Introduction to the Mathematics of Finance

Themodernsubjectofmathematicalfinancehasundergoneconsiderabledevelopment,bothintheoryandpractice,sincetheseminalworkofBlackandScholesappearedathirdofacenturyago.Thisbookisintendedasanintroductiontosomeelementsofthetheorythatwillenablestudentsandresearcherstogoontoreadmoreadvancedtextsandresearchpapers.ThebookbeginswiththedevelopmentofthebasicideasofhedgingandpricingofEuropeanandAmericanderivativesinthediscrete(i.e.,discretetimeanddiscretestate)settingofbinomialtreemodels.Thenageneraldiscretefinitemarketmodelisintroduced,andthefundamentaltheoremsofassetpricingareprovedinthissetting.Toolsfromprobabilitysuchasconditionalexpectation,filtration,(super)martingale,equivalentmartingalemeasure,andmartingalerepresentationareallusedfirstinthissimplediscreteframework.Thisprovidesabridgetothecontinuous(timeandstate)setting,whichrequirestheadditionalconceptsofBrownianmotionandstochasticcalculus.ThesimplestmodelinthecontinuoussettingisthefamousBlack-Scholesmodel,forwhichpricingandhedgingofEuropeanandAmericanderivativesaredeveloped.ThebookconcludeswithadescriptionofthefundamentaltheoremsforacontinuousmarketmodelthatgeneralizesthesimpleBlack-Scholesmodelinseveraldirections.