(1) SA Financial Market Sectors and States: Quantitative Analysis of Cross-correlation Structure    (with matlab covariance matrix toolbox and scaling analysis toolbox)

(2) Scaling and trends in financial time series and prediction  (with scaling analysis toolbox , high-frequency toolbox , nonlinear time-series toolbox, state-space reconstruction toolbox )

RATIONALE

Economics, as a science is not yet well understood (cf. cover articles of The Economist: The puzzling failure of economics (23 August 1997); The world economy and how to rescue it (28 September 2002)). The boom in financial markets which preceded the new millennium, the economic recession which followed, developments in emerging markets such as SA, various financial crises in world financial markets in the 80's and 90's, financial bubbles and crashes, etc, are all the subjects of ongoing topical debate and mathematical analysis (see for example [1-5,13,16-23]; see also New Scientist, 26 Feb 2005, cover story: Too much information by Mark Buchanan; Physics World, Aug 2004, cover Story: Tails of natural hazards by Bruce Malamud).

 In a capital market based economy, understanding the dynamics of financial markets is of practical importance for both investors who wish to deploy capital in a profitable way and for companies who wish to raise capital for business activity. Furthermore, the dynamics of financial markets have impact on national and global economies. Thus, this topic is of scientific interest for the understanding of economies as complex dynamical systems. The development of high speed computing and huge computer storage capacity have converged to make it possible to use empirical methods to conduct scientific studies using financial data to gain insight into the functioning of financial markets.

 It is common in economics to start with a model and then test how the model fits the data. In physics and applied mathematics, one usually starts with empirical data (actual measurements) and then tries to discover the underlying laws or a descriptive model (see  [1-5,13,16-23]). This is the route which has been taken by a several scientists who are making contributions in a broad range of topics in finance and economics, including pricing of derivative securities, modelling of stock prices, modelling the behaviour of investors  (agent based models),  analysis of financial data, estimation of risk and portfolio optimisation (see [references]).

REFERENCES

[1]        L. Bachelier, Théorie de la Spéculation, Doctoral Dissertation, 1900

[2]        M.F.M. Osborne, Brownian Motion in the Stock Market, Operations Research 7, 145-173, 1959

[3]        M.F.M. Osborne, The Stock Market and Finance from a Physicists viewpoint, Crossgar, Minneapolis, 1977 

[4]        B. Mandelbrot, The variation of certain speculative prices, J. Business 36, 394-419, 1963 (see also IBM Research Report NC-87, March 1962)

[5]       B. Mandelbrot, and R. Hudson, The (mis)behaviour of markets - a fractal view of risk, ruin and reward, Basic Books, Cambridge, USA, 2004

[6]        H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, J. Wiley and Sons, New York, 1959

[7]        P. Samuelson, Proof that properly anticipated prices fluctuate randomly, Industrial Management Review 6, 41-49, 1965

[8]        E.Fama, The behaviour of stock market prices, J. Business 38, 34-105, 1965

[9]        P. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Rev. Econ. Stat. 51 (1969) 239-246

[10]      F. Black and M.Scholes, The pricing of options and corporate liabilities, J. Political Economy 81, 637-659, May-June 1973

[11]      R.C. Merton, Continuous-time Finance, Blackwell, 1990

[12]      E. Fama  and K. French, The cross section of expected share returns, J. Finance, 47 (1992) 427- 465

[13]      R.N. Mantegna, and H.E. Stanley, Turbulence and Financial Markets, Nature 383 (1996) 587-588

[14]      P. Bak, M MacZuski and M. Shubik, Price variations in a stock market with many agents, Physica A 246 (1997) 430-453

[15]      M Mehta, Random Matrices, 2nd Edition,  Academic Press, New York, 1991

[16]      A. Lo and A. MacKinlay, A non-random walk down Wall Street, Princeton University Press, Princeton, 1999

[17]      J. D. Farmer, Physicists attempt to scale the Ivory Towers of Finance, International J. Theoretical and Applied Finance, Vol. 3, No. 3, July 2000

[18]      H.E. Stanley, L. A. N. Amaral, P. Gopikrishnan,  Y. Liu, V. Plerou, and B. Rosenow,  Econophysics: what can physicists contribute to economics?, International J. Theoretical and Applied Finance, Vol. 3, No. 3, July 2000)

