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MathematicsofFinance
Chapter1
FinancialMarketsand
Derivatives
1.1FinancialMarkets
A¯nancialmarketconsistsoftradablesecuritiessuchasstocks,bonds,cur-rencies,commodities,orevenindexes(seetheexcerptfromtheWallStreetJournal).Onereasonfortheexistenceof¯nancialmarketsisthattheyfacili-
tatethe°owofcapital.Forexample,ifacompanywantsto¯nancethebuildingofanewproductionfacility,itmightsellsharesofstocktoinvestorswhobuythesharesbasedontheanticipationoffuturerewards,suchasdividendsoiseinthestockprice.
Avarietyofstochasticmodelsisusedinmodelingthepricesofsecurities.Allsuodelsfacetheusualtradeo®,namely,morecomplexmodelstypicallyprovideabetter¯ttodata(althoughthereisthedangerofover¯tting),whereassimplermodelsaregenerallymoretractableanddespitetheirsimplicitycansometimesprovideusefulqualitativeinsights.Findingagoodbalancebetween
arealisticandatractablemodelispartoftheartofstochasticmodeling.
Bothdiscreteandcontinuoustimemodelswillbeconsideredhere.Thetreat-mentofdiscretetimemodelsisnotintendedtobeexhaustive,butisintended
asameansofintroducingnotionssuchashedgingandpricingbyarbitrageinasimplersettingtoprovideabridgetothedevelopmentincontinuoustime.
Binomialtreemodelswillmostlybeusedinthediscretetimesetting.
Anarbitrageopportunityisanopportunityfoiskfreepro¯t.A¯nancialmarketissaidtobeviableiftherearenoarbitrageopportunities.Typically,liquid¯nancialmarketsmoverapidlytoeliminatearbitrageopportunities.Ac-cordingly,inthiscourse,attentionwillfocuson¯nancialmarketmodelsthat
2CHAPTER
1.FINANCIALMARKETSANDDERIVATIVES
areviable.
1.2Derivatives
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