ocal Hurst exponent of the financial time series in the vici

介绍一种策略

PhysicaA387(2008)

4299–4308

http:///locate/physa

ThelocalHurstexponentofthe nancialtimeseriesinthevicinity

ofcrashesonthePolishstockexchangemarket

DariuszGrech ,GrzegorzPamu a

InstituteofTheoreticalPhysics,UniversityofWroc aw,PL-50-204Wroc aw,Poland

Received13November2007;receivedinrevisedform19January2008

Availableonline12February2008

Abstract

Weinvestigatethelocalfractalpropertiesofthe nancialtimeseriesbasedonthewholehistoryevolution(1991–2007)oftheWarsawStockExchangeIndex(WIG),connectedwiththelargestdeveloping nancialmarketinEurope.Calculatingtheso-calledlocaltime-dependentHurstexponentHlocfortheWIGtimeserieswe ndthedependencebetweenthebehaviorofthelocalfractalpropertiesoftheWIGtimeseriesandthecrashes’appearanceonthe nancialmarket.WeformulatethenecessaryconditionsbasedontheHlocbehaviorwhichhavetobesatis ediftheruptureorcrashpointisexpectedsoon.AsaresultweshowthatthesignaltosellorthesignaltobuyonthestockexchangemarketcanbetranslatedintoHlocevolutionpattern.Wealso ndarelationbetweentherateoftheHlocdropandthetotalcorrectiontheWIGindexgainsafterthecrash.Thecurrentsituationonthemarket,particularlyrelatedtotherecentFedinterventioninSeptember’07,isalsodiscussed.

c2008ElsevierB.V.Allrightsreserved.

PACS:89.65.Gh;05.40.Jc;05.45.Vx

Keywords:Econophysics;Timeseries;Scalinglaws;Powerlaws;Hurstexponent;Financialcrashes;Fractals

Themodelsandmechanismsenablingtopredictthefuturebehaviorof nancialmarketsonalong-orshort-termperiodareabigchallengein nancialengineeringandveryrecentlyalsoineconophysics.Inthelattercaseonebelievesthattheapproachbasedontheanalogyofthe nancialmarketwithcomplexdynamicalsystemscouldbeveryfruitful[1–8].Inparticular,thescaleinvarianceofthecomplexsystemsisusedtorevealthelog-periodicoscillationscharacteristicforthesesystemsbeforetheso-calledphasetransitionpointisreached.Thelog-periodicoscillationsprecedingthecrashesorrupturepoints,i.e.themomentswhentheincreasinglong-termtrendisbeingbrokenandstartstoreverse,havereallybeenobservedformainmarketindicesorshareprices(seee.g.Refs.[1,2,9–13]).Anumberofcrashmomentstchavebeenpredictedintheliteraturesofar,someofthembeinginverygoodagreementwiththeactualmomentthecrashtrulytookplace[11–13].

Howeveronehastorememberthatany nancialsystemshouldbeconsideredasanopensystem,whilethescalelessbehaviorandthequantitativedescriptionfollowingfromthisassumptionareabsolutelytrueonlyinthecaseofclosedstatisticalsystems.Thereforeoneshouldbeawarethatthemethodsbasedontheclosedsystemassumptionarevery Correspondingauthor.

E-mailaddress:dgrech@ift.uni.wroc.pl(D.Grech).

c2008ElsevierB.V.Allrightsreserved.0378-4371/$-seefrontmatter

doi:10.1016/j.physa.2008.02.007

介绍一种策略

4300D.Grech,G.Pamu a/PhysicaA387(2008)4299–4308

sensitivetothenumberofdatapoints(information)onetakesintoaccounttomakea twithlog-periodicoscillationparameters.Asaresultthepredictivepowerofsuchmodelscanbeingeneralverylimited[14].

Fewyearsagoanotherapproachbasedonthelocalpropertiesofthetimeserieswasproposed[15].ThemethodusesthelocalHurstexponentHloc[16,17]orthelocalfractaldimensionDlocofthetimeseriesbuiltontheindexvaluesorshareprices.Thesequantitiesarelinkedtogetherbythewell-knownrelation

Dloc=2 Hloc(1)inananalogywiththesimilarequationsatis edfortheglobalDandHvaluesusuallyusedtodescribethemonofractalsignals.

InRef.[15]itwasshownthatthelocalHurstexponentcalculatedfortheDowJones(DJIA)indexformsacharacteristicpatternbeforeanyrecordedcrashontheAmericanstockmarket.TheHlocvaluesdropsigni cantlydownbeforeanyrupturepoint.Themovingaverage Hloc(t) 5ofthelocalHurstexponentcalculatedontheoneweekperiod(5sessions)dropsdownto0.45orevenlessforsessionsimmediatelyprecedingtherupturepoint,thusrevealingthepresenceofthegrowingantipersistenceinthe nancialtimeseriessignalbeforethecrashoccurs.ThesimilarqualitativebehaviorofthelocalHvaluesforother nancialtimeseries(shares)hasalsobeencon rmed[18].

TheadvantageintheuseofHlocoverothermethodsisthatitactuallymeasuresthetemporal uctuationsofthemarketandthereforeitseemstobemoreresistanttothelong-terminaccuraciesordistortionscominge.g.fromtherapidchangeoftheboundaryconditionsaroundthe nancialsystem.ThereforetheHlocmethodmightbealsoappliedtoanopencomplexsystem—the nancialmarketbeingagoodcandidateofsuchsystem.

InthisreportweintendtoapplythetechniqueusedinRef.[15]toinvestigatethemarketindexofthelargestemergingmarketinEurope—theWarsawStockExchangeIndex(WIG)incorporatingmorethan200companies.Ithasalreadya16-yearsoldhistoryuptonowwithalmost3700closuredayvalues.

TheWIGtimeserieshasbeenanalyzedbyuswiththeDetrendedFluctuationAnalysis(DFA)technique[16]toextractthescalingHurstexponentH.Thistechniquewasverywelldescribedintheliteraturesofar[16,17,19]includingthedetaileddiscussionofvariouseffectsonDFAresultsliketrends[20],nonstationarities[21]ornonlinear lters[22].Letusremindhoweverthelessknown‘local’versionofDFAfullyrevealedinRef.[15]andappliedinourcalculationsbelow.

We rstformforagiventradingdayt=iatimesubseriesoflengthNwithpointsintheperiod i N+1,i .Wecallthissubseriestheobservationboxortheobservationtime-window.ThenthestandardDFAprocedureisappliedtothistime-window,i.e.wecoverthesubserieswithsmallernon-overlappingboxesofsizeτstartingfromthegiventradingpointt=iandgoingbackwardsintimeuptot=i N+1.Inordertocoverthewholetime-windowwithτ-sizeboxesweputthedata i N+1,i [N/τ]τ+1 ,where[.]meanstheintegerpart,intothelastbox.Thisboxpartlyoverlapstheprecedingonebutthisdoesnotmodifytheobtainedresults.IneachboxthedetrendedsignalisfoundaccordingtoDFAmethodforthesimplestlineartrendassumedineveryboxofsizeτ.Thedetrendedsignal uctuatesanditsvariance F2(τ) isrelatedtotheboxwidthτbythepower-lawrelationknownforDFA:

F2(τ) ～τ2Hloc.(2)ThusmovingtheobservationboxoflengthNsession-after-sessionweareabletoobservethewholehistoryofHloc(t)changesintime.

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