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1、CHAPTER80MonteCarlosimulationInthisChapter...•therelationshipbetweenoptionvaluesandexpectationsforequities,currencies,commoditiesandindices•therelationshipbetweenderivativeproductsandexpectationswheninterestratesarestochastic•howtodoMonteCarlosimulationstocalculatederivativeprices
2、andtoseetheresultsofspeculatingwithderivatives•howtodosimulationsinmanydimensionsusingCholeskyfactorization80.1INTRODUCTIONThefoundationofthetheoryofderivativepricingistherandomwalkofassetprices,interestratesetc.WehaveseenthisfoundationinChapter3forequities,currenciesandcommoditie
3、s,andtheresultingoptionpricingtheoryfromChapter5onwards.ThisistheBlack–ScholestheoryleadingtotheBlack–Scholesparabolicpartialdifferentialequation.WehavealsoseeninChapter10howthestochasticdifferentialequationmodelforarandomvariableleadstoasimilarequationfortheprobabilitydensityfunc
4、tionfortherandomvariable.InthatchapterIshowedtherelationshipbetweenoptionpricesandthetransitionprobabilitydensity.Inthischapterweexploitthisrelationship,andseehowderivativepricescanbefoundfromspecialsimulationsoftheassetprice,orinterestrate,randomwalks.Briefly,thevalueofanoptionist
5、heexpectedpresentvalueofthepayoff.Thecatchinthisistheprecisedefinitionof‘expected.’Iamgoingtodistinguishbetweenthevaluationofoptionshavingequities,indices,currencies,orcommoditiesastheirunderlyingwithinterestratesassumedtobedeterministicandthoseproductsforwhichitisassumedthatintere
6、stratesarestochastic.First,Ishowtherelationshipbetweenderivativevaluesandexpectationswithdeterministicinterestrates.80.2RELATIONSHIPBETWEENDERIVATIVEVALUESANDSIMULATIONS:EQUITIES,INDICES,CURRENCIES,COMMODITIESRecallfromChapter10thatthefairvalueofanoptionintheBlack–Scholesworldisth
7、epresentvalueoftheexpectedpayoffatexpiryunderarisk-neutralrandomwalkfortheunderlying.1264PartSixnumericalmethodsandprogramsTherisk-neutralrandomwalkforSisdS=rSdt+σSdX.Wecanthereforewriteoptionvalue=e−r(T−t)Epayoff(S)providedthattheexpectationiswithrespecttotherisk-neutralrandomw
8、alk,nottherealone.Thisresultleads
