Matrix H-theory approach to stock market fluctuations

Area of Science:

• Quantitative Finance
• Statistical Physics
• Financial Econometrics

Background:

• Collective behavior in financial markets often arises from complex, multivariate stochastic processes.
• Hierarchical structures and correlations across different timescales are key features of market dynamics.
• Existing models may not fully capture the intricate interplay of these factors.

Purpose of the Study:

• To introduce Matrix H theory, a novel framework for analyzing collective behavior in multivariate stochastic processes with hierarchical structure.
• To provide a mathematical formalism for describing the joint distribution of market signals and their underlying background dynamics.
• To demonstrate the applicability of Matrix H theory to real-world financial market data.

Main Methods:

• Modeling the joint distribution of variables as a compound of large-scale multivariate and background distributions.
• Characterizing the background via hierarchical stochastic evolution of internal degrees of freedom.
• Utilizing Meijer G functions with matrix argument to describe probability distributions within two universality classes (Wishart and inverse Wishart).

Main Results:

• Matrix H theory successfully models the joint distribution of market signals and background fluctuations.
• The theory identifies Wishart and inverse Wishart universality classes for concise distribution descriptions.
• Empirical analysis of S&P 500 daily returns validates the theory's effectiveness in describing market fluctuations.

Conclusions:

• Matrix H theory offers a robust framework for understanding multivariate hierarchical processes in finance.
• The findings contribute to a deeper comprehension of stock market dynamics.
• The theory has potential implications for developing improved portfolio management strategies.