资源描述：
《Monte_Carlo_Methods.handout》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
1、NotesOptionPricingwithMonteCarloMethodsMatthiasBuehlmaier,PhDAssistantProfessorofFinanceTheUniversityofHongKong1NotesPartITheMainIdeaofMonteCarloSimulationMFIN7009,Nov.&Dec.20112ConsideringtheGBMasaSimpleExampleNotes•InthispartofthelecturenotesweconsidertheGeometricBro
2、wnianMotion(GBM)becauseitisrelativelyeasytosimulate•StockpricedynamicsaregivenbythefollowingSDE:dStƹStdtžStdWt•SincethestockpriceisaGBM,wecansolvetheSDEanalytically(¹¡1¾2)tžW•TheanalyticalsolutionisStÆS0e2tforallt2[0,T],whereweassumethatS0Æs0isagivenrealnumber3Simul
3、atingaSingleStockPriceRealizationNotes•ThestockpricereturnisdefinedbypRt:Æln(St/S0),whichimpliesthatRÆ(¹¡1¾2)tžt¢Z,whereZ»N(0,1)t2p•HerewehavereplacedWtwitht¢Z,whichhasthesameprobabilitydistribution•SimulatingStthusrequiresustomakearandomdrawzfromthestandardnormaldistr
4、ibution,thencalculatetherealizationofthepstockreturnrÆ(¹¡1¾2)tžt¢z,andthentocalculatesÆsertt2t0•stisasinglerealizationofthestockpriceSt•ToperformMonteCarlosimulation,weneedmanyrealizationsofthestockprice!MFIN7009,Nov.&Dec.20114TheBasicIdeaofMonteCarloSimulationNotes•W
5、erepeatthiswholeproceduredtimes•Thisrepetitionresultsindrealizationsofthesimulatedstock(1)(2)(d)priceSt,thatis,weobtainst,st,...,st•ThegoalofMonteCarlosimulationistoobtainanapproximationforE[f(St)]•InpracticeweoftenhavetÆTandf(ST)Æ©(ST)•Weobtainthisapproximationbyrepla
6、cingtheexpectationwiththesampleaverageasfollows:1Xd(i)E[f(St)]¼f(st)diÆ1•Wecalldthenumberoftrials5NotesPartIIMonteCarloSimulationbyExample:PricingBinaryOptionsMFIN7009,Nov.&Dec.20116PricingaBinaryOptionwithMonteCarloSimulationNotes•Forexample,considerpricingacash-or-no
7、thingcallonGoogle•ThisisasimpleclaimXÆ©(ST)Æ1STÈK•ThepriceoftheoptionattÆ0isF(0,s¡r(T¡0)Q0)ÆeE0,s0[1STÈK],wherethestockdynamicsunderthemeasureQaredStÆrStdtžStdWt•WecancalculatethepriceofthisclaimusingMonteCarlosimulation:1Xd¡rTQ¡rTeE0,s0[1STÈK]¼e1s(i)ÈKdiÆ1T•Remembert
8、ousetherisk-freeraterinsteadof¹aspartofthedrifttermwhenperformingthesimulation7EstimatingDiffusionParametersNotes•What
