Modeling of Financial Data: Comparison of the Truncated L\'evy Flight and the ARCH(1) and GARCH(1,1) processes

TL;DR

This paper compares the truncated Lévy flight and ARCH/GARCH models in their ability to replicate empirical financial data, highlighting strengths and limitations of each in modeling price changes and volatility.

Contribution

It provides a comparative analysis of two stochastic models, demonstrating the truncated Lévy flight's effectiveness in scaling but limitations in volatility, and ARCH/GARCH's success in distribution but not in scaling.

Findings

TLF describes scaling and its breakdown in data

ARCH/GARCH fit the distribution of price changes

Models differ in capturing short-term scaling properties

Abstract

We compare our results on empirical analysis of financial data with simulations of two stochastic models of the dynamics of stock market prices. The two models are (i) the truncated L\'evy flight recently introduced by us and (ii) the ARCH(1) and GARCH(1,1) processes. We find that the TLF well describes the scaling and its breakdown observed in empirical data, while it is not able to properly describe the fluctuations of volatility empirically detected. The ARCH(1) and GARCH(1,1) models are able to describe the probability density function of price changes at a given time horizon, but both fail to describe the scaling properties of the PDFs for short time horizons.