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

Brand, Rene (2009-03)

Thesis (MBA (Business Management))--University of Stellenbosch, 2009.

Thesis

ENGLISH ABSTRACT: Econophysics is a relatively new branch of physics. It entails the use of models in physics applied to economics. The distributions of financial time series are the aspect most intensely studied by physicists. This study is based on a study by Kleinert and Chen who applied the Boltzmann distribution 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 the distributions 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 econometric benchmark 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 reasonable high 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. If the empirical exponential distribution is therefore interpreted as a Boltzmann distribution, then a market temperature can be calculated from the exponential distribution. Empirical data for this study is in the form of daily closing values of the Johannesburg Stock Exchange (JSE) All Share Index (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. This study differs from the Kleinert and Chen study in that interday data obtained from an emerging market, namely the South African stock market is used. Neither of the aforementioned two differences 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 of the econometric (ARCH/GARCH) method and subject to certain improvements can be utilised successfully. A leptokurtic, non-Gaussian nature was established for daily log-returns of the JSE ALSI 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, annual market temperatures could be calculated for the JSE ALSI and the S & P 500 index. The market temperature 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.

AFRIKAANSE OPSOMMING: Ekonofisika is ‘n relatiewe nuwe studieveld. Dit behels die toepassing van fisiese modelle op finansiële data. Die waarskynlikheidsversdelings van finansiële tydreekse is die aspek wat meeste deur fisisie bestudeer word. Hierdie studie is gebaseer op ‘n studie deur Kleinert en Chen. Hulle het die Boltzmann-verspreiding op ‘n aandele-indeks toegepas en ‘n mark-temperatuur bepaal. Hierdie mark-temperatuur kan deur ontleders gebruik word as waarskuwingsmeganisme teen moontlike aandelebeurs ineenstortings. Die meeste fisisie het die uiterste areas van die verspreidingskurwes geanaliseer omdat hierdie uiterste area risiko in finansiële data verteenwoordig. Die analitiese fokus van hierdie studie, aan die ander kant, is die karakterisering van die die sentrale areas van die waarskeinlikheidsverdeling. Die Boltzmann verspreiding, die hoeksteen van Statistiese Fisika lewer ‘n eksponensiële waarskynlikheidsverdeling. Die doel van hierdie studie is om ‘n ondersoek te doen na die geskiktheid van die gebruik van ‘n ekonofisiese, vooruitskattingsmetode, naamlik die Boltzmann/mark-temperatuur model. As ekonometriese verwysing is die “ARCH/GARCH” metode toegepas. Aandelemark indekse is bekend vir die nie-Gaussiese verspreiding daarvan. Die verspreidingspatroon van ‘n aandelemark indeks met‘n redelike hoë steekproef frekwensie (in die orde van ‘n dag of minder) is leptokurties met breë stert-dele. Mesoskopiese (interdag) verspreidings van finansiële tydreekse is getipeer as eksponensieël. Indien die empiriese eksponensiële-verspreiding as ‘n Boltzmann-verspreiding geinterpreteer word, kan ‘n mark-temperatuur daarvoor bereken word. Empiriese data vir die gebruik in hierdie studie is in die vorm van daaglikse sluitingswaardes van die Johannesburgse Effektebeurs (JSE) se Alle Aandele Indeks (ALSI) en die Standard en Poor 500 (S & P 500) indeks vir die periode 1995 tot en met 2008. Die Kleinert en Chen studie het van intradag data vanuit ‘n ontwikkelde mark gebruik gemaak. Hierdie studie verskil egter van die Kleinert en Chen studie deurdat van interdag data vanuit ‘n opkomende mark, naamlik die Suid-Afrikaanse aandelemark, gebruik is. Nie een van die twee voorafgaande verskille het ‘n beduidende invloed op die resultate van hierdie studie gehad nie. Die JSE ALSI se logaritmiese opbrengs data vertoon nie-Gaussiese eienskappe en die Laplace (dubbeleksponensiële) verspreiding beskryf die data goed. ‘n Grafiek van die mark-temperatuur vertoon duidelik wanneer aandelemarkineenstortings plaasgevind het. Resultate van die ekonofisiese (Boltzmann/mark-temperatuur) metode vergelyk goed met resultate van die ekonometriese (“ARCH/GARCH”) metode en onderhewig aan sekere verbeteringe kan dit met sukses toegepas word. ‘n Leptokurtiese, nie-Gaussiese aard is vir daaglike opbrengswaardes vir die JSE ALSI en die S & P 500 indeks vasgestel. ‘n Laplace (dubbel-eksponensiële) verspreiding kan goed op die jaarlikse logaritmiese opbrengste van die JSE ALSI en die S & P 500 indeks toegepas word. As gevolg van die goeie aanwending van die Laplace-verspreiding kan ‘n jaarlikse mark-temperatuur vir die JSE ALSI en die S & P 500 indeks bereken word. Die mark-temperatuur metode is effektief in die identifisering van aandelemarkineenstorings vir beide indekse, hoewel daar ‘n beperking is op die aantal mark-temperature wat bereken kan word. Die beskikbaarheid van intradag aandele indekswaardes behoort die interval waarvoor mark-temperature bereken kan word te verbeter.

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