Geometric Brownian Motion (GBM) Calculator

Initial Value S0:

Drift (mu):

Volatility (sigma):

Time Horizon T:

Steps n:

Simulations:

Decimal Places: 2 3 5 8

Clear Reset

Introduction

Geometric Brownian Motion models the continuous-time evolution of a variable $S_{t}$ whose rate of change is proportional to its current value, combining deterministic drift $μ$ with random fluctuations governed by volatility $σ$ . Analysing this process provides insight into probabilistic path behaviour, multiplicative growth dynamics, and the distributional properties that arise within stochastic modelling.

What this calculator does

The tool performs an iterative simulation of discrete time steps to model a random walk with specific growth and diffusion characteristics. It requires inputs for the initial value, drift coefficient, volatility, total time horizon, and the number of steps and runs. It outputs detailed sample distribution statistics, including the mean, median, skewness, and excess kurtosis, alongside a stochastic analysis of growth probability and average peak-to-trough declines.

Formula used

The simulation employs the Euler-Maruyama method to approximate the stochastic differential equation. The state update at each time increment $Δ t$ is governed by the exponential of the drift and diffusion components. Here, $S_{t}$ is the value at time $t$ , $μ$ is the drift, $σ$ is the volatility, and $ε$ is a random variable from a standard normal distribution.

S_{t + Δ t} = S_{t} \cdot \exp ((μ - \frac{σ^{2}}{2}) Δ t + σ \sqrt{Δ t} ε)

Theoretical Median = S_{0} \cdot \exp ((μ - \frac{σ^{2}}{2}) T)

How to use this calculator

1. Enter the starting value $S_{0}$ and the drift coefficient.
2. Input the volatility, total time horizon, and specify the number of discrete steps and simulation runs.
3. Select the desired decimal precision for the numerical results.
4. Execute the calculation to view the statistical table, path visualisations, and probability density fits.

Example calculation

Scenario: A researcher in environmental science models the growth of a protected population over one year, assuming a constant growth rate and random fluctuations in environmental conditions.

Inputs: $S_{0} = 100$ , $μ = 0.1$ , $σ = 0.2$ , $T = 1$ .

Working:

Step 1: $Median = S_{0} \cdot e^{(μ - 0.5 σ^{2}) T}$

Step 2: $Median = 100 \cdot e^{(0.1 - 0.5 \cdot {0.2}^{2}) \cdot 1}$

Step 3: $Median = 100 \cdot e^{0.08}$

Step 4: $Median \approx 100 \cdot 1.083287$

Result: 108.33

Interpretation: The theoretical median value suggests that half of the simulated population outcomes will fall above 108.33 after one time unit.

Summary: The simulation demonstrates a positive central tendency despite individual path variance.

Understanding the result

The output provides a comprehensive overview of the terminal distribution. A positive skewness value indicates that the distribution of final states is skewed to the right, a classic property of the lognormal distribution. The comparison between the empirical average path and the theoretical expectation reveals how well the simulation converges as the number of runs increases.

Assumptions and limitations

The model assumes that the drift and volatility parameters remain constant throughout the entire time horizon. It also assumes that the underlying shocks follow a standard normal distribution and that the process is continuous, meaning there are no discrete jumps in value.

Common mistakes to avoid

A frequent error is confusing the drift coefficient with the mean growth of the median; the median grows at a rate of $μ - 0.5 σ^{2}$ . Users should also avoid using extremely high volatility values without a sufficient number of steps, as this can lead to poor approximation of the continuous process.

Sensitivity and robustness

The final mean is highly sensitive to the drift coefficient, while the spread and skewness of the distribution are dominated by the volatility parameter. The model is stable for moderate parameter ranges but requires a high number of simulation runs to produce robust estimates for higher-order moments like kurtosis.

Troubleshooting

If the simulated mean deviates significantly from the theoretical expectation, increase the number of simulation runs to improve convergence. If the paths appear too jagged, increase the number of steps to provide a finer discrete approximation of the continuous time process.

Frequently asked questions

Why is the median lower than the mean?

Because the process follows a lognormal distribution, the mean is influenced by infrequent but very high values in the right tail, whereas the median reflects the 50th percentile.

What does the R-squared value represent?

The R-squared value measures the fit between the empirical average of all simulated paths and the theoretical expected growth curve.

What is the maximum drawdown?

It represents the average of the largest peak-to-trough percentage declines observed across all individual simulated paths.

Where this calculation is used

This statistical modelling technique is widely used in population biology to predict the growth of species under uncertainty and in environmental science to model the diffusion of particles in fluid systems. In academic research, it serves as a fundamental example of a Markov process where the future state depends only on the current value. It is also an essential tool in probability theory for demonstrating the properties of Ito calculus and the relationship between normal and lognormal distributions through stochastic simulation.

Results are based on standard mathematical and statistical methods and may involve rounding or approximation. If precise accuracy is required, please verify results independently. See full disclaimer.