金融工程数值方法分析.ppt

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1、CHAPTER15thebinomialmodelInthisChapterasimplemodelforanassetpricerandomwalkdeltahedgingnoarbitragethebasicsofthebinomialmethodforvaluingoptionsriskneutralityThebinomialmodelisasimpleandfast,accuratenumericalmethodforvaluingoptions.Itrequiresonlybasicarithmeticandnocomplicated

2、stochasticcalculus.Wearegoingtoexamineaverysimplemodelforthebehaviorofastock,andbasedonthismodelseehowtovalueoptions.Wewillhaveastock,andacalloptiononthatstockexpiringtomorrow.Thestockcaneitherriseorfallbyaknownamountbetweentodayandtomorrow.Interestratesarezero.THEOPTIONVALUE

3、Thecorrectansweris...0.5Why?Threequestionsfollowfromtheabovesimpleargument:Whyshouldthis‘theoreticalprice’bethe‘marketprice’?HowdidIknowtosell½ofthestockforhedging?Howdoesthischangeifinterestratesarenon-zero?WHYSHOULDTHIS‘THEORETICALPRICE’BETHE‘MARKETPRICE’?Thisoneissimple,be

4、causeifit’snot,thenthereisrisk-freemoneytobemade.Iftheoptioncostslessthan0.5simplybuyitandhedgetomakeaprofit.Ifitisworthmorethan0.5inthemarketthensellitandhedge,andmakeaguaranteedprofit.HOWDIDIKNOWTOSELL½OFTHESTOCKFORHEDGING?Weintroduceasymbol:TheGeneralFormulaforWecanthinkof

5、asthesensitivityoftheoptiontochangesinthestock.HOWDOESTHISCHANGEIFINTERESTRATESARENON-ZERO?-------multiplyingbyadiscountfactorCOMPLETEMARKETSThepracticalimplicationofcompletemarketsisthatoptionsarehedgeableandthereforecanbepricedwithoutanyneedtoknowprobabilities.Wecanhedgeano

6、ptionwithstocktoreplicate’arisk-freeinvestmentandwecanreplicateanoptionusingstockandarisk-freeinvestment.Wecouldalso,ofcourse,replicatestockwithanoptionandrisk-free.THEREALANDRISK-NEUTRALWORLDSRisk-neutralpricingisaverypowerfultechnique.ANDNOWUSINGSYMBOLSuSV+dSV-SV例1:现有A股票的欧式

7、看涨期权，执行价格为9元，还有一年到期，A股票现在的价格为10元，明年可能涨至12元，也可能跌8元，无风险利率为5%(连续复利),请用二叉树模型计算该期权现在的价值。例2:现有A股票的欧式看跌期权，执行价格为10元，还有0.5年到期，A股票现在的价格为10元，明年可能涨至12元，也可能跌8元，无风险利率为6%(连续复利),请用二叉树模型计算该期权现在的价值。