Basics of Financial Mathematics

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1、TheBasicsofFinancialMathematicsSpring2003RichardF.BassDepartmentofMathematicsUniversityofConnecticutThesenotesarec2003byRichardBass.Theymaybeusedforpersonaluseorclassuse,butnotforcommercialpurposes.Ifyoufindanyerrors,Iwouldappreciatehearingfromyou:bass@math.uconn.edu11.Introduction.Inthiscou

2、rsewewillstudymathematicalfinance.Mathematicalfinanceisnotaboutpredictingthepriceofastock.Whatitisaboutisfiguringoutthepriceofoptionsandderivatives.Themostfamiliartypeofoptionistheoptiontobuyastockatagivenpriceatagiventime.Forexample,supposeMicrosoftiscurrentlysellingtodayat$40pershare.AEurope

3、ancalloptionissomethingIcanbuythatgivesmetherighttobuyashareofMicrosoftatsomefuturedate.Tomakeupanexample,supposeIhaveanoptionthatallowsmetobuyashareofMicrosoftfor$50inthreemonthstime,butdoesnotcompelmetodoso.IfMicrosofthappenstobesellingat$45inthreemonthstime,theoptionisworthless.Iwouldbes

4、illytobuyasharefor$50whenIcouldcallmybrokerandbuyitfor$45.SoIwouldchoosenottoexercisetheoption.Ontheotherhand,ifMicrosoftissellingfor$60threemonthsfromnow,theoptionwouldbequitevaluable.Icouldexercisetheoptionandbuyasharefor$50.Icouldthenturnaroundandselltheshareontheopenmarketfor$60andmakea

5、profitof$10pershare.ThereforethisstockoptionIpossesshassomevalue.Thereissomechanceitisworthlessandsomechancethatitwillleadmetoaprofit.Thebasicquestionis:howmuchistheoptionworthtoday?ThehugeimpetusinfinancialderivativeswastheseminalpaperofBlackandScholesin1973.Althoughmanyresearchershadstudiedt

6、hisquestion,BlackandScholesgaveadefinitiveanswer,andagreatdealofresearchhasbeendonesince.Thesearenotjustacademicquestions;todaythemarketinfinancialderivativesislargerthanthemarketinstocksecurities.Inotherwords,moremoneyisinvestedinoptionsonstocksthaninstocksthemselves.Optionshavebeenaroundfor

7、alongtime.Theearliestoneswereusedbymanu-facturersandfoodproducerstohedgetheirrisk.Afarmermightagreetosellabushelofwheatatafixedpricesixmonthsfromnowratherthantakeachanceonthevagariesofmarketprices.Similarlyasteelrefinerymightwanttolockinthepriceofironoreat