Monte Carlo Methods to Financial Problems

This repository contains a collection of Monte Carlo simulation techniques applied in the field of financial mathematics and statistical analysis. It includes simulations for generating financial returns and estimating order statistics from a Cauchy distribution, highlighting different methodologies and computational strategies.

• Modeling Financial Returns
• Cauchy Distribution Order Statistics
• Normal Random Variables Generation Using Box-Muller Method

• Description: This script simulates financial returns using different probability distributions, namely the Cauchy and logistic distributions. It demonstrates the process of parameterization, simulation, and graphical analysis of financial returns.
• Methodology: The simulation involves generating returns, adjusting parameters to fit certain statistical properties, and analyzing the behavior of these returns over time.
• Usage: Execute modeling_financial_returns.m in MATLAB. The script includes parameter setting, generating returns, and plotting the cumulative and averaged returns.

This section presents two Monte Carlo methods for estimating order statistics from a Cauchy distribution, focusing on computational efficiency and statistical theory application.

• Description: Generates samples from a uniform distribution, transforms them to a Cauchy distribution, and then sorts them to find specific order statistics.
• Methodology: Involves direct application of distribution transformations and sorting.

• Description: Uses the Beta distribution to directly sample specific order statistics for a Cauchy distribution.
• Methodology: Leverages the statistical properties of the Beta distribution.

Usage: Execute CauchyOrderStatsSamplingComparison.m in MATLAB for a comparison with the traditional method.

• Description: This script generates pairs of independent, normally distributed random variables using the Box-Muller transformation. It contrasts typical uniform random number generation with a custom linear congruential generator defined by $x_{n+1} = (383x_{n} + 263)$ mod 10000. This method highlights the versatility of Monte Carlo methods in generating samples from a desired distribution, given a source of uniform random numbers.
• Methodology: A custom linear congruential generator is used to produce uniform random numbers, which are then transformed into normal random variables using the Box-Muller algorithm. The script generates 5,000 pairs of normal random variables and visualizes them to assess their independence.
• Usage: Run the script GenerateNormalRNG_BoxMuller.m in MATLAB. The figures produced include a scatter plot of the generated pairs and histograms for each variable, allowing for visual inspection of the normality and independence of the samples.
• Validation: The script includes timing of the generation process to evaluate the performance of using a custom RNG with the Box-Muller method. The histograms and scatter plot provide a visual means to assess the quality and independence of the generated samples.

• Clone the repository and navigate to the specific project folder. Each folder contains MATLAB scripts with descriptive comments.

We welcome contributions. Please follow the existing style and add clear comments. For major changes, open an issue first to discuss your proposed changes.

This project is licensed under the MIT License - see the LICENSE file for details.