Principal Curves for Statistical Divergences and an Application to Finance

Area of Science:

• Quantitative Finance
• Financial Econometrics
• Statistical Modeling

Background:

• Traditional asset pricing models often assume Gaussian returns, which may not accurately reflect real-world financial markets.
• The mean-variance framework, while foundational, has limitations when dealing with non-normal return distributions.
• Existing models may not fully capture the complex behavior and divergences present in asset returns.

Purpose of the Study:

• To propose a generalized beta pricing model that accounts for non-Gaussian asset returns.
• To develop a robust optimization framework for financial pricing models using principal curves and divergence measures.
• To extend the Markowitz mean-variance approach to a mean-divergence framework for portfolio optimization.

Main Methods:

• Generalization of the mean-variance model using principal curves to define a divergence model.
• Application of the q-exponential model to characterize asset return distributions and their divergences.
• Derivation of the minimum divergence portfolio and the Capital Asset Pricing Model (CAPM) using information geometry.

Main Results:

• A novel beta pricing model is derived, accommodating non-Gaussian returns.
• The minimum divergence portfolio is established, generalizing the Markowitz approach.
• The Capital Asset Pricing Model (CAPM) is reformulated using information geometrical aspects of asset distributions.

Conclusions:

• The proposed method offers a more comprehensive approach to asset pricing under non-Gaussian conditions.
• The integration of principal curves and divergence models provides a powerful tool for financial modeling.
• The framework has potential applications in adaptive procedures for optimizing financial applications.