معادلات شولز ـ سیاه پیوسته با هزینه معاملات در نظام "پخش کننده فرعی" حرکت براونی جزئی

In this paper, we study the problem of continuous time option pricing with transaction costs by using the homogeneous subdiffusive fractional Brownian motion (HFBM) Z(t)=X(Sα(t))Z(t)=X(Sα(t)), 0<α<10<α<1, here dX(τ)=μX(τ)(dτ)2H+σX(τ)dBH(τ)dX(τ)=μX(τ)(dτ)2H+σX(τ)dBH(τ), as a model of asset prices, which captures the subdiffusive characteristic of financial markets. We find the corresponding subdiffusive Black–Scholes equation and the Black–Scholes formula for the fair prices of European option, the turnover and transaction costs of replicating strategies. We also give the total transaction costs.

TheclassicalandstillmostpopularmodelofthemarketistheBlack–Scholesmodelbasedonthediffusionprocesscall geometricBrownianmotion(GBM)where are constants, and is the Brownian motion. In the presence of transaction costs (TC), Leland [ ] first examinedoptionreplicationinadiscretetimesetting,andposeamodifiedreplicatingstrategy,whichdependsuponthe leveloftransactioncostsandupontherevisioninterval,aswellasupontheoptiontobereplicatedandtheenvironment. Sincethen,alotofauthorsstudythisproblem,butallinadiscretetimesetting[ TheoptionpricingtheoryasdevelopedbyBlack–Scholes[ ]restsonanarbitrageargument:bycontinuouslyadjusting aportfolioconsistingofastockandarisk-freebond,aninvestorcanexactlyreplicatethereturnstoanyoptiononthestock. Itleadsusnaturallytoposethefollowingquestion. Inthepresenceoftransactioncosts,isthereanalternativereplicatingstrategydependingupontheleveloftransaction costsandatechniqueleadingtotheBlack–Scholesequationinacontinuoustimesetting?Doestheperfectreplicationincur aninfiniteamountoftransactioncosts? TheBlack–Scholes(BS)modelisbasedonthediffusionprocesscalledgeometricBrownianmotion(GBM).However,the empiricalstudiesshowthatmanycharacteristicpropertiesofmarketscannotbecapturedbytheBSmodel,suchas:long- rangecorrelations,heavy-tailedandskewedmarginaldistributions,lackofscaleinvariance,periodsofconstantvalues,etc. Therefore,inrecentyearsoneobservesmanygeneralizationsoftheBSmodelbasedontheideasandmethodsknownfrom statisticalandquantumphysics.Inthispaper,wedealwiththeassetpriceexhibitingsubdiffusivedynamics, 1,inwhichthe priceofanasset X (τ) followsastochasticdifferentialequation where isthefractionalBrownianmotion(FBM)withHurstexponen ,and istheinverse -stable subordinatordefinedasbelo isastrictlyincreasing -stable process[ ]withLaplacetransformgivenby .When reducestothe‘‘objectivetime’’t.Here,weapplythesubdiffusivemechanismoftrappingeventsinorderto describefinancialdataexhibitingperiodsofconstantvaluesasinRef.[ 15 Fromthe Appendix ,wecanexpressthemodel intothefollowingform where isthefractionalGaussiannoise,heuristically .TheFBMhastwouniqueproperties:self- similarityandstationaryincrements[ ].TheautocorrelationfunctionoffractionalGaussiannoiseisthememoryker whichisrightlythefractionaloperatorin .ThismodelwasfirstproposedbyKoutosimulatethefluctuationofthedistanc betweenafluorescein–tyrosinepairwithinasingleproteincomplex[ ].Eq. canbeconvertedtoanequationforthe timecorrelationfunction bymultiplying andtakingexpectation,yields Thelastterm 0for isorthogonalto inthephasespace[ ].TheLaplacetransformofEq. gives where is the Mittag-Leffler function [ ]. So, by computing the covariance of the real data, then using to approximateit,wecangetthevalueof .Fromthecorrelationfunction ,wecangetthemodelislong-dependentfor 1.Notingthat,theself-similarity,stationaryincrementsandlongdependenceareallimportantpropertiesin financialmarket. Thispaperisorganizedasfollows.InSection ,byusingadeltahedgingstrategyinitiatedbyLeland[ ],wededucethe Black–Scholesequationwithtransactioncostsincontinuoustimesettingfortheassetprice where follows and isdefinedin .IntheSection ,weobtainthecorrespondingBlack–Scholesformula.In Section ,weestimateturnoverandtransactioncostsofreplicatingstrategies.InSection ,wegivethetotaltransaction costs.

UsingthestrategyofLeland[ 3 ],thispaperhasdevelopedatechniqueforreplicatingoptionreturnsinacontinuoustime settinginthepresenceoftransactioncostsandobtainthecorrespondingBlack–Scholesequations,Black–Scholesformulas andtotaltransactioncostswithtransactioncostsbothinsubdiffusiveregimeandinrealtime.