Hostname: page-component-7dd5485656-frp75 Total loading time: 0 Render date: 2025-10-30T03:52:30.764Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1234 - 1239
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Andersen, T. (1996). Return volatility and trading volume. An information flow interpretation of stochastic volatility. J. Finance 51, 169–204.Google Scholar
Arthur, W. B. (1995). Complexity in economic and financial markets. Complexity 1, 20–25.Google Scholar
Bak, P. (1996). How Nature Works. Copernicus. Springer, New York.Google Scholar
Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, New York.Google Scholar
Clark, P. K. (1973). A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41, 135–155.Google Scholar
Ding, Z., and Granger, C. W. J. (1996). Modeling volatility persistence of speculative returns: a new approach. J. Econometrics 73, 185–215.Google Scholar
Fisher, A., Calvet, L., and Mandelbrot, B. (1997). Multifractality of the Deutschmark/US Dollar exchange rates. Cowles Foundation Discussion Paper No. 1165.Google Scholar
Granger, C. W. J., and Ding, Z. (1996). Varieties of long memory models. J. Econometrics 73, 61–77.Google Scholar
Heyde, C. C. (1997). Quasi-Likelihood and its Application. General Theory of Optimal Parameter Estimation. Springer, New York.Google Scholar
Heyde, C. C., and Dai, W. (1996). On the robustness to small trends of estimation based on the smoothed periodogram. J. Time Series Anal. 17, 141–150.Google Scholar
Heyde, C. C., and Yang, Y. (1997). On defining long-range dependence. J. Appl. Prob. 34, 939–944.Google Scholar
Hurst, S. R., and Platen, E. (1997). The marginal distribution of returns and volatility. In L1-Statistical Procedures and Related Topics, Dodge, ed. Y. (IMS Lecture Notes-Monograph Series 31). IMS, Hayward, CA, pp. 301–314.Google Scholar
Hurst, S. R., Platen, E., and Rachev, S. R. (1997). Subordinated Markov models: a comparison. Fin. Eng. Japanese Markets 4, 97–124.Google Scholar
Lo, A. W. (1991). Long-term memory in stock market prices. Econometrika 59, 1279–1313.Google Scholar
Mandelbrot, B., and Taylor, H. (1967). On the distribution of stock price differences. Operat. Res. 15, 1057–1062.Google Scholar
Mandelbrot, B., Fisher, A., and Calvet, L. (1997). A multifractal model of asset returns. Cowles Foundation Discussion Paper No. 1164.Google Scholar
Mikosch, T. (1997). Heavy tails in finance. Bull. Int. Statist. Inst. 57, Book 2, 109–112.Google Scholar
Peters, E. E. (1991). Chaos and Order in the Capital Markets. John Wiley, New York.Google Scholar
Peters, E. E. (1994). Fractal Market Analysis. John Wiley, New York.Google Scholar
Resnick, S. (1997). Heavy tail modeling and teletraffic data. Ann. Statist. 25, 1805–1869.Google Scholar
Rogers, L. C. G. (1997). Arbitrage from fractional Brownian motion. Math. Finance 7, 95–105.Google Scholar
Willinger, W., Taqqu, M., and Teverovsky, V. (1999). Stock market prices and long-range dependence. Finance and Stochastics 3, 1–13.Google Scholar
Yang, Y. (1998). Long range dependence in the study of time series with finite or infinite variances. Ph.D. Dissertation, Columbia University, New York, NY.Google Scholar