量子金融工程模型——金融物理學(xué)論文系列

1、PhysicaA389(2010)57695775ContentslistsavailableatScienceDirectPhysicaAjournalhomepage:AquantummodelforthestockmarketChaoZhanga,LuHuangbabSchoolofPhysicsandEngineering,SunYat-senUniversity,Guangzhou510275,ChinaSchoolofEconomicsandBusinessAdministration,ChongqingUniversity,Chongqing400044,Chinaarticle

2、infoabstractBeginningwithseveralbasichypothesesofquantummechanics,wegiveanewquantummodelineconophysics.Inthismodel,wedefinewavefunctionsandoperatorsofthestockmarkettoestablishtheSchrödingerequationforstockprice.Basedonthistheoreticalframework,anexampleofadriveninfinitequantumwellisconsidered,in

3、whichweuseacosinedistributiontosimulatethestateofstockpriceinequilibrium.AfteraddinganexternalfieldintotheHamiltoniantoanalyticallycalculatethewavefunction,thedistributionandtheaveragevalueoftherateofreturnareshown.©2010ElsevierB.V.Allrightsreserved.Articlehistory:Received12April2010Receivedinr

4、evisedform27August2010Availableonline18September2010Keywords:EconophysicsQuantumfinanceStockmarketQuantummodelStockpriceRateofreturn1.IntroductionThestudyofeconophysicsoriginatedinthe1990s1.Somephysicistsfoundthatafewmodelsofstatisticalphysicscouldbeusedtodescribethecomplexityoffinancialmarkets2,3.N

5、owadaysmostoftheeconophysicstheoriesareestablishedonthebasisofstatisticalphysics.Statisticalphysicsisonlyonebranchofphysics.Afterseveralyearsdevelopmentofeconophysics,somephysicistsbegantouseotherphysicaltheoriestostudyeconomics.Thequantumtheoryisoneofthemostimportanttheoriesincontemporaryphysics.It

6、wasthefirsttimethatthequantumtheorywasappliedtothefinancialmarketswhensomeoneusedthequantumfieldtheorytomakeportfoliosafinancialfield4,5,inwhichpathintegralsanddifferentialmanifoldswereintroducedasthetoolstodescribethechangeoffinancialmarketsaftergaugetransformation.Thisideaisthesameastheessenceofth

7、estochasticanalysisinfinance.Therearesomeotherinterestingquantummodels.Forinstance,Schadenoriginallydescribedassetsandcashheldbytheinvestorasawavefunctiontomodelthefinancialmarkets,whichwasdifferentfromtheusualfinancialmethodsusingthechangeoftheassetpricetobethedescription6.Inaddition,peoplepaidmore

8、attentiontothequantumgametheoryandthatwasusefulintradingstrategies7,8.Inrecentyears,anincreasingnumberofquantummodelshavebeenappliedtofinance915,whichattractedgreatattention.Inthispaper,westartfromanewapproachtoexplorethequantumapplicationtothestockmarket.InSection2webeginfromseveralbasichypotheseso

9、fquantummechanicstoestablishanewquantumfinancemodel,whichcanbeusedtostudythedynamicsofstockprice.InSection3asimpleHamiltonianofastockisgiven.Bysolvingthecorrespondingpartialdifferentialequation,wequantitativelydescribethevolatilityofthestockintheChinesestockmarketunderthenewframeworkofquantumfinance

10、theory.AconclusionisillustratedinSection4.Correspondingauthor.E-mailaddress:zhang_chao(C.Zhang).0378-4371/$seefrontmatter©2010ElsevierB.V.Allrightsreserved.doi:10.1016/j.physa.2010.09.0085770C.Zhang,L.Huang/PhysicaA389(2010)576957752.ThequantummodelQuantummechanicsisthetheorydescribingthemicro-

11、world.Nowthistheoryistobeappliedinthestockmarket,inwhichthestockindexisbasedonthestatisticsofthesharepricesofmanyrepresentativestocks.Ifweregardtheindexasamacro-scaleobject,itisreasonabletotakeeverystock,whichconstitutestheindex,asamicrosystem.Thestockisalwaystradedatcertainprices,whichpresentsitsco

12、rpuscularproperty.Meanwhile,thestockpricefluctuatesinthemarket,whichpresentsthewaveproperty.Duetothiswaveparticledualism,wesupposethemicro-scalestockasaquantumsystem.Rulesaredifferentbetweenthequantumandclassicalmechanics.Inordertodescribethequantumcharactersofthestock,wearegoingtobuildapricemodelon

