Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T14:56:37.442Z Has data issue: false hasContentIssue false
  • English
  • Français

Scaling and Multiscaling in Financial Series: A Simple Model

Part of: Mathematical finance Stochastic processes Mathematical economics

Published online by Cambridge University Press:  04 January 2016

Alessandro Andreoli ,
Francesco Caravenna ,
Paolo Dai Pra  and
Gustavo Posta
Alessandro Andreoli*
Affiliation:
Università Politecnica delle Marche
Francesco Caravenna*
Affiliation:
Università degli Studi di Milano-Bicocca
Paolo Dai Pra*
Affiliation:
Università degli Studi di Padova
Gustavo Posta*
Affiliation:
Politecnico di Milano
*
∗ Postal address: Dipartimento di Management, Università Politecnica delle Marche, Piazzale Martelli 8, 60121 Ancona, Italy. Email address: alessandro.andreoli@univpm.it
∗∗ Postal address: Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via Cozzi 53, I-20125 Milano, Italy. Email address: francesco.caravenna@unimib.it
∗∗∗ Postal address: Dipartimento di Matematica, Università degli Studi di Padova, via Trieste 63, I-35121 Padova, Italy. Email address: daipra@math.unipd.it
∗∗∗∗ Postal address: Dipartimento di Matematica, Politecnico di Milano, Piazzale Leonardo da Vinci 32, I-20133 Milano, Italy. Email address: gustavo.posta@polimi.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We propose a simple stochastic volatility model which is analytically tractable, very easy to simulate, and which captures some relevant stylized facts of financial assets, including scaling properties. In particular, the model displays a crossover in the log-return distribution from power-law tails (small time) to a Gaussian behavior (large time), slow decay in the volatility autocorrelation, and multiscaling of moments. Despite its few parameters, the model is able to fit several key features of the time series of financial indexes, such as the Dow Jones Industrial Average, with remarkable accuracy.

Keywords

Financial indextime seriesscalingmultiscalingBrownian motionstochastic volatilityheavy tailmultifractal model

MSC classification

Primary: 60G44: Martingales with continuous parameter
Secondary: 91B25: Asset pricing models 91G70: Statistical methods, econometrics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 4 , December 2012 , pp. 1018 - 1051
Copyright
© Applied Probability Trust 

References

Accardi, L. and Lu, Y. G. (1993). A continuous version of de Finetti's theorem. Ann. Prob. 21, 1478–1493.CrossRefGoogle Scholar
Andreoli, A. (2011). Scaling and multiscaling in financial indexes: a simple model. , University of Padova. Available at http://www.matapp.unimib.it/∼fcaraven/c.html.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues (Appl. Math. 51), 2nd edn. Springer, New York.Google Scholar
Ané, T. and Geman, H. (2000). Order flow, transaction clock, and normality of asset returns. J. Finance 55, 2259–2284.Google Scholar
Baillie, R. T. (1996). Long memory processes and fractional integration in econometrics. J. Econometrics 73, 5–59.Google Scholar
Baldovin, F. and Stella, A. (2007). Scaling and efficiency determine the irreversible evolution of a market. Proc. Nat. Acad. Sci. USA 104, 19741–19744.Google Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. R. Statist. Soc. B 63, 167–241.Google Scholar
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31, 307–327.Google Scholar
Bollerslev, T. and Mikkelsen, H. O. (1996). Modeling and pricing long memory in stock market volatility. J. Econometrics 31, 151–184.Google Scholar
Bollerslev, T., Kretschmer, U., Pigorsch, C. and Tauchen, G. (2009). A discrete-time model for daily S&P500 returns and realized variations: Jumps and leverage effects. J. Econometrics 150, 151–166.Google Scholar
Bonino, M. (2011). Portfolio allocation and monitoring under volatility shocks. , University of Padova. Available at http://www.matapp.unimib.it/∼fcaraven/c.html.Google Scholar
Calvet, L. E. and Fisher, A. J. (2008). Multifractal Volatility. Academic Press.Google Scholar
Calvet, L. and Fisher, A. (2001). Forecasting multifractal volatility. J. Econometrics 105, 27–58.Google Scholar
Calvet, L., Fisher, A. and Mandelbrot, B. (1997). A multifractal model of asset returns. Discussion paper 1164, Yale University. Available at http://cowles.econ.yale.edu.Google Scholar
Calvet, L., Fisher, A. and Mandelbrot, B. (1997). Large deviations and the distribution of price changes. Discussion paper 1165, Yale University. Available at http://cowles.econ.yale.edu.Google Scholar
Clark, P. K. (1973). A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41, 135–155.Google Scholar
Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quant. Finance 1, 223–236.CrossRefGoogle Scholar
Di Matteo, T., Aste, T. and Dacorogna, M. M. (2005). Long-term memories of developed and emerging markets: using the scaling analysis to characterize their stage of development. J. Banking Finance 29, 827–851.Google Scholar
Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of variance of United Kingdom inflation. Econometrica 50, 987–1007.Google Scholar
Fisher, L., Calvet, A. and Mandelbrot, B. (1997). Multifractality of Deutschemark/US dollar exchange rates. Discussion paper 1166, Yale University. Available at http://cowles.econ.yale.edu.Google Scholar
Freedman, D. A. (1963). Invariants under mixing which generalize de Finetti's theorem: continuous time parameter. Ann. Math. Statist. 34, 1194–1216.Google Scholar
Galluccio, S., Caldarelli, G., Marsili, M. and Zhang, Y.-C. (1997). Scaling in currency exchange. Physica A 245, 423–436.Google Scholar
Ghashghaie, S. et al. (1996). Turbulent cascades in foreign exchange markets. Nature 381, 767–770.Google Scholar
Hull, J. C. (2009). Options, Futures and Other Derivatives. Pearson/Prentice Hall.Google Scholar
Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Klüppelberg, C., Lindner, A. and Maller, R. (2004). A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour. J. Appl. Prob. 41, 601–622.Google Scholar
Klüppelberg, C., Lindner, A. and Maller, R. (2006). Continuous time volatility modelling: COGARCH versus Ornstein-Uhlenbeck models. In From Stochastic Calculus to Mathematical Finance, eds Kabanov, Y., Lipster, R. and Stoyanov, J., Springer, Berlin, pp. 393–419.Google Scholar
Pigato, P. (2011). A multivariate model for financial indexes subject to volatility shocks. , University of Padova. Available at http://www.matapp.unimib.it/∼fcaraven/c.html.Google Scholar
Shephard, N. and Andersen, T. G. (2009). Stochastic volatility: origins and overview. In Handbook of Financial Time Series, Springer, Berlin, pp. 233–254.Google Scholar
Shiryaev, A. N. (1995). Probability (Graduate Texts Math. 95), 2nd edn. Springer, New York.Google Scholar
Stella, A. L. and Baldovin, F. (2008). Role of scaling in the statistical modeling of finance. Pramana 71, 341–352.Google Scholar
Vassilicos, J. C., Demos, A. and Tata, F. (1993). No evidence of chaos but some evidence of multifractals in the foreign exchange and the stock market. In Applications of Fractals and Chaos, eds. Crilly, A. J., Earnshaw, R. A. and Jones, H., Springer, Berlin, pp. 249–265.Google Scholar
Weiss, L. (1955). The stochastic convergence of a function of sample successive differences. Ann. Math. Statist. 26, 532–536.CrossRefGoogle Scholar