Monte Carlo Simulation for Probability, Integration, and Price Paths
Summary
This MATLAB note introduces Monte Carlo estimation through repeated random sampling. It uses the Monty Hall problem to illustrate estimating a win probability by switching doors, and compares simulated frequencies with the theoretical probability. It then describes estimating the area between two curves and a three-dimensional volume by sampling points in bounding regions and counting how many fall inside the target region.
The trading examples simulate stock-price paths using geometric Brownian motion and show how normally distributed increments can generate many possible outcomes. A further example builds price paths from log-price increments under Black–Scholes assumptions. The note presents sample simulations and histograms, but does not assess model fit, parameter estimation, or forecasting accuracy. Its stock models rely on simplified stochastic assumptions, and the simulation outputs are illustrative rather than evidence that real market prices follow those processes. More trials can stabilize estimates, but do not correct a misspecified model.
Key ideas
- Monte Carlo methods estimate probabilities by repeating random experiments and measuring outcome frequencies.
- Point sampling can approximate areas and volumes by counting samples inside a target region.
- Geometric Brownian motion can be used to generate simulated stock-price paths.
- A Black–Scholes-style log-price process produces many possible paths from assumed drift and volatility.
- Simulation accuracy depends on both sample size and the suitability of the model assumptions.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.