Applied Statistics for Finance

A comprehensive course implementation covering advanced statistical methods and computational techniques for financial modeling and derivatives pricing.

Professors: Alice Pignatelli di Cerchiara & Luca Fraone
Institution: Università Cattolica del Sacro Cuore
Languages: R, Python

Course Overview

This repository contains a complete implementation of statistical methods applied to finance, ranging from Monte Carlo simulation to advanced option pricing models. The course bridges theoretical finance with practical computational implementation, emphasizing hands-on coding and real-world applications.

Key Technologies & Methods

Statistical Computing

• Monte Carlo Methods - Variance reduction techniques, antithetic sampling
• Stochastic Process Simulation - Wiener processes, Geometric Brownian Motion
• Parameter Estimation - MLE, Quasi-MLE, Method of Moments
• Time Series Analysis - Change-point detection, volatility modeling

Financial Models

• Black-Scholes Framework - European & American option pricing
• Lévy Processes - Variance Gamma, Meixner models
• Advanced SDEs - CIR, CKLS, Ornstein-Uhlenbeck processes
• FFT Pricing - Fast Fourier Transform for complex derivatives

Programming Implementation

• R Packages: quantmod, tseries, sde, yuima, stats4
• Python Libraries: Scientific computing and numerical optimization
• Numerical Methods: Euler-Maruyama, Milstein discretization schemes

Getting Started

Usage Options

1. Download HTML Report - Complete analysis with visualizations
2. Render Quarto Document - For reproducible research workflow
3. Run R Scripts - Execute individual components without Quarto

Core Topics Implemented

1. Random Number Generation & Monte Carlo

• Inverse transformation and acceptance-rejection methods
• Pseudo-random number generation with reproducible seeds
• Monte Carlo integration for financial derivatives pricing

2. Stochastic Processes Simulation

3. Advanced Parameter Estimation

• Maximum Likelihood Estimation for complex distributions
• Quasi-MLE for model misspecification robustness
• Method of Moments for distributions without closed-form densities

4. European Options Pricing

• Risk-neutral valuation framework
• Multiple risk-neutralization methods (Esscher, Mean-correcting martingale)
• Variance reduction techniques implementation

5. Lévy Processes & Jump Models

• Beyond Gaussian assumptions in financial modeling
• Variance Gamma and Meixner process implementations
• Lévy-Khintchine representation and characteristic functions

6. American Options & Longstaff-Schwartz method implementation

7. Volatility Modeling & Market Microstructure

• Implied volatility extraction from market prices
• Volatility smile/skew analysis
• Liquidity effects and bid-ask spread considerations

Practical Applications

Real Market Analysis

• Data Sources: Yahoo Finance, FRED, multiple asset classes
• Assets Covered: Equity indices (SPY, QQQ), individual stocks, commodities, bonds
• Market Phenomena: Volatility clustering, fat tails, asymmetric returns

Risk Management Tools

• Greeks Calculation - Delta, Gamma, Vega, Theta, Rho
• Sensitivity Analysis - Linear and quadratic approximations
• Model Validation - Kolmogorov-Smirnov tests, AIC model selection

Learning Outcomes Demonstrated

1. Computational Finance Proficiency - Advanced R/Python implementation
2. Statistical Modeling - From basic distributions to complex stochastic processes
3. Derivatives Pricing - Multiple methodologies and model comparisons
4. Market Data Analysis - Real-world financial time series processing
5. Risk Assessment - Quantitative risk metrics and scenario analysis

Technical Highlights

Advanced Numerical Methods

• FFT Pricing for models without closed-form solutions
• Numerical Integration for complex payoff structures
• Optimization Algorithms (Nelder-Mead, BFGS) for parameter estimation

Statistical Rigor

• Change-point Detection for structural breaks in time series
• Model Diagnostics - QQ plots, density fitting, goodness-of-fit tests
• Variance Reduction techniques for Monte Carlo efficiency

Industry-Relevant Implementation

• Market Data Integration via quantmod and tseries packages
• Production-Ready Code with error handling and optimization
• Comprehensive Documentation with mathematical foundations

Key Insights & Findings

• Non-normality in Returns - Empirical evidence across asset classes
• Model Limitations - When Black-Scholes assumptions break down
• Computational Trade-offs - Accuracy vs. efficiency in numerical methods
• Market Microstructure - Impact of liquidity on option pricing

Technical Skills Showcased

R Programming

• Advanced statistical computing and financial modeling
• Package development and optimization techniques
• Reproducible research with Quarto integration

Financial Engineering

• Derivatives pricing across multiple methodologies
• Risk-neutral measure transformations
• Advanced stochastic calculus implementation

Data Science

• Large-scale financial data processing
• Time series analysis and forecasting
• Model validation and backtesting frameworks

Note: This repository represents a comprehensive exploration of computational finance, combining theoretical rigor with practical implementation. The code demonstrates both academic understanding and industry-applicable skills in quantitative finance.

Academic Integrity: All work represents original analysis and implementation based on course materials and established financial mathematics literature.