A Quantum Approach to Stock Price Fluctuations.....pdf

《A Quantum Approach to Stock Price Fluctuations.....pdf》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、AQuantumApproachtoStockPriceFluctuations†MartinSchadenNewYorkUniversity,4WashingtonPlace,NewYork,NewYork10003AbstractAsimplequantummodelexplainstheL´evy-unstabledistributionsforin-dividualstockreturnsobservedbyref.[1].Theprobabilitydensityfunc-tionofthereturnsiswrittenasthesquaredmodulusofan

2、amplitude.ForshorttimeintervalsthisamplitudeisproportionaltoaCauchy-distributionandsatisfiestheSchr¨odingerequationwithanon-hermitianHamiltonian.Theobservedpowerlawtailsofthereturnfluctuationsimplythatthe”decayrate”,γ(q)asymptoticallyisproportionalto

3、q

4、,forlarge

5、q

6、.Thewavenumberq,theFourier-co

7、njugatevariabletothereturnx,isinterpretedasaquantitativemeasureof”marketsentiment”.Onatimescaleoflessthanafewweeks,thedistributionofreturnsinthisquantummodelisshapestableandscales.Themodelquantitativelyreproducestheobservedcumulativedistributionfortheshort-termnor-malizedreturnsover7ordersof

8、magnitudewithoutadjustableparame-ters.ThereturnfluctuationsoverlargetimeperiodsultimaptelybecomeGaussianifγ(q∼0)∝q2.Theansatzγ(q)=b22Tm+qisfoundtodescribethepositivepartoftheobservedhistoricprobabilityofnormal-izedreturnsfortimeperiodsbetweenT=5minandT∼4yearsovermorethan4ordersofmagnitudeinte

9、rmsofoneadjustableparametersT=mbT∝T.TheSharperatioofastockinthismodelhasafi-nitelimitastheinvestmenthorizonT→0.Implicationsforshort-terminvestmentsarediscussed.1IntroductionThequestforaquantitativestatisticaldescriptionofstockpricesbeganmorethanacenturyagowithBachelier’sthesis2inwhichhedescri

10、bedpricemove-mentsbyarandomwalk.AlthoughBachelier’srandomwalkhassincebeenmodified,theefforttomodelstockpricemovementsbyastochasticprocesscontinues.Computerizedhistoricfinancialrecordshaverecentlymadeahigh-precisionstatisticalanalysisofshort-termreturnspossible.FortimescalesthatarXiv:physics/020

11、5053v2[physics.soc-ph]9May2003arelessthanafewweeksitisincreasinglydifficulttoreconcilethiswealthofnewinformationwiththeconjecturethatpricefluctuationsaredescribedbyastochasticprocesswithindependentandidenticallydistributedincrements(iid).†Emailaddress