options an example of arbitrage pricing

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1、Chapter5Options:AnExampleofArbitragePricing5.1OptionsLetXbethepayoffofanassetatt=1andSitspriceatt=0.Inwhatfollows,weoftencalltheasset“stock”.定义5.1AnEuropeancall(put)optiononthestockwithstrikepriceKandmaturitydatetisasecuritythatgivesthebuyertherighttobuy(sell)thest

2、ockatmaturityatpriceK.LetcbethepayoffofaEuropeancalloptionatt=1,pbethepayoffofaEuropeanputoptionatt=1,thencXK[]max[XK，0],pKX[]=max[KX,0]（5.1）5-1定义5.2AnAmericancall(put)optiononthestockwithstrikepriceKandmaturitydatetisasecuritythatgivesthebuyert

3、herighttobuy(sell)thestockatanytimebeforeandincludingthematuritydateatpriceK.5-2称X-K为看涨期权的intrinsicvalue，如果一份期权的内在价值大于、等于或小于0，相应地称之为inthemoney、atthemoney和outofthemoney。LetcSK(,)bethepriceofaEuropeancalloptionatt=0andp(,)SKthepriceofaEuropeanputoption;letCSK(,)bethep

4、riceofanAmericancalloptionatt=0andPSK(,)thepriceofanAmericanputoption.5-35.2SomeArbitragePricingRelationsOnOptionsBynoarbitrage,iffortwoassetsAandB,X()Xi(),(..,eXX),thenABABSS.Usingthearbitrageargument,wehavethefollowingpricingrelationsforoptions.AB定理5.1cSK

5、(,)andp(,)SKarenon-negative.cXK[]0,[]pKX0定理5.2cSK(,)isnon-increasinginK,andp(,)SKisnon-decreasinginK.Proof:KK,max[0,XK]max[0,XK].Thus,cSK(,)(,)cSK.1212125-4定理5.3cSK(,)andp(,)SKareconvexinK.Proof:KKand(0,1),12max[0,XK((1)K)]max[0,(X

6、K)(1)(XK)]1212max[0,XKX](1)max[0,K]12Inotherwords,max[0,XK]isaconvexfunctionofK.Soismax[0,KX].FromcSK(,)SXK([])andS()islinear,weknowthatcSK(,)isconvexinK.定理5.4Let0beaportfolioofNassets.Then,TTT

7、TcSK(,)iiiicSKpSK(,)i,(,)iipSK(,)iThus,anoptiononaportfolioisworthlessthanaportfolioofoptionsontheassetsinthatportfolio.5-5Proof:股票i的价格为S，iN1,,，以其为基础资产的期权的执行价为K。股票iiiT及其相对应的期权的头寸为。0时股票组合的价值为S，1时的支付为X。iThepayoffofanoptionontheportfolioismax[0,(XK

8、X)T]max[0,(K)]iiiiThepayoffofaportfolioofoptionsisTmax[0,XKX]max[0,K]iiiiFromtheconvexityofthepayofffunction,wehavemax[0,