Notes on scaling behaviour in financial and commodity markets
Cian Tomas O'Criodain, Donal Holland, Misha Volkov and Khurshid Ahmad
Trinity College, Dublin, Ireland
Contents
Abstract .......................................................................................................................... 2
Background .................................................................................................................... 2
Decomposing a time series .................................................................................... 2
The Promise of Wavelet Analysis .......................................................................... 3
Risk Analysis ............................................................................................................ 3
Statement of Problem: ................................................................................................ 3
Oil Future Contracts – High Frequency and Low frequency movements ...... 3
Oil Company Stocks ................................................................................................ 4
Method .......................................................................................................................... 4
Data and Case Studies ................................................................................................ 4
Decomposition of CL1 ............................................................................................. 4
Wavelet Variance ..................................................................................................... 4
Predicting CL1 .......................................................................................................... 4
Systemic Risk ........................................................................................................... 4
Afterword ....................................................................................................................... 4
References .................................................................................................................... 4
Notes on scaling behaviour in financial and commodity markets
Cian Tomas O'Criodain, Donal Holland, Misha Volkov and Khurshid Ahmad
Trinity College, Dublin, Ireland
Abstract
Oil shocks have had a profound effect on a global scale and yet oil prices show a
periodicity that is remarkable in itself. Wavelet analysis has been used to decompose
an oil futures contract (Cushing Light Oil – CL1) over a 10 year period covering wars,
presidential elections, successful peace processes and ambitious economic and social
programmes, and major changes in transportation technology. The decomposition is
helps in computing wavelet variance at different scales and following Gencay et
al.[23] leads to an estimate of systemic risk in oil future prices. We demonstrate a
method that can be used to predict price of oil contracts over a limited horizon.
Background
One of the key properties of complex systems is scale invariance. Consider systems
in financial and commodity markets for example: These systems are intrinsically nonlinear, have a multi-scale character in both space and time, and comprise components
that show dynamic and interdependent relationships amongst each other: The systems,
nevertheless, show a remarkable degree of stability characterised by cyclic change in
the values of idiosyncratic variable and by the scale-invariant power law behaviour of
control parameters, for instance, trading interval, or the size of an actor – firms large
and small. The systemic stability can be ‘shocked’ into linear or non-linear trends for
varying periods of time; this may produce volatility clusters, and there maybe longterm effects of the shock observed sometimes by the emergence of variance breaks in
financial time series. The shock may be endogenous, caused by the complexity of
interaction between constituent parts, or exogenous, caused by external or
environmental stimuli.
Decomposing a time series
Wavelet analysis has ‘formalized old notions of decomposing a series into trend,
seasonal and business cycle components’ [38]. The result of the analysis is the reexpression of a highly correlated time series in terms of the combinations of the uncorrelated wavelet coefficients [12][32][36]. These coefficients are associated with
oscillations of different periods. In economics and finance the analysis of the raw
time series at different scales can be used with some degree of success: (a) to
deseasonalise and detrend raw foreign exchange data [21] [39]; (b) to study the
scaling behaviour of economic indices [33]; (c) to conduct (Granger-) causality tests
of different macroeconomic variables like money supply and real economy variables
output [22] which shows no conclusive indication of causality at the raw data level but
clear indication of causal connection between the variables at finer scales; (d) to
investigate heterogeneous trading in commodity markets and the effect of other
financial instrument on the prices [11] (e) to evaluate risk using multiscale beta
estimation [23]. Wavelets have been used to test for the onset of trends [24], for
stationarity [44], and for predicting the behaviour of financial markets [40] or as a preprocessor of time series for input to fuzzy logic-based prediction system[37]. Note
that the wavelet-based approach is similar in spirit to the well-known ARCH and
GARCH models [4][5][14][15][43] However, unlike the approach adapted in
(G)ARCH models, the wavelet approach breaks down each volatile time series into
several time series of very high to very low volatility: these time series can then be
studied and interpreted separately. This study appears to the discovery of newer
relationships and insights in the analysed data, which are not afforded by the GARCH
and ARCH models [39].
The Promise of Wavelet Analysis
The potential of multiscale wavelet decompositions in providing new insights in
subjects as diverse as hydrology, astronomy, physics and geology, suggests the
analysis of economic and financial data will perhaps benefit from wavelet analysis
[36]. The roots of the wavelet analysis lie within filtering methods that deal with the
identification and extraction of certain features (e.g., trends, seasonalities) from a time
series, which are important in terms of modelling and inference. Traditionally, filters
in economics and finance are used to extract components of a time series such as
trends, business cycles, seasonalities, and noise. Two most commonly used methods
of decomposition of a macroeconomic time series are the Beveridge-Nelson
procedure [1] and Hodrick-Prescott filter [27]. Canova critically evaluated a set of
popular filters used in detrending macroeconomic time series and showed that
different detrending methods provide different stylised facts of the U.S. business
cycle [7][8]. Burnside claimed that preferring a commonly used filtering method in
macroeconomic analysis does not induce any lack of power [6]. Niemira and Klein
studied forecasting financial and economic cycles in general [35], and Weigend and
Gerschenfeld investigated time series prediction in natural and physical phenomena
[45]. Kaiser and Maraval [28], and Diebold and Rudebusch [13] contained a detailed
analysis of business cycle measurements and forecasts.
