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A risky asset model with strong dependence through fractal activity time

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

C. C. Heyde
C. C. Heyde*
Affiliation:
Australian National University and Columbia University
*
∗Postal address: School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia. Email address: chris@maths.anu.edu.au
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Abstract

The geometric Brownian motion (Black–Scholes) model for the price of a risky asset stipulates that the log returns are i.i.d. Gaussian. However, typical log returns data shows a leptokurtic distribution (much higher peak and heavier tails than the Gaussian) as well as evidence of strong dependence. In this paper a subordinator model based on fractal activity time is proposed which simply explains these observed features in the data, and whose scaling properties check out well on various data sets.

Keywords

Risky asset modelBlack–Scholes modelheavy tailslong-range dependencefractal activity timeself-similarity

MSC classification

Secondary: 62M10: Time series, auto-correlation, regression, etc. 60G18: Self-similar processes
Type
Short Communications
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1234 - 1239
Copyright
Copyright © Applied Probability Trust 1999 

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