13、thebasichypothesesofquantummechanics.2.1.StatevectorintheHilbertspaceInthefirsthypothesisofquantummechanics,thevectorcalledwavefunctionintheHilbertspacedescribesthestateofthequantumsystem.Beingdifferentfrompreviousquantumfinancemodels6,9,herewetakethesquaremodulusofthewavefunction(,t)asthepricedistr

14、ibution,wheredenotesthestockpriceandtisthetime.Duetothewavepropertyofthestock,thewavefunctionintheso-calledpricerepresentationcanbeexpressedbyDiracnotationsas|=n|ncn,(1)where|nisthepossiblestateofthestocksystemandthecoefficientcn=n|.Itisexactlythesuperpositionprincipleofquantummechanics,whichhasbeen

15、studiedbyShi16andPiotrowskiandSladkowski17inthestockmarket.Asaresult,thestateofthestockpricebeforetradingshouldbeawavepacket,orratheradistribution,whichisthesuperpositionofitsvariouspossiblestateswithdifferentprices.Undertheinfluenceofexternalfactors,investorsbuyorsellstocksatsomeprice.Suchatradingp

16、rocesscanbeviewedasaphysicalmeasurementoranobservation.Asaresult,thestateofthestockturnstobeoneofthepossiblestates,whichhasacertainprice,i.e.thetradingprice.Inthiscase,|cn|2denotestheappearanceprobabilityofeachstate.Inthestatisticalinterpretationofthewavefunction,|(,t)|2istheprobabilitydensityofthes

17、tockpriceattimet,andP(t)=b|(,t)|2dpa(2)istheprobabilityofthestockpricebetweenaandbattimet.Actuallythefluctuationofthestockpricecanbeviewedastheevolutionofthewavefunction(,t)andwewillpresentthecorrespondingSchrödingerequationinthefollowingtext.2.2.HermitianoperatorforthestockmarketInquantummecha

18、nics,physicalquantitiesthatareusedtodescribethesystemcanbewrittenasHermitianoperatorsintheHilbertspace,whichdeterminetheobservablestates.Thevaluesofphysicalquantitiesshouldbetheeigenvaluesofcorrespondingoperators.Whileinthestockmarket,eachHermitianoperatorrepresentsaneconomicquantity.Forexample,inth

19、epriceoperator(hereweapproximatethepriceasacontinuous-variable)correspondstothepositionoperatorxquantummechanics,whichhasbeenoriginallyusedintheBrownianmotionofthestockprice18.Thereforethefluctuationofthestockpricecanbeviewedasthemotionofaparticleinspace.Moreover,theenergyofthestock,whichrepresentst

20、heintensityofthepricesmovement,canbedescribedbytheHamiltonianthatplaysakeyroleintheSchrödingerequation.2.3.UncertaintyprincipleTherelationoftwovariablesthatdonotcommutewitheachothercanbedemonstratedbytheuncertaintyprinciple.Forexample,thepositionandthemomentumaretwofamiliarconjugatevariablesinq

21、uantummechanics.Theproductoftheirstandarddeviationisgreaterthanorequaltoacertainconstant.Thismeansonecannotsimultaneouslygetaccuratevaluesofbothpositionandmomentum.Themorepreciselyonevariableismeasured,thelesspreciselytheotheronecanbeknown.Asismentionedabove,thestockpricecorrespondstotheposition.Mea

22、nwhilethereshouldbeanothervariableTcorrespondingtothemomentum.Asguidanceinquantumtheory,thecorrespondenceprinciplefiguresoutthatwhenthelawswithintheframeworkofthemicro-worldextendtomacroscope,theresultsshouldbeconsistentwiththeoutcomesoftheclassicallaws.Inthemacrosystem,themomentumcanbewrittenasthem

23、asstimesthefirst-ordertimederivativeofthepositioninsomespecialcases.Asaresult,inourquantumfinancemodel,(3)dtwherewecalltheconstantm0themassofstock.Tisavariabledenotingtherateofpricechange,whichcorrespondstothetrendofthepriceinthestockmarket.Inourmodel,theuncertaintyprinciplethuscanbewrittenasT=m0d&#

24、983051;󰀋Th¯2,(4)C.Zhang,L.Huang/PhysicaA389(2010)576957755771where󰀋and󰀋Tarethestandarddeviationsofthepriceandthetrendrespectively,andh¯isthereducedPlanckconstantinquantummechanics.TheequalityisachievedwhenthewavefunctionofthesystemisaGaussiandistributionfunction.Mea

