具有时变参数的分数布朗运动下欧式双向期权的定价中期报告

具有时变参数的分数布朗运动下欧式双向期权的定价中期报告Abstract摘要Inthisreport,wepresentamethodforpricingEuropeandouble-barrieroptionsunderatime-varyingparameterfractionalBrownianmotion(fBm).Thetime-varyingparameterfBmisausefulmodelinfinanceasitcancapturenumerousstylizedfactssuchaslong-rangedependence,volatilityclustering,andscalingbehavior.Theoptionweconsiderhasapayoffthatdependsonwhethertheunderlyingassetcrosseseitheroftwobarriersbeforeexpiration.WeproposeaFouriertransform-basedmethodforpricingtheoption,wherethecharacteristicfunctionoftheunderlyingassetisderived.Wealsoprovidenumericalexamplestodemonstratetheeffectivenessofourproposedmethod.在本报告中，我们提出了一种针对具有时变参数分数布朗运动（fBm）模型的欧式双向障碍期权定价方法。时变参数fBm模型在金融建模中非常有用，因为它可以捕捉到许多特定的统计特征，例如长程依赖、波动率聚集和比例行为等。所考虑的期权具有依赖于标的资产在到期前是否可以突破两个障碍的收益。我们提出了一种基于Fourier变换的期权定价方法，并推导出标的资产的特征函数。我们还提供了数字实例，以展示我们提出的方法的有效性。Introduction引言ThefractionalBrownianmotion(fBm)isapopularstochasticprocessusedinfinancetomodelassetprices.UnliketheBrownianmotion(BM),whichisstationaryandhasindependentincrements,fBmisnonstationaryandhasdependentincrements.ThismakesthefBmausefultoolformodellingfinancialphenomena,suchasvolatilityclusteringandlong-rangedependence.However,financialdataoftenexhibittime-varyingvolatility,whichisnotaccountedforbystandardfBmmodels.Toaddressthisissue,weintroduceatime-varyingparameterfBmmodel.Thismodelallowsthevolatilityparametertovaryovertime,bettercapturingthechangingnatureoffinancialdata.Inthisreport,weconsiderthepricingofEuropeandouble-barrieroptionsunderthetime-varyingparameterfBmmodel.Specifically,wederiveaFouriertransform-basedmethodforpricingtheoption,wherethecharacteristicfunctionoftheunderlyingassetisderived.LiteratureReview文献综述Thepricingoffinancialderivativesisafundamentalprobleminmathematicalfinance.Variousmethodshavebeenproposedforpricingoptionsunderdifferentstochasticprocesses.Forexample,theBlack-ScholesmodeliswidelyusedforpricingoptionsundertheassumptionofgeometricBrownianmotion.However,thisassumptiondoesnotalwaysholdinpractice,leadingtoinaccuratepricing.ToaddressthelimitationsoftheBlack-Scholesmodel,severalalternativemodelshavebeendeveloped.Forexample,theHestonmodelandthestochasticvolatilitymodelarepopularmodelsthatallowfortime-varyingvolatility.Thesemodelshavebeenextendedtoincludefeaturessuchasjumpsinpricesandstochasticinterestrates.ThefBmmodelisanotherpopularmodelusedinmathematicalfinance.Ithasbeenshowntocapturevariousstylizedfactsoffinancialtimeseries,suchaslong-rangedependenceandvolatilityclustering.ThepricingofoptionsunderthefBmmodelhasbeenstudiedextensivelyintheliterature.However,thestandardfBmmodelassumesaconstantvolatility.Toaccountfortime-varyingvolatility,severalmodelshavebeenproposed,suchasthestochasticvolatilityfBmandthemultifractionalBrownianmotion.Thesemodelshavebeenappliedtovariousfinancialapplications,suchascreditriskmodelsandoptionpricingmodels.Methodology方法WeconsiderthepricingofaEuropeandouble-barrieroptionunderthetime-varyingparameterfBmmodel.Theoptionhasapayoffthatdependsonwhethertheunderlyingassetcrosseseitheroftwobarriersbeforeexpiration.Theunderlyingassetfollowsatime-varyingparameterfBm.WeproposeaFouriertransform-basedmethodforpricingtheoption.Specifically,wederivethecharacteristicfunctionoftheunderlyingasset.ThecharacteristicfunctionisthenusedtocalculatetheoptionpriceviatheinverseFouriertransform.Results结果Weprovidenumericalexamplestodemonstratetheeffectivenessofourproposedmethod.Thenumericalexamplesarebasedonasimulatedtime-varyingparameterfBmprocess.WecomparetheresultsofourproposedmethodtothoseofthestandardBlack-ScholesmodelandtheHestonmodel.TheresultsshowthatourproposedmethodismoreaccuratethantheBlack-ScholesmodelandcomparabletotheHestonmodel.Furthermore,ourproposedmethodiscomputationallyefficientandcanbeusedforreal-timeoptionpricing.Conclusion结论WeproposeaFouriertransform-basedmethodforpricingEuropeandouble-barrieroptionsunderatime-varyingparameterfBmmodel.OurproposedmethodismoreaccuratethanthestandardBlack-Scholesmodelandiscomputationallyef