Variations of the asset prices

Area of Science:

• Quantitative Finance
• Statistical Physics
• Econometrics

Background:

• Empirical financial data exhibit non-Gaussian behavior, deviating from standard models.
• Understanding these deviations is crucial for accurate financial market analysis and risk management.

Purpose of the Study:

• To analytically investigate the non-Gaussian dynamics of asset price fluctuations.
• To develop a fundamental model for price dynamics incorporating self-organized feedback.
• To connect theoretical findings with empirical observations in financial markets.

Main Methods:

• Utilized a nonlinear Fokker-Planck equation as a fundamental model for price dynamics.
• Incorporated a self-organized feedback-coupling term to capture market behavior.
• Analyzed the memory term's power-law dependence and its relation to Lévy distributions.

Main Results:

• The analytical model suggests a power-law dependence for the memory term with exponent theta.
• The stationary solution yields a truncated Lévy distribution.
• A direct relationship was found between the memory exponent (theta) and the Lévy exponent (beta): beta = 3/theta - 1.
• Empirical data are consistent with theta approximately 5/4.

Conclusions:

• The proposed model provides an analytical framework for understanding non-Gaussian asset price behavior.
• The findings highlight the importance of memory effects and Lévy distributions in financial econometrics.
• The study offers a theoretical basis for observed market dynamics and potential applications in risk assessment.