the fractal market hypothesis-the multifractal markets

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1、Chapter10Thefractalmarkethypothesis10.1Fractalstructureinthemarkets10.1.1Introducingfractalanalysis10.1.1.1AbriefhistoryTheHurstExponent(H)proposedbyHurst[1951]andtestedbyHurstetal.[1965]relatestotheautocorrelationsofthetimeseries,andtherateatwhichthesed

2、ecreaseasthelagbetweenpairsofvaluesincreases.Itisusedasameasureoflong-termmemoryoftimeseries,classifyingtheseriesasarandomprocess,apersistentprocess,orananti-persistentprocess.Hurstmeasuredhowareservoirlevel(NileRiverDamproject)fluctuatedarounditsaveragel

3、evelovertime,andfoundthattherangeofthefluctuationwouldchange,dependingonthelengthoftimeusedfor1measurement.Iftheserieswererandom,therangewouldincreasewiththesquarerootoftimeT2.Tostandardisethemeasureovertime,hecreatedadimensionlessratiobydividingtherangeb

4、ythestandarddeviationoftheobservation,obtainingtherescaledrangeanalysis(R=Sanalysis).Hefoundthatmostnaturalphenomenafollowabiasedrandomwalk,thatis,atrendwithnoisewhichcouldbemeasuredbyhowtherescaledrangescaleswithtime,orhowhightheexponentHisabove1.2In197

5、5Mandelbrot(seeMandelbrot[1975][2004])introducedthetermfractalasageometricshapethatisself-similarandhasfractionaldimensions,leadingtotheMandelbrotsetin1977and1979.AnobjectforwhichitstopologicaldimensionislowerorequaltoitsHausdorff-Besicovitch(H-B)dimensi

6、onisafractal.Bydoingso,hewasabletoshowhowvisualcomplexitycanbecreatedfromsimplerules.Amoregeneraldefinitionstatesthatafractalisashapemadeofpartssimilartothewholeinsomeway(seeFeder[1988]).TwoexamplesoffractalshapesaretheSierpinskiTriange(seeFigure(10.1))an

7、dKochCurve(seeFigure(10.2)).TheaboveFiguresarebi-dimensionalanddeterministicfractal.Focusingonfinancialtimeseries,Figure(10.3)illustratesaone-dimensionalnon-deterministicfractalrepresentingthirtyreturnsfromtheCAC40Indexatdifferenttimescales.Withoutanylabe

8、lsontheXYscales,itisimpossibletoknowwhichoneisdaily,weekly,ormonthlyreturns.ThefractaldimensionDmeasuresthesmoothnessofasurface,or,inourcase,thesmoothnessofatimeseries.AsdiscussedbyPeters[1994],theimportanceofthefractaldi