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Additive-multiplicative stochastic models of financial mean-reverting processes
1Departamento de Física, Pontifícia Universidade Católica do Rio de Janeiro, CP 38071, 22452-970, Rio de Janeiro, Brazil. celia@cbpf.br
Summary
This study introduces a generalized stochastic model incorporating mean reversion and Wiener processes for financial volatilities. The model accurately describes empirical distributions, offering a flexible framework for financial modeling.
Area of Science:
- Quantitative Finance
- Stochastic Modeling
- Financial Mathematics
Background:
- Stochastic models are crucial for understanding financial market dynamics.
- Mean reversion describes the tendency of financial variables to return to a historical average.
- Existing models often lack the flexibility to capture complex volatility behaviors.
Purpose of the Study:
- To introduce and analyze a generalized stochastic model for financial volatilities.
- To incorporate both mean reversion and additive-multiplicative Wiener processes.
- To assess the model's capability in describing empirical financial data distributions.
Main Methods:
- Development of a generalized stochastic model based on Itô-Langevin equations.
- Analysis of the long-time probability density function (PDF).
- Inclusion of multiplicative (internal) and additive (exogenous) Wiener processes.
Main Results:
- The generalized model encompasses various existing mean-reverting financial process models.
- A rich spectrum of probability density function shapes was observed, dependent on model parameters.
- The additive-multiplicative process demonstrated realistic fitting for empirical distributions across various datasets.
Conclusions:
- The proposed generalized stochastic model offers a flexible and realistic approach to modeling financial volatilities.
- The inclusion of additive and multiplicative Wiener processes enhances the model's descriptive power.
- This framework provides a valuable tool for quantitative finance and risk management.