 [19]      J.-P. Bouchaud and M. Potters, Theory of Financial Risks -  from statistical physics to risk management, Cambridge University Press, Cambridge, 2000

 [20]      R.N. Mantegna and H.E. Stanley, Introduction to Econophysics, - correlations and complexity in finance, Cambridge University Press, Cambridge, 2000

[21]      D. Sornette, Why Stock Markets Crash – critical events in complex financial systems, Princeton University Press, Princeton, 2003

[22]      R. Cont, Empirical properties of asset returns: stylized facts and statistical issues, Quantitative Finance 1, 223-236, 2001

[23]      V. Plerou, P. Gopikrishnan, B. Rosenow, L. A..N. Amaral, H. E. Stanley, Econophysics: financial time series from a statistical physics point of view, Physica A, 279 (2000) 443-456

[24]      P. Mirowski, More Heat than Light, Cambridge University Press, 1989

[25]      P. Mirowski, Machine Dreams - Economics becomes a cyborg science, Cambridge University Press, 2002

[26]      T. Gebbie, In quants we trust (and hope), GARP Risk Review, Issue 11( 2003) 13-17

[27]     T. Gebbie, Business Report, Market Column, December 17, 2004, There is more to the stock market than raising capital, 

[28]     T. Gebbie, Business Reports, Market Column, October 26, 2004, Chaos in markets rules out long-term forecasts, http://www.busrep.co.za/index.php?fSectionId=564&fArticleId=2274219

[29]     T. Gebbie ,Business Reports, Market Column, September 13, 2004, Market crashes can provide critical clues,  http://www.busrep.co.za/index.php?fSectionId=564&fArticleId=2222158

[30]      D.Wilcox and T. Gebbie, On the analysis of cross correlations in South African market data, Physica A, to appear, available online 21 July 2004,   http://www.sciencedirect.com/science/journal/03784371

[31]      D.Wilcox and T. Gebbie, An analysis of cross correlations in South African market data, submitted to Physica A, e-print http://arxiv.org/abs/cond-mat/0402389

[32]     D.Wilcox and T. Gebbie, Periodicity and scaling of eigenmodes in an emerging market, submitted to International Journal of Theoretical and Applied Finance, e-print http://arxiv.org/abs/cond-mat/0404416

[33]     Various, http://www.unifr.ch/econophysics

[TO BE ADDED]  More mathematics of finance references

QUANTITATIVE ANALYSIS OF FINANCIAL MARKET DATA

(1) SA Financial Market Sectors and States: Quantitative Analysis of Cross-correlation Structure 

problem identification

It is common to consider financial markets as made up of stocks which can be grouped or clustered according to similar characteristics, for example some stocks offer low returns with respect to improvement in their value but pay high dividends, while others offer small dividends but increase in value relatively quickly. The understanding of grouping characteristics is important for understanding the overall dynamics of the market in which such clustering takes place. The appearance of grouping or clustering behaviour may be due to random effects (noise) since there are bound to be overlapping properties when the number of stocks is high. Hence it is necessary to investigate methods for measuring possible noise contributions to clustering. Furthermore, grouping may change over time so that it is necessary to identify the time horizons for which clusters are stable. Is it possible to identify sectors, groups of stocks which display similar behaviour with respect to returns, and states, time periods for which the market behaves similarly, in SA financial data, by purely quantitative methods under the constraint that noise and temporal stability are understood? 

 in collaboration with Tim Gebbie, Chanel Malherbe.

[1]  On the analysis of cross correlations in South African market data (with T.Gebbie), Physica A, Volume 344, Issues 1-2 , 

      1 December 2004, Pages 294-298 [Applications of Physics in Financial Analysis 4 (APFA4)] (link)

[2]  An analysis of cross correlations in South African market data (with T.Gebbie),
         http://arxiv.org/abs/cond-mat/0402389, 

      under review for publication
 [3] Periodicity and scaling of eigenmodes in an emerging market (with T. Gebbie),
         http://arxiv.org/abs/cond-mat/0404416, 

      under review for publication
 

[ ] further work is in progress

This project is supported by the NRF Grant 2054394 (Thuthuka programme)

(2) Scaling and trends in financial time series  and prediction

problem identification

Financial timeseries are not Gaussian.

[ ] work in progress

This project is partially supported by the NRF Grant 2054394 (Thuthuka programme)

 in collaboration with Tim Gebbie, Uli Kirchner, ...