Stochastic Processes in Financial Mathematics: A New Approach to Option Pricing

Abstract

This paper introduces a novel approach to option pricing, leveraging advanced stochastic processes to address the limitations of traditional models like Black-Scholes and the Binomial model. Classical approaches, while foundational, often fail to capture the complexities of real-world financial markets, such as stochastic volatility, fat-tailed distributions, and market jumps. The proposed model incorporates a generalized hyperbolic Lévy process and a stochastic volatility component to better reflect these market realities. By doing so, it enhances the accuracy and robustness of option pricing, particularly in volatile and non-Gaussian market environments. The paper details the theoretical foundation of the new approach, discusses its implementation using numerical methods, and conducts a comparative analysis with classical models. The results demonstrate that the new model provides superior pricing accuracy and stability across various market conditions. Practical applications and case studies are presented, showcasing the model's effectiveness in real-world scenarios. The paper concludes with suggestions for future research, including extending the model to other derivative types and further improving its computational efficiency. This new approach represents a significant advancement in the field of financial mathematics, offering a more flexible and reliable framework for option pricing.