Complex Systems Approaches to Financial Markets: Scaling Properties, Correlations and Physics Inspired Modeling

This Master's Thesis project is about various fundamental tools coming from statistical physics and complexity science in general, that are used to analyse economical and financial systems. In particular the dissertation focuses on the empirical study of financial data, employing different methods and ideas inspired by a physical approach to the analysis of complex systems: scaling relations, correlations and physics inspired modeling. Investigating the scaling properties of financial time series can give crucial insights about the underlying processes generating the empirical observations and can provide useful tools to detect consistent patterns across different scales. The analysis of the statistical relation between the price of different assets can offer the possibility to quantify their interactions and to extract the important information contained in the correlation structure of financial markets. A physics inspired modeling approach to financial data may offer a unique perspective which can provide interesting insights into the underlying mechanisms and dynamics of the system. The data used consist of a set containing all the daily closing prices from 1990 to 2022 of the stocks comprised, as of November 2022, in the S&P500 index. In the first part of the project, some of the main empirical statistical properties of financial time series found in literature are retrieved on the data: heavy tails, aggregational Gaussianity, absence of autocorrelations and volatility clustering. Particular attention is placed on the estimate of the tails exponents of the distribution of the returns, hallmark of the non-Gaussianity of financial data. The second part of the work is about the main ideas behind the emergence of scaling laws in complex systems and their study. Particular focus is placed on the multifractal analysis of financial markets, its theoretical foundation and the methods to apply it. The Generalized Hurst Exponent method, as well as some of its extensions, is presented and applied to the data set to extract the so called multiscaling proxy and the Hurst exponents of the time series. In the third part of the thesis different kind of correlation measures are presented and exploited to study the statistical relations between the time series of the stocks in the data set, both statically and dynamically. Moreover, the correlation structure of the market is represented with a complete graph and an information filtering technique taken from network theory (Minimum Spanning Tree) is employed to highlight peculiar clustering properties of the data. The last part of the project is devoted to the study of a relatively new random walk model by Takayasu et al. (2010), named the PUCK model, in which the random walker is subjected to a potential centered at its moving average position. The model is presented and its application as a novel type of time series data analysis tool, characterizing the time-dependent stability of markets, is shown. Finally, its scaling properties are investigated through the previously presented methodology and various relations between its parameters and the scaling exponents are devised.