25、nwhile,infinance,theGaussiandistributionusuallymayapproximatelydescribetherateofreturnoftheassetinthebalancedmarket18.TakingYuan(thecurrencyunitinChina)astheunitofpriceintherestoftext,wemayestimatethestandarddeviationofthepriceas󰀋=103Yuan19.Whenthetotalvariationofthestockpriceissmall,thestan

26、dardddeviationofdinthetrend(3)canbeapproximatedast󰀋ddt222ddd=.dtdtdt(5)Meanwhile,theaveragerateofstockpricechangecanbeevaluatedas102YuanpertensecondsintheChinesestockmarket.Viatheuncertaintyprinciple(4)weestimatethemagnitudeofm0isabout1028.Althoughtheunitofthismass,whichcontainsunitsofmass,l

27、engthandcurrency,isdifferentfromtherealmass,itdoesnotaffectthecalculationofthewavefunctionwhichisnon-dimensionalandwestillcallitmassinthispaper.Itshouldbeanintrinsicpropertyandrepresentstheinertiaofastock.Whenthestockhasabiggermass,itspriceismoredifficulttochange.Ingeneral,stockshavinglargermarketca

28、pitalizations,alwaysmoveslowerthanthesmallermarketcapitalizationsones.Thus,themassofthestockmaybeconsideredasaquantityrepresentingthemarketcapitalization.Theuncertaintyprinciple(4)canbeoftenseeninfinance.Forexample,atacertaintimesomeoneknowsnothingbuttheexactpriceofastock.Asaresult,hecertainlydoesno

29、tknowtherateofpricechangethenexttimeandthedirectionofthepricemovement.Inotherwords,theuncertaintyofthetrendseemstobeinfinite.Howeverintherealstockmarket,weknowmorethanthestockpriceitselfatanytime.Wecanalwaysgettheinformationabouthowmanybuyersandsellerstherearenearthecurrentprice(e.g.investorsinChina

30、areabletoseefiveortenbidandaskpricesandtheirvolumesonthescreenviastocktradingsoftware).Itisactuallyadistributionofthepricewithinacertainrangeinsteadofanexactprice.Asaresult,wecanevaluateastandarddeviationoftheprice.Thusthetrendofthestockpricemaybepartlyknownviatheuncertaintyprinciple(4).Forexample,i

31、fatraderseesthenumberofbuyersisfarmorethanthenumberofsellersnearthecurrentprice,hemaypredictthatthepricewillrisethenexttime.Infinance,thestandarddeviationoftheassetpriceisusuallyanindicatorofthefinancialrisks.Introducingtheuncertaintyprincipleofthequantumtheorymaybehelpfulinthestudyoftheriskmanageme

32、nttheory.2.4.SchrödingerequationWiththeassumptionsofthewavefunctionandtheoperatorabove,letusconsideradifferentialequationtocalculatetheevolutionofthestockpricedistributionovertime.Inthequantumtheory,itshouldbetheSchrödingerequationwhichdescribestheevolutionofthemicro-world.Correspondingtoo

33、urmodel,itcanbeexpressedas(,t),(,t)=H(6)t=H(,T,t)isthefunctionofprice,trendandtime.Whenweknowtheinitialstateoftheprice,wheretheHamiltonianHih¯bysolvingthepartialdifferentialequation(6)wecangetthepricedistributionatanytimeinthefuture.ThedifficultyhereistheconstructionoftheHamiltonianbecausethere

34、arelotsoffactorsimpactingthepriceandthetrendofthestock,suchastheeconomicenvironment,themarketinginformation,thepsychologyofinvestors,etc.Itisnoteasytoquantifythemandimportthemintoanoperator.NextwewillusethetheoriesabovetoconstructasimpleHamiltoniantosimulatethefluctuationofthestockpriceintheChinesem

35、arketunderanidealperiodicimpactofexternalfactors.3.ThestockinaninfinitehighsquarewellIntheChinesestockmarketthereisapricelimitrule,i.e.therateofreturninatradingdaycannotbemorethan±10%comparedwiththepreviousdaysclosingprice,whichisappliedtomoststocksinChina.Thisleadsustosimulatethefluctuationoft

36、hestockpricebetweenthepricelimitsinaone-dimensionalinfinitesquarewell.Weconsideraninfinitewellwithwidthd0=0×20%,where0isthepreviousdaysclosingpriceofastock.Afterthetransformationofcoordinate=0,(7)thequantumwellbecomesasymmetricinfinitesquarewellwithwidthd0,andthestockpriceistransformedintotheab

