Investigating Levy's model in financial series prediction(case of vanilla option)

In recent years, there has been growing interest in the application of stochastic processes to model financial markets, particularly in the pricing and prediction of derivative instruments such as options. One of the more advanced models that has emerged for capturing the dynamics of financial time series is the Lévy process, which generalizes the traditional Brownian motion by incorporating jumps and heavy tails, features often observed in real financial data. This paper investigates the applicability of Lévy processes in predicting the evolution of financial series, with a specific focus on vanilla option pricing. In our methodology,By reviewing the theoretical underpinnings of Lévy processes, highlighting key aspects such as the characteristic function and the variance-gamma process, we calibrate a Lévy-based model to 77 mid-prices of a set of European call options on the S&P 500 Index at the close of the market on 11 April 2022. We employ maximum likelihood estimation (MLE) and the expectation-maximization (EM) algorithm to fit the parameters of the Lévy process. Our results indicate that the Lévy process model provides a significantly better fit to market data than the Black-Scholes model, particularly in capturing the heavy tails and jump behavior observed in option price movements. Additionally, the Lévy model demonstrates superior predictive performance in out-of-sample testing, improving the accuracy of option pricing and hedging strategies. These findings suggest that Lévy processes hold substantial promise for enhancing financial series prediction and derivative pricing in markets characterized by volatility clustering and sudden jumps.