Abstract
In general, the absolute majority of financial market models is based on the stochastic properties of the asset returns, while the properties of the related asset quantities play a minor role. Starting from these remarks, in this paper we propose a system of nonlinear and stochastic difference equations in which the asset price behaviour and the corresponding asset quantity one are jointly taken into account. More precisely, in order effectively to represent the properties of the real asset price variations, we assume that (also on the basis of well known empirical evidences) their dynamics is distinguished by different stochastic processes alternating each other: the “classical” standard Brownian one, the fractional Brownian motion (which is able to represent the dependence among the returns), and the Pareto-Lévy stable one (which is able to represent the the non-Gaussian distributional features). All these processes are characterized by the same “fractal” quantity, the exponent of Hurst, which is properly utilized in the proposed dynamical model in order to represent the different stochastic properties of the asset price changes. Finally, because of the possible “bad” analytical peculiarities of the system itself, we investigate its dynamics by means of an agent-based approach developed in the Swarm software environment.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Beran, J.“Statistics for Long-Memory Processes” Chapman & Hall, 1994.
Campbell, J.Y., Lo, A.W. and MacKinlay, A.C.“The Econometrics of Financial Markets” Princeton University Press, 1997.
Cheung, Y.-W., and Lai, K.S., “Do Gold Market Returns Have Long Memory?”The Financial Review28(2), 181–202, 1993.
Cootner, P.H. (ed.)“The Random Character of Stock Market Prices”The M.I.T. Press, 1964.
Corazza, M., “Long-Term Memory Stability in the Italian Stock Market”Economics 6 Complexity1(1), 19–28, 1996.
Corazza, M., “Merton-like Theoretical Frame for Fractional Brownian Motion in Finanza”, in Canestrelli, E. (ed.)“Current Topics in Quantitative Finance”Physica-Verlag, 37–47, 1999.
Corazza, M. and Malliaris, A.G. “Multifractality in Foreign Currency Markets”Quaderno del Dipartimento di Matematica Applicata e Informatica dell’Universit¨¤ degli Studi di Venezia49/97, 1997a.
Corazza, M., Malliaris, A.G. and Nardelli, G., “Searching for fractal Structure in Agricultural Futures Markets”The Journal of Futures Markets17(4), 433–473, 1997b.
Falconer, K.“fractal Geometry”John Wiley & Sons, 1990.
L¨¦vy, P.“Calcul des Probabilities”Gauthier-Villar, 1925.
Lee, C.M.C. and Swaminathan, B., “Price Momentum and Trading Volume”The Journal of Finance55(5), 2017–2069, 2000.
Lo, A.W., “Long-Term Memory in Stock Market Prices”Econometrica59(5), 1279–1313, 1991.
Lo, A.W. and Wang, J., “Trading Volume: Definitions, Data Analysis, and Implications of Portfolio Theory”The Review of Financial Studies13(2), 257–300, 2000a.
Lo, A.W., Mamaysky, H. and Wang, J., “Foundations of Technical Analysis: Computational Algorithms, Statistical Inference, and Empirical Implementation”The Journal of Finance55(4), 1705–1765, 2000b.
Luna, F. and Stefansson, B. (eds.)“Economic Simulations in Swarm: Agent-Based Modelling and Object Oriented Programming”Kluwer Academic Publishers, 2000.
Kopp, E., “Fractional Brownian Motion and Arbitrage”Mimeo - University of Hull(England), 1995.
Kunimoto, N. “Long-Term Memory and Fractional Brownian Motion in financial markets”Revised version of Discussion Paper at Faculty of Economics - University of Tokyo(Japan) 92#F#12, 1993.
Mandelbrot, B.B. “The Variation of Certain Speculative Prices”IBM Research Report NC-87, 1962.
Mandelbrot, B.B. and Van Ness, J.W., “Fractional Brownian Motion, Fractional Noises and Applications”, SIAM Review, 10(4), 422–437, 1968.
Pancham, S., "Evidence of the Multifractal Market Hypothesis Using Wavelet Transforms", Mimeo - Florida International University (Florida, U.S.A.), 1994.
Peitgen, H.-O., J’urgens, H. and Saupe, D.“Chaos and fractals. New Frontiers of Science”Springer-Verlag, 1992.
Rachev, S. and Mittnik, S.“Stable Paretian Models in Finance”Wiley, 2000.
Rogers, L.C.G., “Arbitrage with Fractional Brownian Motion”Mathematical Finance7(1), 95–105, 1997.
Samorodnitsky, G., and Taqqu, M.S.“Stable Non-Gaussian Random Processes”Chapman & Hall, 1994.
Taqqu, M.S., “A Bibliographical Guide to Self-Similar Processes and Long-Range Dependence”, in Eberlein, E. and Taqqu, M.S. (eds.)“Dependence in Probability and Statistics”Birkhauser, 137–162, 1986.
Willinger, W., Taqqu, M.S. and Teverovsky, V. “Stock Market Prices and Long-Range Dependence”Finance and stochastics 3(1), 1–13, 1999.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media New York
About this chapter
Cite this chapter
Corazza, M., Perrone, A. (2002). Simulating Fractal Financial Markets. In: Luna, F., Perrone, A. (eds) Agent-Based Methods in Economics and Finance. Advances in Computational Economics, vol 17. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0785-7_6
Download citation
DOI: https://doi.org/10.1007/978-1-4615-0785-7_6
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-5238-9
Online ISBN: 978-1-4615-0785-7
eBook Packages: Springer Book Archive