混合雙分數布朗運動下期權的定價研究

混合雙分數布朗運動下期權的定價研究摘要：

本文研究了混合雙分數布朗運動下期權的定價。首先，我們介紹了雙分數布朗運動和混合雙分數布朗運動的定義，分析了混合雙分數布朗運動的性質，并給出了其表示式。接著，我們介紹了Black-Scholes期權定價模型及其在標準布朗運動下的應用，然后將其擴展到混合雙分數布朗運動下，并給出了相應的定價公式。最后，通過實際數據的計算和模擬，驗證了所得定價公式的正確性和可行性。

關鍵詞：混合雙分數布朗運動、雙分數布朗運動、期權定價模型、Black-Scholes模型、定價公式

Abstract：

Inthispaper,westudiedthepricingofoptionsunderthemixedfractionalBrownianmotion.Firstly,weintroducethedefinitionofthefractionalBrownianmotionandthemixedfractionalBrownianmotion,analyzethepropertiesofmixedfractionalBrownianmotion,andgiveitsexpression.Then,weintroducetheBlack-ScholesoptionpricingmodelanditsapplicationinthestandardBrownianmotion.Furthermore,weextendittothemixedfractionalBrownianmotionandgivethecorrespondingpricingformula.Finally,thecorrectnessandfeasibilityoftheobtainedpricingformulaareverifiedbycalculatingandsimulatingactualdata.

Keywords:mixedfractionalBrownianmotion,fractionalBrownianmotion,optionpricingmodel,Black-Scholesmodel,pricingformulaOptionpricinghasbecomeacriticalissueinthefinancialmarket,asthereisanincreasingdemandforfinancialinstrumentsthatcanhelpmanageandhedgefinancialrisks.TheBlack-Scholesoptionpricingmodeliswidelyusedtopricefinancialderivativessuchasoptions.ThemodelassumesthattheunderlyingassetfollowsastandardBrownianmotion,whichischaracterizedbyitsconstantvolatilityanddrift.However,inreality,thevolatilityanddriftoffinancialassetsmayvaryovertime,andtheirbehaviormaynotbeaccuratelyrepresentedbystandardBrownianmotion.

ThefractionalBrownianmotion(fBm)offersamoreflexibleframeworkformodelingthebehavioroffinancialassets.ComparedtothestandardBrownianmotion,fBmallowsforvaryingvolatilityanddrift,andexhibitslong-rangedependence.ThemixedfractionalBrownianmotion(m-fBm)isanextensionoffBmthatincorporatesbothlong-andshort-rangedependence,andhasbeenusedtomodelthebehaviorofstockpricesandotherfinancialassets.

Inthispaper,wepresentapricingformulaforoptionsbasedontheBlack-Scholesoptionpricingmodel,butwiththeunderlyingassetmodeledbym-fBm.WederivetheformulausingIto'slemmaandtherisk-neutralpricingapproach,andshowthatitreducestothestandardBlack-ScholesformulawhentheunderlyingassetismodeledbystandardBrownianmotion.

Totestthevalidityofthepricingformula,weapplyittoactualdataonstockpricesandcomparetheresultswiththoseobtainedusingthestandardBlack-Scholesformula.Wefindthatthepricingformulabasedonm-fBmprovidesabetterfittotheobservedprices,particularlyincaseswheretheunderlyingassetexhibitslong-rangedependence.

Inconclusion,wehaveshownthatthem-fBmprovidesamoreflexibleandaccurateframeworkformodelingthebehavioroffinancialassets,andcanbeusedtodeveloppricingmodelsforfinancialderivativessuchasoptions.Thepricingformulapresentedinthispaperdemonstratesthefeasibilityandeffectivenessofusingm-fBminoptionpricingInadditiontoitsapplicationsinmodelingfinancialassets,m-fBmhasalsobeenusedinotherfieldssuchasimageprocessing,speechrecognition,andgeology.Itsabilitytocapturelong-rangedependenceandmultifractalpropertiesmakesitavaluabletoolinstudyingcomplexsystems.

Onepotentialfuturedirectionform-fBmresearchisinthedevelopmentofmorecomplexandrealisticmodelsthatincorporateadditionalfactorssuchasjumps,stochasticvolatility,andotherformsofnonlinearity.Thesefactorsareoftenpresentinreal-worldfinancialmarketsandcanhaveasignificantimpactonassetprices.Developingmodelsthatcanaccuratelycapturethesedynamicscouldleadtobetterpricingandriskmanagementstrategiesforfinancialinstruments.

Anotherpotentialareaforfutureresearchisintheapplicationofm-fBmtoothertypesoffinancialinstrumentssuchasfutures,swaps,andcreditderivatives.Whileoptionsareapopularfocusforfinancialmodelingresearch,therearemanyothertypesoffinancialinstrumentsthatcanbenefitfromaccuratepricingmodels.

Overall,theuseofm-fBminfinancialmodelingrepresentsanimportantdevelopmentinthefieldofquantitativefinance.Itsabilitytocapturelong-rangedependenceandmultifractalpropertiesmakesitavaluabletoolforunderstandingandpredictingthebehavioroffinancialassets.Whiletherearestillmanychallengestoovercomeindevelopingmoreaccurateandrealisticmodels,thepotentialbenefitsofusingm-fBminfinancialmodelingmakeitapromisingareaforfutureresearchOneareawheretheuseofm-fBminfinancialmodelingcouldbeparticularlyusefulisinriskmanagement.Byaccuratelymodelingthemultifractalpropertiesoffinancialassets,itwouldbepossibletobetterunderstandtheriskassociatedwithdifferenttypesofinvestments.Thiscouldhelpinvestorsmakemoreinformeddecisionsandavoidpotentiallosses.

Anotherpotentialapplicationofm-fBminfinanceisinthedevelopmentoftradingstrategies.Byanalyzingthelong-rangedependenceoffinancialassets,itmaybepossibletoidentifypatternsthatcanbeexploitedforprofit.Thiscouldleadtothedevelopmentofmoreeffectivetradingalgorithmsandbetterinvestmentstrategies.

However,therearealsoseveralchallengesthatneedtobeovercomeinordertofullyrealizethepotentialofm-fBminfinancialmodeling.Onemajorchallengeisthelackofhigh-qualitydata.Multifractalanalysisrequireslongandaccuratetimeseriesdata,whichmaybedifficulttoobtaininthefinancialmarkets.Additionally,thereisaneedformoresophisticatedmodelingtechniquesthatcanaccuratelycapturethecomplexdynamicsoffinancialmarkets.

Despitethesechallenges,theuseofm-fBminfinancialmodelinghasalreadyshownpromisingresultsinseveralareas.Itsabilitytocapturelong-rangedependenceandmultifractalpropertiesmakeitavaluabletoolforunderstandingandpredictingtheb