Risk Analysis
Financial time series do not vary in isolation from one another and their movements
are indeed correlated [25]. The capital asset pricing model (CAPM), a tenet of modern
financial economics, provides a method to quantify correlations amongst stock prices
with a statistic called systematic risk or beta [41][30][29]. The CAPM essentially
works on the principle that the expected return of a particular asset is dependent upon
the total risk involved in holding that asset: that is, the average return of high beta
stocks is higher than the average return of low beta stocks and that this relationship is
roughly linear [3][18]. Again, recognizing the effect of time-scale on the beta or risk
of an asset, several researchers have studied the stability of beta over different time
horizons [26][20][2]. Since, wavelet analysis is a natural tool for studying the timescale properties of time series, its application for the calculation of betas or systematic
risks at various time horizons within a CAPM framework are eminent. Gencay et al.
provide a comprehensive wavelet-based methodology to calculate systematic risk at
various time-scales and show that the relationship between the return of a portfolio
and its beta becomes stronger as the time-scale increases [23]. This offers newer
research avenues for assessing the risk and value of different cash flows in financial
markets. In SCALES, we want to study the risk-return tradeoffs at different timescales and want to investigate their relationship with market sentiment.
Statement of Problem:
Oil Future Contracts – High Frequency and Low frequency
movements
Oil Company Stocks
Method
Data and Case Studies
Decomposition of CL1
Wavelet Variance
Predicting CL1
Systemic Risk
Afterword
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
Beveridge, S., and C. R. Nelson. 1981. “A new approach to decomposition of
economic time series into permanent and transistory components with particular
attention to measurement of the business cycle.” Journal of Monetary
Economics 7: 151-174.
Bjornson B., H. S. Kim, and K. Lee. 1999. “Low and high frequency
macroeconomic forces in asset pricing.” Q. Rev. Economics Finance 39: 77-100.
Black, F., M. Jensen, and M. Scholes. 1972. “Capital asset pricing model: some
empirical tests.” Studies in the Theory of Capital Markets. (ed.) M Jensen, New
York: Praeger.
Bollerslev, T. 1990. “Modelling the coherence in short run nominal exchange
rates: A multivariate generalized ARCH model.” Reiew of Economics and
Statistics 72: 498-505.
Bollerslev.
T.
1986.
“Generalized
Autoregressive
Conditional
Heteroscedasticity.” Journal of Econometrics 31: 307-327.
Burnside, C. 1998. “Detrending and business cycle facts: A comment.” Journal
of Monetory Economics 41: 513-532.
Canova, F. 1998a. “Detrending and business cycle facts.” Journal of Monetary
Economics 41: 475-512.
Canova, F. 1998b. “Detrending and business cycle facts: A user’s guide.”
Journal of Monetary Economics 41: 533-540.
Chan, F. K., A. W. Fu, and C. Yu. 2003. “Haar Wavelets for Efficient Similarity
Search of Time-Series: With and Without Time Warping.” IEEE Transactions
on Knowledge and Data Engineering 15 (3): 686-705.
Cho, Young-Hye and Robert F. Engle. 2000. Time-Varying Betas and
Asymmetric Effects of News: Empirical Analysis of Blue Chip Stocks.
Connor, J. and R. Rossiter. 2005. “Wavelet Transforms and Commodity
Prices”. Studies in Nonlinear Dynamics & Econometrics. Vol 9 (No.1, Article
6). (available at http://www.bepress.com/snde).
Daubechies, I. 1992. “Ten Lectures on Wavelets”. Society for Industrial and
Applied Mathematics.
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
Diebold, F. X., and G. D. Rudebusch. 1999. “Business Cycles: Durations,
Dynamics, and Forecasting.” Princeton University Press, Princeton, New
Jersey.
Engle, R. F. 1982. “Autoregressive Conditional Heteroscedasticity with
Estimates of Variance of United Kingdom Inflation.” Econometrica 50: 9871008.
Engle, R. F. and K. F. Kroner. 1995. “Multivariate simultaneous generalized
ARCH.” Econometric Theory 11: 122-150.
Engle, R. F., and K. N. Victor. 1993. “Measuring and testing the impact of news
on volatility.” Journal of Finance 48 (5): 1749-1778.
Engle, R.F. & Ng, V. K. (1993). “Measuring and Testing the Impact of News
on Volatility’. Journal of Finance. Vol. 48 (5), pp 1749-1788.
Fama, E. F., and J. MacBeth. 1973. “Risk, return and equilibrium: empirical
tests.” J. Political Economy 71: 607-36.
