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1、Chapter1IntroductionFairedesmathematiques,c'estdonnerlem^emenomadeschosesdi erentes.
2、HenriPoincareWiththenumerousbooksonmathematical nancepublishedeachyear,theusefulnessofanewonemaybequestioned.Thisisthe rstbooksettingouttheapplicationsofadvancedanalytical
3、andgeometricalmethodsusedinrecentphysicsandmathematicstothe nancial eld.Thismeansthatnewresultsareobtainedwhenonlyapproximateandpartialsolutionswerepreviouslyavailable.Wepresentpowerfultoolsandmethods(suchasdi erentialgeometry,spectraldecomposition,supersymmet
4、ry)thatcanbeappliedtopracticalproblemsinmathematical nance.Althoughencounteredacrossdi erentdomainsintheoreticalphysicsandmathematics(forexampledi erentialgeometryingeneralrelativity,spectraldecompositioninquantummechanics),theyremainquiteunheardofwhenappliedt
5、o nanceandallowtoobtainnewresultsreadily.Weintroducethesemethodsthroughtheproblemofoptionpricing.Anoptionisa nancialcontractthatgivestheholdertherightbutnottheobligationtoenterintoacontractata xedpriceinthefuture.ThesimplestexampleisaEuropeancalloptionthatgive
6、stherightbutnottheobligationtobuyanassetata xedprice,calledstrike,ata xedfuturedate,calledmaturitydate.SincetheworkbyBlack,Scholes[65]andMerton[32]in1973,ageneralprobabilisticframeworkhasbeenestablishedtopricetheseoptions.Inthisframework,the nancialvariablesin
7、volvedinthede nitionofanop-tionarerandomvariablesandtheirdynamicsfollowstochasticdi erentialequations(SDEs).Forexample,intheoriginalBlack-Scholes-Mertonmodel,thetradedassetsareassumedtofollowlog-normaldi usionprocesseswithconstantvolatilities.Thevolatilityisth
8、estandarddeviationofaprobabil-itydensityinmathematical nance.Theoptionpricesatis esa(parabolic)partialdi erentialequation(PDE),calledtheKolmogorov-Black-Scholespric-ingequation,dependingonthestochasticdi erentialequationsintroducedtomodelthemarket.Themarketmod
9、eldependsonunobservableorobservableparameterssuchasthevolatilityofeachasset.Theyarechosen,wesaycalibrated,inordertoreproducethepriceofliquidoptionsquotedonthemarketsuchasEuro-1©200
