Lévy-based Stochastic Models for Financial Engineering

In recent decades, attention to the study and improvement of mathematical modeling applied to the financial world has grown exponentially. This dissertation focuses on mathematical analysis, calibration and evaluation of option pricing methods and density functions through the lens of advanced stochastic processes, particularly the Lévy processes. Starting with the Black-Scholes model and identifying its limitations, the need to use more advanced models to better capture the complexity that describe the financial market is discussed. Despite its wide popularity, the Black-Scholes model is unable to explain large and abrupt price changes (jumps) and the empirical observation of volatility smiles, making it necessary to explore alternative processes. Following, the mathematical complexities of jump processes are explored, starting with the Poisson process and ending with more complex Lévy processes. Key concepts such as the Lévy-Khintchine formula and the Lévy-Itô decomposition are discussed. Lévy processes are classified into jump diffusion models and pure jump models. Merton’s and Kou’s models are explored, highlighting their ability to incorporate jumps and improve the limitations of the Black-Scholes model. Infinite activity models, such as the Variance Gamma model and the Normal Inverse Gaussian model, are also examined for their efficiency in capturing market behavior. All of these processes are fundamental to modeling asset price dynamics more accurately. Continuing, the Fourier transform and the fast Fourier transform (FFT) algorithm are presented as powerful tools for option pricing. These methods allow options prices to be calculated efficiently, offering a substantial improvement over traditional numerical techniques. In addition, parallel to the theoretical study, practical applications obtained through programming in Python are presented. All functions used to reproduce the processes described are reported. Through the use of Python, the calibration and analysis of the chosen models and datasets is of crucial importance. Mean absolute percentage error (MAPE) is used to evaluate the accuracy of the various option pricing models, starting from the Black-Scholes model to Merton, Kou, Variance Gamma and Normal Inverse Gaussian to studying the densities associated with these processes. This thesis highlights the value of using advanced stochastic models, particularly those that incorporate jumps, to better capture the dynamics of financial markets. MAPE-based comparative analysis demonstrates the superiority of newer models in fitting to market data, making these advanced models practical and robust for financial markets.