Risky Asset Models with Tempered Stable Fractal Activity Time

Abstract

Two constructions of fractal activity time in the fractal activity time geometric Brownian motion (FATGBM) model for a risky asset are discussed. Both constructions produce tractable dependence structure that includes long-range dependence. One construction uses Ornstein-Uhlenbeck type processes and leads to stationary log returns with exact normal tempered stable distribution and pricing formula that is based on the asymptotic self-similarity of the activity time. The second construction uses convoluted subordinator with Holmgren-Liouville kernel and leads to exact pricing formula that is based on exact tempered stable distribution of the activity time. Option prices computed using our model and pricing formulae are remarkably close to the real market values.

Keywords:

• Random time change
• Fractal activity time
• Ornstein-Uhlenbeck type processes
• Lévy processes
• Fractional Lévy motion
• Tempered stable distribution
• Normal tempered stable distribution

Mathematics Subject Classification:

• Primary 60G99
• Secondary: 60G22 60G51
• 60E70
• 60H30