medium-and-smallscale-analysis-of-financial-data中小規(guī)模的金融數(shù)據(jù)分析外文翻譯學(xué)士學(xué)位論文

Mediumandsmall-scaleanalysisoffinancialdataAbstractAstochasticanalysisoffinancialdataispresented.Inparticularweinvestigatehowthestatisticsoflogreturnschangewithdifferenttimedelayst.Thescale-dependentbehaviouroffinancialdatacanbedividedintotworegions.Thefirsttimerange,thesmall-timescaleregion(intherangeofseconds)seemstobecharacterisedbyuniversalfeatures.Thesecondtimerange,themedium-timescalerangefromseveralminutesupwardscanbecharacterisedbyacascadeprocess,whichisgivenbyastochasticMarkovprocessinthescaleτ.AcorrespondingFokker–Planckequationcanbeextractedfromgivendataandprovidesanon-equilibriumthermodynamicaldescriptionofthecomplexityoffinancialdata.Keywords:Econophysics;Financialmarkets;Stochasticprocesses;Fokker–Planckequation1.IntroductionOneoftheoutstandingfeaturesofthecomplexityoffinancialmarketsisthatveryoftenfinancialquantitiesdisplaynon-Gaussianstatisticsoftendenotedasheavytailedorintermittentstatistics.Tocharacterizethefluctuationsofafinancialtimeseriesx(t),mostcommonlyquantitieslikereturns,logreturnsorpriceincrementsareused.Here,weconsiderthestatisticsofthelogreturny(τ)overacertaintimescalet,whichisdefinedasy(τ)=logx(t+τ)-logx(t),(1)wherex(t)denotesthepriceoftheassetattimet.Acommonproblemintheanalysisoffinancialdataisthequestionofstationarityforthediscussedstochasticquantities.Inparticularwefindinouranalysisthatthemethodsseemtoberobustagainstnonstationarityeffects.Thismaybeduetothedataselection.Notethattheuseof(conditional)returnsofscaleτcorrespondstoaspecificfilteringofthedata.Neverthelesstheparticularresultschangeslightlyfordifferentdatawindows,indicatingapossibleinfluenceofnonstationarityeffects.Inthispaperwefocusontheanalysisandreconstructionoftheprocessesforagivendatawindow(timeperiod).TheanalysispresentedismainlybasedonBayerdataforthetimespanof1993–2003.ThefinancialdatasetswereprovidedbytheKarlsruherKapitalmarktDatenbank(KKMDB).2.Small-scaleanalysisOneremarkablefeatureoffinancialdataisthefactthattheprobabilitydensityfunctions(pdfs)arenotGaussian,butexhibitheavytailedshapes.Anotherremarkablefeatureisthechangeoftheshapewiththesizeofthescalevariableτ.Toanalysethechangingstatisticsofthepdfswiththescaletanon-parametricapproachischosen.Thedistancebetweenthepdfp(y(τ))onatimescaleτandapdfpT(y(T))onareferencetimescaleTiscomputed.Asareferencetimescale,T=1sischosen,whichisclosetothesmallestavailabletimescaleinourdatasetsandonwhichtherearestillsufficientevents.Inordertobeabletocomparetheshapeofthepdfsandtoexcludeeffectsduetovariationsofthemeanandvariance,allpdfsp(y(τ))havebeennormalisedtoazeromeanandastandarddeviationof1.Asameasuretoquantifythedistancebetweenthetwodistributionsp(y(τ))andpT(y(T)),theKullback–Leiblerentropyisused.dK(τ)=(2)TheevolutionofdKwithincreasingtisillustrated.Thisquantifiesthechangeoftheshapeofthepdfs.Fordifferentstockswefoundthatfortimescalessmallerthanabout1minalineargrowthofthedistancemeasureseemstobeuniversallypresent.IfanormalisedGaussiandistributionistakenasareferencedistribution,thefastdeviationfromtheGaussianshapeinthesmall-timescaleregimebecomesevident.ForlargertimescalesdKremainsapproximatelyconstant,indicatingaveryslowchangeoftheshapeofthepdfs.3.MediumscaleanalysisNextthebehaviourforlargertimescales(τ>1min)isdiscussed.Weproceedwiththeideaofacascade.itispossibletograspthecomplexityoffinancialdatabycascadeprocessesrunninginthevariableτ.InparticularithasbeenshownthatitispossibletoestimatedirectlyfromgivendataastochasticcascadeprocessintheformofaFokker–Planckequation.Theunderlyingideaofthisapproachistoaccessstatisticsofallordersofthefinancialdatabythegeneraljointn-scaleprobabilitydensitiesp(y1,τ1;y2,τ2;…;yN,τN).Hereweusetheshorthandnotationy1=y(τ1)andtakewithoutlossofgeneralityτi<τi+1.Thesmallerlogreturnsy(τi)arenestedinsidethelargerlogreturnsy(τi+1)withcommonendpointt.Thejointpdfscanbeexpressedaswellbythemultipleconditionalprobabilitydensitiesp(yi,ti│yi+1,ti+1;...;yN,tN).Thisverygeneraln-scalecharacterisationofadataset,whichcontainsthegeneraln-pointstatistics,canbesimplifiedessentiallyifthereisastochasticprocessint,whichisaMarkovprocess.Thisisthecaseiftheconditionalprobabilitydensitiesfulfilthefollowingrelations:p(y1,τ1│y2,τ2;y3,τ3;...;yN,τN)＝p(y1,τ1│y2)(3)Consequently,p(y1,τ1;…;yN,τN)=p(y1,τ1│y2)……p(yN-1,τN-1│yN,τN)·p(yN,τN)(4)holds.Eq.(4)indicatestheimportanceoftheconditionalpdfforMarkovprocesses.Knowledgeofp(y,τ│y0,τ0)(forarbitraryscalesτandτ0withτ<τ0)issufficienttogeneratetheentirestatisticsoftheincrement,encodedintheN-pointprobabilitydensityp(y1,τ1;y2,τ2;…;yN,τN).ForMarkovprocessestheconditionalprobabilitydensitysatisfiesamasterequation,whichcanbeputintotheformofaKramers–MoyalexpansionforwhichtheKramers–MoyalcoefficientsD(K)(y,τ)aredefinedasthelimit△τ→0oftheconditionalmomentsM(K)(y,τ,△τ):(5)(6)Forageneralstochasticprocess,allKramers–Moyalcoefficientsaredifferentfromzero.AccordingtoPawula’stheorem,however,theKramers–Moyalexpansionstopsafterthesecondterm,providedthatthefourthordercoefficientD(4)(y,τ)vanishes.Inthatcase,theKramers–MoyalexpansionreducestoaFokker–Planckequation(alsoknownasthebackwardsorsecondKolmogorovequation):(7)D(1)isdenotedasdriftterm,D(2)asdiffusionterm.Theprobabilitydensityp(y,τ)hastosatisfythesameequation,ascanbeshownbyasimpleintegrationofEq.