Fractal Activity Time Models for Risky Asset with Dependence and Generalized Hyperbolic Distributions

Abstract

Risky asset models with the dependence through fractal activity time are described. The construction of the fractal activity time is implemented via superpositions of Ornstein-Uhlenbeck type processes driven by Lévy noise. The model features both tractable dependence structure and desired marginal distributions of the returns from the generalized hyperbolic class: the Variance Gamma and normal inverse Gaussian. These distributions provide good fit to real financial data. Pricing formulae for the proposed models are derived.

Keywords:

• Fractal activity time
• Geometric Brownian motion
• Lévy noise
• Normal inverse Gaussian distribution
• Ornstein-Uhlenbeck type processes
• Self-similarity
• Variance Gamma distribution

Mathematics Subject Classification:

• Primary 60G18
• Secondary 60G10
• 62M10

Acknowledgments

This work is supported in part by the Commission of the European Communities grant PIRSES-GA-2008-230804 within the program. “Marie Curie Actions.”

The authors thank the referee for the comments and suggestions that helped improve the manuscript.