2.00 | 0.02 | 120 |
The Euler Maruyama update approximates a stochastic differential equation of the form dX = a(X,t)dt + b(X,t)dW with the stepwise recursion:
Xn+1 = Xn + a(Xn, tn)Δt + b(Xn, tn)ΔWn
Here, ΔWn is a normal random increment with mean zero and variance Δt, commonly generated as √Δt · Z where Z ~ N(0,1).
a(X,t)=μX, b(X,t)=σXa(X,t)=θ(m-X), b(X,t)=σa(X,t)=μ, b(X,t)=σEuler–Maruyama accuracy depends on the ratio between horizon and step size. Smaller Δt reduces local truncation error and improves weak approximation for expected values. In practical simulations, halving the step can change tail behavior when volatility is large. Analysts compare grids to verify convergence before using outputs in valuation, stress testing, or demonstrations.
Drift controls the deterministic direction of the process. Positive drift shifts the average path upward over time, while negative drift lowers the expected trajectory. In geometric Brownian motion, drift scales with the current state and produces compounding behavior. In additive noise models, drift contributes a constant increment each step. Reviewing mean path values beside a representative path helps separate trend from fluctuation.
Volatility determines the amplitude of random shocks entering each increment. As σ increases, the gap between minimum and maximum simulated paths widens, and terminal standard deviation rises. High volatility can generate skewed outcomes in multiplicative models. Comparing terminal mean, standard deviation, and range provides a summary of uncertainty created by stochastic forcing.
When the Ornstein–Uhlenbeck option is selected, θ sets the speed of reversion toward the long-run mean m. Larger θ values pull the process back aggressively after shocks, reducing sustained deviations. This behavior is useful for rates, spreads, temperatures, and quantities that oscillate around equilibrium. The calculator shows how the mean path aligns with the target level.
A single stochastic path can mislead because one random draw may sit far above or below the expected pattern. Running many paths reveals the distribution of possible outcomes. With more paths, terminal statistics become more stable and less sensitive to one realization. This matters when the simulation is used to estimate averages, compare parameter scenarios, or communicate risk to stakeholders.
The results panel combines numerical summaries, tabular checkpoints, and a Plotly line graph so users can interpret simulations from several angles. Exporting CSV supports analysis in spreadsheets or statistical tools, while PDF output preserves a documented scenario. A professional workflow records assumptions, compares parameter sets, and explains whether conclusions rely on expected value, dispersion, or worst-case path behavior.
It approximates stochastic differential equations by replacing continuous random evolution with discrete time steps and random normal increments.
Step size controls discretization error. Smaller values usually improve approximation quality, but they also increase computation time and output length.
Use it for processes that tend to move back toward a long-run mean, such as spreads, temperatures, or some interest-rate style series.
It displays one simulated realization. It is useful for intuition, but the mean and dispersion statistics give a more reliable summary.
Multiple paths reduce dependence on one random draw and provide better estimates of expected behavior, dispersion, and terminal extremes.
Yes. The calculator includes CSV export for data analysis and PDF export for documentation, sharing, or archived scenario reviews.
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.