Simulating Brownian Motion Paths for Monte Carlo Pricing
Summary
The article explains how to simulate standard Brownian motion and a process with constant drift and volatility using discretized time steps. It applies the recursive update to many paths at once with vectorized arrays, then plots the paths and estimates the distribution of their ending values. This connects stochastic process simulation to Monte Carlo methods commonly used to value options.
For the standard process, the sampled increments are scaled by the square root of the time step; the extended process adds a drift increment and scales the random term by volatility. In the example, a kernel density estimate and sample statistics provide a rough check against the expected zero mean and unit variance at the end of the interval. The author notes that the estimate is based on only 50 paths, so observed values differ from theoretical ones; more paths should improve the estimate. The tutorial assumes constant parameters and equally spaced steps, leaving time-varying dynamics and other Brownian motion variants for later work.
Key ideas
- Brownian motion paths can be generated recursively from independent standard normal increments scaled by the square root of each time step.
- A constant drift adds a deterministic increment at each step, while volatility scales the random increment.
- Vectorized arrays can generate and update many paths together.
- The distribution of simulated terminal values can be compared with theoretical expectations as a basic diagnostic.
- A small number of paths produces noisy sample statistics, and the examples assume constant parameters and equal time steps.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.