An econophysical investigation : using the Boltzmann distribution to determine market temperature as applied to the JSE all share index

ENGLISH ABSTRACT: Econophysics is a relatively new branch of physics. It entails the use of models in physics applied toeconomics. The distributions of financial time series are the aspect most intensely studied byphysicists. This study is based on a study by Kleinert and Chen who applied the Boltzmanndistribution to stock exchange data to define a market temperature that may be used by investors to indicate an impending stock market crash. Most econophysicists’ analysed the tail regions of thedistributions as the tails represent risk in financial data. This study’s focus of analysis, on the other hand is the characterisation of the central portion of the probability distribution. The Boltzmann distribution, a cornerstone in statistical physics, yields an exponential distribution. The objective of this study is to investigate the suitability of using a market volatility forecasting method from econophysics, namely the Boltzmann/market temperature method. As econometricbenchmark the ARCH/GARCH method is used. Stock market indices are known to be non-normally(non-Gaussian) distributed. The distribution pattern of a stock market index of reasonablehigh sampling frequency (typically interday or intraday) is leptokurtic with heavy tails. Mesoscopic (interday) distributions of financial time series have been found to be exponential distributions. Ifthe empirical exponential distribution is therefore interpreted as a Boltzmann distribution, then amarket temperature can be calculated from the exponential distribution. Empirical data for thisstudy is in the form of daily closing values of the Johannesburg Stock Exchange (JSE) All ShareIndex (ALSI) and the Standard & Poor 500 (S & P 500) index for the period 1995 through to 2008.The Kleinert and Chen study made use of intraday data obtained from established markets. Thisstudy differs from the Kleinert and Chen study in that interday data obtained from an emergingmarket, namely the South African stock market is used. Neither of the aforementioned twodifferences had a significant influence on the results of this study. The JSE ALSI log-return data displays non-Gaussian properties and the Laplace (double exponential) distribution fit the data well.A plot of the market temperature provided a clear indication of when stock market crashes occurred.Results of the econophysical (Boltzmann/market temperature) method compared well to results ofthe econometric (ARCH/GARCH) method and subject to certain improvements can be utilisedsuccessfully. A leptokurtic, non-Gaussian nature was established for daily log-returns of the JSEALSI and the S & P 500 index. The Laplace (double exponential) distribution fit the annual logreturns of the JSE ALSI and S & P 500 index well. As a result of the good Laplace fit, annualmarket temperatures could be calculated for the JSE ALSI and the S & P 500 index. The markettemperature method was effective in identifying market crashes for both indices, but a limitation of the method is that only annual market temperatures can be determined. The availability of intraday stock index data should improve the interval for which market temperature can be determined.