دانلود مقاله ISI انگلیسی شماره 19826
ترجمه فارسی عنوان مقاله

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

عنوان انگلیسی
Continuous time Black–Scholes equation with transaction costs in subdiffusive fractional Brownian motion regime
کد مقاله سال انتشار تعداد صفحات مقاله انگلیسی
19826 2012 10 صفحه PDF
منبع

Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)

Journal : Physica A: Statistical Mechanics and its Applications, Volume 391, Issue 3, 1 February 2012, Pages 750–759

ترجمه کلمات کلیدی
- فرمول سیاه شولز - معادله سیاه شولز جزء به جزء - هزینه های معامله -
کلمات کلیدی انگلیسی
Subdiffusion, Black–Scholes formula, Fractional Black–Scholes equation, Transaction costs,
پیش نمایش مقاله

#### چکیده انگلیسی

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.

#### مقدمه انگلیسی

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.

#### نتیجه گیری انگلیسی

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