(7).4.DiscussionTheresultsindicatethatforfinancialdatatherearetwoscaleregimes.Inthesmall-scaleregimetheshapeofthepdfschangesveryfastandameasureliketheKullback–Leiblerentropyincreaseslinearly.Attimescalesofafewsecondsnotallavailableinformationmaybeincludedinthepriceandprocessesnecessaryforpriceformationtakeplace.Neverthelessthisregimeseemstoexhibitawell-definedstructure,expressedbytheverysimplefunctionalformoftheKullback–Leiblerentropywithrespecttothetimescaleτ.Theupperboundaryintimescaleforthisregimeseemstobeverysimilarfordifferentstocks.Basedonastochasticanalysiswehaveshownthatasecondtimerange,themediumscalerangeexists,wheremulti-scalejointprobabilitydensitiescanbeexpressedbyastochasticcascadeprocess.Here,theinformationonthecomprehensivemulti-scalestatisticscanbeexpressedbysimpleconditionedprobabilitydensities.Thissimplificationmaybeseeninanalogytothethermodynamicaldescriptionofagasbymeansofstatisticalmechanics.Thecomprehensivestatisticalquantityforthegasisthejointn-particleprobabilitydensity,whichdescribesthelocationandthemomentumofalltheindividualparticles.Oneessentialsimplificationforthekineticgastheoryisthesingleparticleapproximation.TheBoltzmannequationisanequationforthetimeevolutionoftheprobabilitydensityp(x;p;t)inone-particlephasespace,wherexandparepositionandmomentum,respectively.InanalogytothiswehaveobtainedforthefinancialdataaFokker–Planckequationforthescaletevolutionofconditionalprobabilities,p(yi,τi│yi+1,τi+1).Inourcascadepicturetheconditionalprobabilitiescannotbereducedfurthertosingleprobabilitydensities,p(yi,τi),withoutlossofinformation,asitisdoneforthekineticgastheory.Asalastpoint,wewouldliketodrawattentiontothefactthatbasedontheinformationobtainedbytheFokker–Planckequationitispossibletogenerateartificialdatasets.Theknowledgeofconditionalprobabilitiescanbeusedtogeneratetimeseries.Oneimportantpointisthatincrementsy(τ)withcommonrightendpointsshouldbeused.Bytheknowledgeofthen-scaleconditionalprobabilitydensityofally(τi)thestochasticallycorrectnextpointcanbeselected.Wecouldshowthattimeseriesforturbulentdatageneratedbythisprocedurereproducetheconditionalprobabilitydensities,asthecentralquantityforacomprehensivemulti-scalecharacterisation.AndreasP-Nawroth,JoachimPeinke.Carl-von-Ossietzky奧爾登堡大學(xué),D-26111奧爾登伯格，德國[J].2008年3月30日.