Farmer, D. J., and A. W. Lo. 1999. “Frontiers of finance: Evolution and efficient
markets.” Proc. Natl. Acad. Sci. 96 (18): 9991-9992.
Garcia, R., and E. Ghysels. 1998. “Structural change and asset pricing in
emerging markets.” J. Int. Money Finance 17: 455–73.
Gençay, R., F. Selcuk, and B. Whitcher. (2001). ‘Differentiating intraday
seasonalities through wavelet multi-scaling’. Physica A. Vol. 289, pp 543-556.
Gençay, R., F. Selcuk, and B. Whitcher. 2002. “An Introduction to Wavelets
and Other Filtering Methods in Finance and Economics.” Academic Press, New
York.
Gencay, R., F. Selcuk, and B. Whitcher. 2003. “Systematic risk and timescales.”
Quantitative Finance 3: 108-116.
Gilbert, S.D. 1999. “Testing for the onset of trends, using wavelets”. Journal of
Time Series Analysis. Vol 20 (No.5), pp 513-526.
Granger, C., and R. Engle. 1987. “Cointegration and Error Correction:
Representation, Estimation and Testing.” Econometrica 55: 251-76.
Harvey, C. R. 1989. “Time-varying conditional covariances in tests of asset
pricing models.” J. Financial Economics 24: 289-317.
Hodrick, R. J., and E. Prescott. 1997. “Post-war U.S. business cycle: An
empirical investigation.” Journal of Money, Credit, and Banking 29: 1-16.
Kaiser, R., and A. Maraval. 2000. “Measuring Business Cycles in Economic
Time Series.” Springer-Verlag, New York.
Khan M. A., and Y. Sun. 1997. “The capital-asset-pricing model and arbitrage
pricing theory: A unification.” Proc. Natl. Acad. Sci. 94: 4229-4232.
Lintner, J. 1965. “The valuation of risky assets and the selection of risky
investments in stock portfolios and capital budgets.” Rev. Economics Statistics
47: 13-37.
Lux T., and M. Marchesi. 1999. “Scaling and criticality in a stochastic multiagent model of a financial market.” Nature 397: 498-500.
Mallat, S. 1989. “A Theory for Multiresolution Signal Decomposition: The
Wavelet Representation.” IEEE Transactions on Pattern Analysis and Machine
Intelligence 11: 674–93.
Mantegna, R. N., and H. E. Stanley. 1995. “Scaling Behaviour in the Dynamics
of an Economic Index.” Nature 376: 46-49.
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
McQueen, G., & Vorkink, K. (2004). “Whence GARCH? A Preference-based
Explanation for Conditional Volatility’. The Review of Financial Studies. Vol.
17 (No.4), pp 915-949.
Niemira, M. P., and P. A. Klein. 1994. “Forecasting Financial and Economic
Cycles.” John Wiley & Sons, New York.
Percival, D. B., and A. T. Walden. 2000. “Wavelet Methods for Time Series
Analysis.” Cambridge University Press.
Popoola, A., Ahmad, S., and Ahmad, K. 2004. “A Fuzzy-Wavelet Method for
Analyzing Non-Stationary Time Series.” Proc. of The 5th Int. Conf. on Recent
Advances in Soft Computing (December 16-18, 2004, Nottingham, UK).
(http://www.computing.surrey.ac.uk/grid/fingrid/papers.html)
Ramsey, J. B. 1999. “The contribution of wavelets to the analysis of economic
and financial data.” Phil. Trans. R. Soc. Lond. A 357: 2593-2606.
Ramsey, J. B., and Z. Zhang. 1997. “The analysis of foreign exchange data
using waveform dictionaries.” Journal of Empirical Finance 4: 341-372.
Renaud, O., J. L. Starck, and F. Murtagh. 2003. “Prediction Based on a
Multiscale Decomposition.” International Journal of Wavelets, Multiresolution
and Information Processing. 1 (2): 217-232.
Sharpe, W. 1964. “Capital asset prices: a theory of market equilibrium under
conditions of risk.” J. Finance 19: 425-42.
Stanley, H. E., L. A. N. Amaral, S. V. Buldyrev, P. Gopikrishnan, V. Plerou,
and M. A. Salinger. 2002. “Self-organized complexity in economics and
finance.” Proc. Natl. Acad. Sci. 99 (1): 2561-2565.
Tse, Y. K. and A. K. Tsui. 2002. “A multivariate generalized autoregressive
conditional heteroscedasticity model with time varying correlations.” Journal of
Business and Economic Statistics 20: 351-362.
von Sachs, R., and M.H. Neumann. 2000. “A wavelet-based test for
stationarity”. Journal of Time Series Analysis. Vol 21 (No.5) pp 597-613.
Weigend, A., and N. Gerschenfeld. 1994. “Time Series Prediction: Forecasting
the Future and Understanding the Past.” Addison-Wesley, Reading: MA.
Yamada, Hiroshi. 2005. 'Wavelet-based beta estimation and Japanese industrial
stock prices'. Applied Economics Letters, Volume 12(2), pp. 85-88