MATLAB Code Implementation for Monte Carlo Simulation

Monte Carlo simulation is a numerical computation method based on random sampling, widely applied in fields such as finance, engineering, and physics. The core concept involves approximating solutions to deterministic or stochastic problems through extensive random experiments. Implementing Monte Carlo simulations in MATLAB typically involves the following key steps and programming logic: Problem Modeling The first step requires defining the target problem and converting it into a probability model. For example, when calculating the value of π, one can estimate it by randomly scattering points within a unit square and counting the proportion that falls inside the inscribed quarter circle. In MATLAB code, this involves defining geometric boundaries and creating logical conditions for point classification. Random Number Generation MATLAB provides comprehensive random number functions (such as `rand` for uniform distribution and `randn` for normal distribution) to generate random samples. By adjusting parameters of these functions, developers can simulate input variables under different scenarios. For instance, `rand(1,n)` generates n uniformly distributed random numbers between 0 and 1, while `randn(m,n)` creates an m×n matrix of standard normal random variables. Looping and Vectorization Monte Carlo simulations typically require numerous repeated experiments. Beginners can implement this using `for` loops, but MATLAB recommends vectorized operations for efficiency. For example, instead of processing points individually, generate million-point matrices directly for computation. Vectorized code like `points = rand(2,N);` creates a 2×N matrix of random coordinates, allowing simultaneous processing of all points. Result Statistics and Analysis Each experiment's results need recording and statistical analysis. Common operations include calculating mean values, variances, and confidence intervals. MATLAB's statistical toolbox functions (such as `mean`, `std`, and `quantile`) simplify this process. For reliability analysis, developers can use functions like `prctile` to determine confidence bounds or `var` to compute result variability. Visualization Use functions like `plot`, `scatter`, or `histogram` to create distribution charts or convergence curves that visually demonstrate simulation effects. For example, as sampling次数 increases, the estimated value of π gradually approaches the true value, which can be visualized using `plot(1:N, cumulative_estimates)` to show convergence trends. Extension Strategies Application Scenarios: Beyond classic cases, try using Monte Carlo for option pricing models (using geometric Brownian motion), signal noise analysis, or complex system reliability assessments. Implement financial simulations using stochastic differential equations with Euler-Maruyama discretization. Optimization Techniques: Improve large-scale simulation efficiency through parallel computing (`parfor` loops) or GPU acceleration (`gpuArray` functions). For option pricing, combine with antithetic variates for variance reduction. Error Control: Dynamically adjust sampling sizes using while-loops with precision checks to ensure results meet preset accuracy thresholds, avoiding unnecessary computations. Implement adaptive sampling algorithms that stop when the standard error falls below a specified tolerance. For beginners, start with simple geometric probability problems and gradually progress to more complex real-world applications, while referring to MATLAB documentation to learn efficient programming practices such as preallocation of arrays and anonymous function usage.