中小規(guī)模的金融數(shù)據(jù)分析摘要財(cái)務(wù)數(shù)據(jù)隨機(jī)分析已經(jīng)被提出，特別是我們探討如何統(tǒng)計(jì)在不同時(shí)間里記錄返回的變化。財(cái)務(wù)數(shù)據(jù)的時(shí)間規(guī)模依賴行為可分為兩個(gè)區(qū)域：第一個(gè)時(shí)間范圍是被描述為普遍特征的小時(shí)就區(qū)域（范圍秒）。第二個(gè)時(shí)間范圍是增加了幾分鐘的可以被描述為隨機(jī)的級(jí)聯(lián)過程的中期時(shí)間范圍。相應(yīng)的Fokker-Planck方程可以從特定的數(shù)據(jù)提取，并提供了一個(gè)非平衡熱力學(xué)描述的復(fù)雜的財(cái)務(wù)數(shù)據(jù)。關(guān)鍵詞：經(jīng)濟(jì)物理學(xué)；金融市場(chǎng)；隨機(jī)過程；Fokker-Planck方程前言復(fù)雜的金融市場(chǎng)的其中一個(gè)突出特點(diǎn)是資金數(shù)量顯示非高斯統(tǒng)計(jì)往往被命名為重尾或間歇統(tǒng)計(jì)。描述金融時(shí)間序列x(t)的波動(dòng)，最常見的就是log函數(shù)或價(jià)格增量的使用。在這里我們認(rèn)為，log函數(shù)y(τ)超過一定時(shí)間t的統(tǒng)計(jì)，被定義為：y(r)=logx(t+r)-logx(t)（1）其中x(t)是指在時(shí)間t時(shí)資產(chǎn)的價(jià)格。在財(cái)務(wù)分析數(shù)據(jù)中一個(gè)常見的問題是討論隨機(jī)數(shù)量的平穩(wěn)性，尤其是我們發(fā)現(xiàn)在我們的分析中采用什么樣的方法似乎是強(qiáng)大的非平穩(wěn)性的影響，這可能是由于數(shù)據(jù)的選擇。請(qǐng)注意，有條件的應(yīng)用τ相當(dāng)于一個(gè)特定的數(shù)據(jù)過濾。盡管如此，特殊的結(jié)果略微改變了不同的數(shù)據(jù)窗口，顯示出非平穩(wěn)性影響的可能性。在本文中，對(duì)于一個(gè)特定的數(shù)據(jù)窗口（時(shí)間段）我們側(cè)重于分析和重建進(jìn)程。目前已有的分析主要是基于1993至2003年的拜耳數(shù)據(jù)，財(cái)務(wù)數(shù)據(jù)集是由KapitabmarktDatenbank（KKMDB）提供。第二章小規(guī)模分析財(cái)務(wù)數(shù)據(jù)的一個(gè)突出特點(diǎn)是事實(shí)上概率密度函數(shù)（pdfs）不是Gaussian，而是展覽重尾形狀。另一個(gè)顯著的特點(diǎn)是形狀伴隨著可變規(guī)模τ的大小而變化。分析pdfs伴隨著規(guī)模τ的變化的統(tǒng)計(jì)，非參數(shù)方法是一種選擇。Pdfp(y(τ))的時(shí)間T和PT（y(T)）的參考時(shí)間T之間的差距是可以計(jì)算的。作為一個(gè)參考的時(shí)間，在我們的數(shù)據(jù)集上接近最小的可用時(shí)間但仍然有足夠的活動(dòng)，T=1s是選擇。為了能夠比較pdfs，并排除由于不同的均值和方差的影響，所有的pdfsp(y(τ))正?；癁榱闫骄瑯?biāo)準(zhǔn)偏差為1。作為衡量量化兩個(gè)分布p(y(τ))和PT(y(T))之間的距離，需使用Kullback–Leibler：dK(τ)=（2）dK隨著t的增加而變化，量化的改變pdfs的形狀。對(duì)于不同的股票，目前我們發(fā)現(xiàn)時(shí)間小于1分鐘的線性增長(zhǎng)的距離測(cè)度似乎是普遍的。如果正?；腉aussian分布是作為參考分布的，在小型時(shí)間表制度中快速偏離Gaussian變得很明顯。對(duì)于較大的時(shí)間規(guī)模dK仍然接近常數(shù)，這表明pdfs的形狀改變的非常緩慢。第三章中等規(guī)模的分析接下來，對(duì)于較大的時(shí)間尺度（τ﹥1分鐘）進(jìn)行討論。我們從級(jí)聯(lián)觀點(diǎn)著手，有可能通過級(jí)聯(lián)運(yùn)行過程中的變量τ掌握復(fù)雜的財(cái)務(wù)數(shù)據(jù)，尤其是它已被證明，有可能從給出的隨機(jī)級(jí)聯(lián)過程Fokker-Planck方程的形式中直接估計(jì)數(shù)據(jù)。這一做法的基本意圖是為了獲取所有的財(cái)務(wù)數(shù)據(jù)的一般性聯(lián)合正規(guī)模概率密度p(y1,τ1;y2,τ2;…;yN,τN)的訂單統(tǒng)計(jì)。在這里，我們使用速記符號(hào)y1=y(τ1),采取完整的概括性的τi＜τi+1，包含在較大的y(τi+1)中的較小的y(τi)都取決于t。復(fù)合的pdfs可由多個(gè)條件概率密度p(yi,τi│yi+1,τi+1;...;yN,τN)來表達(dá)，包含眾多點(diǎn)n的數(shù)據(jù)集n大概的數(shù)值范圍，基本上可以簡(jiǎn)化為馬爾可夫過程中τ的一個(gè)隨機(jī)變化過程。這種情況下，如果條件密度符合下列關(guān)系：p(y1,τ1│y2,τ2;y3,τ3;...;yN,τtN)＝p(y1,τ1│y2)（3）因此，p(y1,τ1;…;yN,τN)=p(y1,τ1│y2)……p(yN-1,τN-1│yN,τN)·p(yN,τ