This project implements and compares different numerical methods for simulating Geometric Brownian Motion (GBM), a stochastic process commonly used in financial mathematics and quantitative finance.
Geometric Brownian Motion is described by the stochastic differential equation (SDE):
Where:
-
$X_t$ is the process value at time t -
$\mu$ is the drift coefficient (expected return) -
$\sigma$ is the volatility (diffusion coefficient) -
$dW_t$ is the increment of a Wiener process (standard Brownian motion)
The project compares two numerical methods:
- Exact solution - using the analytical formula
- Euler-Maruyama method - a first-order numerical scheme for SDEs
The implementation is written in Rust using the Peroxide library for numerical computations. It includes:
- A
GBMstruct with parameters for the stochastic process - Methods for generating sample paths using both exact and Euler-Maruyama methods
- Data output to Parquet format for further analysis
Visualization is handled through a Python script (pq_plot.py) which:
- Reads the simulation data from the Parquet file
- Creates a plot comparing the exact solution, Euler-Maruyama approximation, and the drift component
- Uses the scienceplots package for publication-quality plots
cargo run --releaseThis generates a GBM.parquet file containing the simulation results.
python pq_plot.pyThis creates a plot.png file visualizing the simulation results.
The current simulation uses the following parameters:
- Initial value (x0): 1.0
- Drift coefficient (μ): 1.0
- Volatility (σ): 0.5
- Time step (dt): 1e-4
- Number of steps: 100,000
- Random seed: 42
- peroxide: For numerical computation and data handling
- pandas: For data manipulation
- matplotlib: For plotting
- scienceplots: For publication-quality plot styles
- pyarrow: For reading Parquet files
