Parameter Estimation for Geometric Lévy Processes with Constant Volatility

Abstract

In finance, various stochastic models have been used to describe price movements of financial instruments. Following the seminal work of Robert Merton, several jump-diffusion models have been proposed for option pricing and risk management. In this study, we augment the process related to the dynamics of log returns in the Black–Scholes model by incorporating alpha-stable Lévy motion with constant volatility. We employ the sample characteristic function approach to investigate parameter estimation for discretely observed stochastic differential equations driven by Lévy noises. Furthermore, we discuss the consistency and asymptotic properties of the proposed estimators and establish a Central Limit Theorem. To further demonstrate the validity of the estimators, we present simulation results for the model. The utility of the proposed model is demonstrated using the Dow Jones Industrial Average data, and all parameters involved in the model are estimated. In addition, we delved into the broader implications of our work, discussing the relevance of our methods to big data-driven research, particularly in the fields of financial data modeling and climate models. We also highlight the importance of optimization and data mining in these contexts, referencing key works in the field. This study thus contributes to the specific area of finance and beyond to the wider scientific community engaged in data science research and analysis.