Quantitative Finance

13Generalized Lévy Processes, Long Range Correlations, and Memory Effects

13.1 Introduction

In this chapter we introduce the stable distributions and generalized Lévy processes. In order to introduce generalized Lévy models, we first present some background in stable distributions. In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. The family of stable distributions is also referred to as the Lévy alpha‐stable distribution named after Paul Lévy. In this chapter, we will present some forms of stable distributions which includes the Lévy flight models.

The Black–Scholes is not appropriate for the study of high frequency data or for the study of financial indices or asset prices when a Market Crash takes place. For these financial data, other models are more appropriate, such as the Lévy‐like stochastic processes. The generalized Lévy models like the Range Scale Analysis, Detrended Fluctuation Analysis (DFA) (see Chapter 12 of this book) and Diffusion Entropy Analysis (DEA) (see [178]) are very convenient methodologies for the analysis of extreme events, like financial crashes and earthquakes. The DEA method determines the correct scaling exponent in a time series even when the statistical properties, as well as the dynamic properties, are irregular.

We will conclude this chapter by presenting some applications to financial ...