Abstract
Despite the classical hypothesis states that the asset returns are (log-Normally) identically and independently distributed, in many financial market is detectable significative empirical evidence that there are dependence inside such returns. From a distributional point of view, this dependence can be modelled by the so-calledfractionalBrownian (fB) motion which is a Gaussian stochastic process whose increments are (long-term) dependent with each other. Although there exists an increasing empirical literature about this topic, from a theoretical standpoint there is not an equivalent number of results concerning with the relationships between the fB motion and the financial markets.
Starting from these remarks, in this work we propose a Merton-like system of economic-financial assumptions on the dynamical behaviour of financial asset price by which it is possible to deduce the consistency between the fB motion and the discrete-time trading. Moreover, we also prove the “convergence” of the fB motion to the standard Brownian (sB) one when the discrete-time trading tends to the continuous-time one.
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
Ambrose B.W., Weinstock Ancel E. and Griffiths M.D. (1993) Fractal Structure in the Capital Markets Revisited.Financial AnalystJournal, MayJune:73–77.
Beran J. (1994)Statistics for Long-memory Processes 1st edn. Chapman & Hall.
Berg L, Lyhagen J. (1995) Short and Long Run Dependence in Swedish Stock Returns.Mimeo University of Uppsala - Sweden.
Booth G.G., Kaen F.R.Koveos P.E. (1982) R/S Analysis of Foreign Exchange Rates under Two International Monetary Regimes. Journalof Monetary Economics10:407–415.
Campbell J.Y., Lo A.W. and Mackinlay A.C. (1997)The Econometrics of Financial Markets1st edn. Princeton University Press.
Cheung Y.-W. Lai K.S. (1993) R/S Analysis of Foreign Exchange Rates under Two International Monetary Regimes. Journalof Monetary Economics10:407–415.
Corazza M. and Malliaris A.G. (1997) MultiFractality in Foreign Currency Markets. Quaderno del Dipartimento diMatematicaApplicata edInformatica,University of Venice -Italy, 49/97.
Corazza M., Malliaris A.G. and Nardelli C. (1997) Searching for Fractal Structure in Agricultural Futures Markets.The Journal of Futures Markets17(4):433–473.
Evertsz C.J.G. (1995) Self-similarity of High-frequency USD-DEM Exchange Rates. In:Proceedings of the First International Conference on High Frequency Datain Finance, Zurich (Switzerland).
Evertsz C.J.G. and Berkner K. (1995) Large Deviation and Self-similarity Analysis of Graphs: DAX Stock Prices.Chaos Solitons f4 Fractals6:121–130.
Falconer K. (1990)Fractal Geometry. Mathematical FoundationsandApplications,1st edn.John Wiley&Sons.
Greene M.T. and Fielitz B.D. (1977) Long-Term Dependence in Common Stock Returns. Journalof Financial Economics4:339–349.
Hodges S. (1995) Arbitrage in a Fractal Brownian Motion Market.Mimeo University of Warwick - England.
Kopp E. (1995) Fractional Brownian Motion and Arbitrage.Mimeo University of Hull - England.
Kunimoto N. (1993) Long-term Memory and Fractional Brownian Motion in Financial Markets.Revised version of DiscussionPaperat Faculty of Economics,University of Tokyo, 92-F-12.
Lo A.W. and Mackinlay A.C. (1988) Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test.Review of Financial Studies1:41–66.
Lo A.W. (1991), Long-term Memory in Stock Market Prices.Econometrica59(5):1279–1313.
Mandelbrot B.B. and Van Ness J.W. (1968) Fractional Brownian Motions, Fractional Noises and Applications.SIAM Review10(4):422–437.
Merton R.C. (1982) On the Mathematics and Economics Assumptions of Continuous-Time Models. In: Sharpe W.F. and Cootner C.M. (Eds.)Financial Economics: Essays in Honor of Paul Cootner1st edn. Prentice Hall.
Merton R.C. (1990)Continuous-time Finance 1st ed. Basil Blackwell.
Neftci S.N. (1996) AnIntroduction to the Mathematics ofFinancialDerivatives,1st ed.Academic Press.
Poterba J.M. and Summers L.H. (1988) Mean Reversion in Stock Prices. Evidence and Implications. JournalofFinancialEconomics, 22:27–59.
Rogers L.C.G. (1995) Arbitrage with Fractional Brownian Motion.Mimeo University of Bath - England.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Corazza, M. (1999). Merton-like Theoretical Frame for Fractional Brownian Motion in Finance. In: Canestrelli, E. (eds) Current Topics in Quantitative Finance. Contributions to Management Science. Physica, Heidelberg. https://doi.org/10.1007/978-3-642-58677-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-58677-4_4
Publisher Name: Physica, Heidelberg
Print ISBN: 978-3-7908-1231-2
Online ISBN: 978-3-642-58677-4
eBook Packages: Springer Book Archive