37、solutereturn.Goonandletr=,0(8)thusthevariableofcoordinateturnstobetherateofthereturn.Atthesametime,thewidthofthewellbecomesd=20%.IfweapproximatelytaketheaveragestockpriceinChinatobe10Yuan,toevaluatethemassofthestockagaininthenew5772C.Zhang,L.Huang/PhysicaA389(2010)57695775Fig.1.(a)Intheinfinitesquar

38、ewellwithwidthd=0.2,theprobabilitydensityoftherateofstockreturncanbeapproximatelydescribedbytheGaussiandistribution(dashedline).Hereweuseacosine-squarefunction(solidline)tosimulatethedistributionoftherateofreturninequilibrium.(b)WhenaperiodicfieldisputintotheHamiltonian,thebottomofthequantumwellbegi

39、nstoslopeanditchangesperiodically.coordinatesystem,thereshouldbeadivisionby102Yuanfromtheleftsideoftheuncertaintyprinciple(4).Comparedwithm0,themassdenotedbymhereshouldbeapproximatelyevaluatedas1030,inwhichthedimensionofcurrencyvanishesandthisismoreacceptableinphysics.Usually,whenthemarketstaysinthe

40、stateofequilibrium,thereturndistributioncanbedescribedapproximatelybytheGaussiandistribution18ormorepreciselybytheLévydistribution20.WhileintheChinesestockmarketwiththepricelimitrule,thedistributionmaybemorecomplicatedduetotheboundaryconditions.However,hereweonlywanttoquantitativelydescribethev

41、olatilityofthestockreturnbythequantummodel,sowechooseacosine-squarefunction,whoseshapeisclosetotheGaussiandistribution,toapproximatelysimulatethereturndistributioninequilibrium.Thereasonforsuchaselectionisthegroundstateofthesymmetricinfinitewelldiscussedaboveisaconcavecosinefunctionwithnozeroexceptf

42、ortwoextremepoints21,whichisusuallydenotedby0(r)=2dcosrd(9)withcorrespondingeigenenergyE0=h¯222md2.(10)Accordingtothepropertyofthewavefunction,thesquaremodulusofthestate(9)istheprobabilitydensityofthedistributionoftherateofthereturn.AsisshowninFig.1(a),whatisthesameastheGaussiandistributionorth

43、eLévydistributionisthatinthecenterofthewell,whichcorrespondstozeroreturn,theprobabilitydensityhasthemaximalvalueanditdecreasessymmetricallyandgraduallytowardstheleftandrightsides.Themainreasonwhythisdistributionisnotpreciseenoughisitdoesnothavefattailsandasharppeak.Thecosinedistributionfunction

44、equalstozeroatr=±d/2duetotheboundaryconditionoftheinfinitesquarewell.Judgingfromtheshape,thecosinedistributionisagoodapproximationfortheGaussiandistributionwithalargevarianceortheLévydistributionwithagreatparameter.Thereisalwaysalotofmarketinformationaffectingthestockprice.Thetotaleffectof

45、informationappearingatacertaintimeisusuallyconduciveeithertothestockpricesriseortothestockpricesdecline.Inordertodescribetheevolutionoftherateofreturnundertheinformation,weconsideranidealizedmodel,inwhichweassumetwotypesofinformationappearperiodically.Wemayuseacosinefunctioncosttosimulatethefluctuat

46、ionoftheinformation,whereistheappearancefrequencyofdifferentkindsofinformationandhereweassume=104s1.ThatmeanstheinformationfluctuatesinasinglecycleofaboutfourtradingdaysinChina,whichmaybereasonable.Thevalueofthefunctioncostchangesbetween1,1overtime,wheretheinformationisadvantageoustothepricesrisewhe

47、nthecosinefunctionislessthanzero,andadvantageoustothepricesdeclinewhenitisgreaterthanzero.Thestockhereissimilartoachargedparticlemovinginanelectromagneticfield,wherethedifferenceisthattheexternalfieldofthestockmarketisconstructedbytheinformation.Thestockpricemaybeinfluencedbysuchafield.Underthedipol

48、eapproximation,thepotentialenergyofthestockcanbesimilarlyexpressedaseFrcost,whereeisaconstantwiththesameorderofmagnitudeasanelementarycharge,andFdenotestheamplitudeoftheexternalfield.TheHamiltonianofthiscoupledsystemcanbewrittenash¯22H=+eFrcost,2mr2(11)wherethefirsttermisthekineticenergyofthest

49、ockreturn,whichrepresentssomepropertiesofthestockitself.Thesecondonecorrespondingtothepotentialenergyreflectsthecyclicalimpactthestockfeelsintheinformationfield.Inordertoobservethecharactersofthissystementirely,wemakethemagnitudedifferencebetweenthekineticenergy(i.e.groundstateenergyoftheinfinitesqu

50、arewellwithoutexternalfield)andthepotentialenergysmall,thustheestimateF=1019C.Zhang,L.Huang/PhysicaA389(2010)576957755773here.BecausethesecondtermofHamiltoniancontainstherateofreturnr,atiltappearsatthebottomoftheinfinitesquarewell,whichisshowninFig.1(b).Theslopeofthebottomfluctuatesperiodicallydueto

51、thechangeofcostovertime,whichmeansthewellisnolongersymmetric.Thisreflectstheimbalanceofthemarket,andthedistributionoftherateofreturnbecomessymmetrybreakingfromthestate(9)inequilibrium.AfterconstructingsuchanidealHamiltonian(11),weneedtogetthesolutionofthecorrespondingSchrödingerequationh¯2

52、2ih+eFrcost(r,t).¯(r,t)=t2mr2(12)However,theanalyticalsolutionforasimilarequationhasbeenstudiedbefore22.Thereforewejusthaveabriefreview.Atfirst,let(r,t)=(,t)(r,t)withvariablesubstitutionm2Inthewavefunction(13),let(13)=reFcost.(14)iEctieFrsintie2F2(2tsin2t)(r,t)=exp,hh8h¯¯¯m3(15)w

53、hereEcdenotestheenergyofthedrivensystem.AfterbothsidesofEq.(12)aredividedby(r,t),wefind(,t)satisfiesthetime-dependentSchrödingerequationh¯222m(,t)E(,t)=ih¯2l=(,t)t(16)anditssolutioncanbewrittenas(,t)=Alexp±i2mEh¯exp(ilt),(17)wheretheenergymaybeexpressedasE=Ec±lh¯.(

54、18)ThecentricenergyEcisdeterminedbytheboundaryconditionsandclosetotheenergy(10)ofthegroundstateofthestationarySchrödingerequationwithoutexternalimpact.TheenergysplitsaroundEcwiththeintegralmultipleofenergyunith¯andeverynewenergycorrespondstoapossiblestateofthesystem.FromEqs.(13),(15)and(17

55、),thewavefunctionoftherateofstockreturncanbewrittenasthesuperpositionofallthosestates(r,t)=exp×iEcth¯ieFrsinth¯ie2F2(2tsin2t)Alexp(ilt)expikl8h¯m3reFcostm2l=eFcost+()lexpiklr,m2(19)wherethewavevectoriskl=2m(Ec+lh¯)/h¯2.(20)Inthewavefunction(19),Aldenotestheamplitudefore

56、achpossiblestate,whicharesomeconstantstobedetermined.InEq.(19),weusetheFourierexpansionfortheexponentialfunctionexpikeFkleFcostm2=n=inJnkleFm2exp(int),(21)whereJn(m2)isthenthBesselfunctionandletEq.(19)satisfytheboundaryconditionoftheinfinitequantumwelld2,t=,t2d=0.(22)Accordingtophysics,thismustbesat

57、isfiedatanytime.BytheorthogonalityofBesselfunction23l=Jnl(u)Jml(u)=m,n,(23)5774C.Zhang,L.Huang/PhysicaA389(2010)57695775Fig.2.Numericalsimulationsoftheprobabilitydensityoftherateofreturnat(a)t=0(solidline),(b)t=1000s(dottedline)and(c)t=25,000s(dashedline),inwhichparametersarerespectivelye=1019,m=103

58、0,=104,F=1019andd=0.2.Att=0thedistributioncorrespondstotheinitialstateofthesystem.Theexternalfieldmakesthedistributionsimbalanceatt=1000sandt=25,000s.thesummationoverninEq.(21)canbereduced,andEq.(19)attheboundaryturnstobe(i)lAlexp(ikld/2)+()lexp(ikld/2)Jn+lkleFm2=0.(24)l=Bydefiningtwoparametersv=q=h¯Eck0eFm2(25)andexpandingthewavevectorkl=k0maybefound22Al=il1ltothesecondorderofv,thenon-normalizedapproximatesolutionofAl3q2v32Jl(q)+q(q22)v264Jl+1(q)Jl1(q)q3v264Jl+2(q)Jl2(q)9q4v22048whichmeetsq2v232Jl+