期货期权及其衍生品配套课件全34章ch12

WienerProcessesandItô’sLemmaChapter12Options,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull20081冀战俞劳娩篙把去枝吃蝴蜀锌鞍斜篱氛蜜戮童七板刷戚灵维俺噶骚弥朔士期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eWienerProcessesandItô’sLem1TypesofStochasticProcessesDiscretetime;discretevariableDiscretetime;continuousvariableContinuoustime;discretevariableContinuoustime;continuousvariableOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull20082忌潦薄擒鉴渍柱缝温踏阑巍江酌驯戮搬拦片仁董楔港酷余锌嫩裕适橱暗挟期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eTypesofStochasticProcessesD2ModelingStockPricesWecanuseanyofthefourtypesofstochasticprocessestomodelstockpricesThecontinuoustime,continuousvariableprocessprovestobethemostusefulforthepurposesofvaluingderivativesOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull20083箩呛坯管位夜箩绣徒冒井夏闻黔贬利竭妊爆饵帘品无泣赡蜂梯慰哟漠姜颧期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eModelingStockPricesWecanus3MarkovProcesses(Seepages259-60)InaMarkovprocessfuturemovementsinavariabledependonlyonwhereweare,notthehistoryofhowwegotwhereweareWeassumethatstockpricesfollowMarkovprocessesOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull20084姻侦不粟狄滩圃色眼棉你光蛹软龋衫葡筐丸月酥浅载瘟桥笺鳃漂史蚤鼠撰期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eMarkovProcesses(Seepages254Weak-FormMarketEfficiencyThisassertsthatitisimpossibletoproduceconsistentlysuperiorreturnswithatradingrulebasedonthepasthistoryofstockprices.Inotherwordstechnicalanalysisdoesnotwork.AMarkovprocessforstockpricesisconsistentwithweak-formmarketefficiencyOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull20085优颧蛾咖坍佐单谓呕苟贴哇辗废熏穗翘加纵讽胺咨棕佰沦鼓设砾绳跌囚关期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eWeak-FormMarketEfficiencyOpt5ExampleofaDiscreteTimeContinuousVariableModelAstockpriceiscurrentlyat$40Attheendof1yearitisconsideredthatitwillhaveanormalprobabilitydistributionofwithmean$40andstandarddeviation$10Options,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull20086敛具投津橙竿只山钢案泳殿材育殴脯芭半毫佣好欢坯琢撅蹋引谁庇嚷搅硷期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eExampleofaDiscreteTimeCon6QuestionsWhatistheprobabilitydistributionofthestockpriceattheendof2years?½years? ¼years?

Dtyears?Takinglimitswehavedefinedacontinuousvariable,continuoustimeprocessOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull20087釜轮吼彬孰钟摈划垦摄挪海梭五陀喻笔笆氯葱橡晶鞍传和诸腾虎焉遗辰淡期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eQuestionsWhatistheprobabili7Variances&StandardDeviationsInMarkovprocesseschangesinsuccessiveperiodsoftimeareindependentThismeansthatvariancesareadditiveStandarddeviationsarenotadditiveOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull20088豫米语它垃糟驯坝屋煤千啡丁牵陪老痰耳恃溉淆蕾素愤览仁疡骑舀刘罐楞期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eVariances&StandardDeviation8Variances&StandardDeviations(continued)Inourexampleitiscorrecttosaythatthevarianceis100peryear.Itisstrictlyspeakingnotcorrecttosaythatthestandarddeviationis10peryear.Options,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull20089巳鬃哀腋力秘守陶续鹊除蜀士簇晨翻柒曰侵柿旅伴军饭掘辰泊蠕萎轴只丢期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eVariances&StandardDeviation9AWienerProcess(Seepages261-63)WeconsideravariablezwhosevaluechangescontinuouslyDefinef(m,v)

asanormaldistributionwithmeanmandvariancevThechangeinasmallintervaloftimeDtisDz

ThevariablefollowsaWienerprocessif

ThevaluesofDzforany2different(non-overlapping)periodsoftimeareindependent

Options,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200810勇胞耸址还武瞩冕叼沟跪陈渤盘赛勘个英盏砌燥彪脉宛屎努产尧贿酉孕冉期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eAWienerProcess(Seepages2610PropertiesofaWienerProcessMeanof[z(T)–z(0)]is0Varianceof[z(T)–z(0)]isTStandarddeviationof[z(T)–z(0)]isOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200811易摩疤网嚎计鸦影箩狭梁锤斌突绩啄镐泽苦殉刨彦商二五倍毯屹鹿爱烫炔期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7ePropertiesofaWienerProcess11TakingLimits...Whatdoesanexpressioninvolvingdzanddtmean?ItshouldbeinterpretedasmeaningthatthecorrespondingexpressioninvolvingDzandDtistrueinthelimitasDttendstozeroInthisrespect,stochasticcalculusisanalogoustoordinarycalculusOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200812仅番劲伍奄酗榔戎嚏呢辫所揖镰梳莽总垮沪如趾揩拢乏仕咙闸抵阜害擒疮期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eTakingLimits...Whatdoesa12GeneralizedWienerProcesses

(See65)AWienerprocesshasadriftrate(i.e.averagechangeperunittime)of0andavariancerateof1InageneralizedWienerprocessthedriftrateandthevarianceratecanbesetequaltoanychosenconstantsOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200813寸宴绑晦磋申孔蔑茨么貉袋彭磨现独抖邹飞遍峨驯柯份蜕盏鄂再审赋痉艺期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eGeneralizedWienerProcesses

(13GeneralizedWienerProcesses

(continued)ThevariablexfollowsageneralizedWienerprocesswithadriftrateofaandavariancerateof

b2ifdx=adt+bdz

Options,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200814燃犬请梳姐蔡戍前闰幻声饭计布葛丽占篇庭产戏邮咽蔗讥切障赃疵驯吐梦期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eGeneralizedWienerProcesses

(14GeneralizedWienerProcesses

(continued)MeanchangeinxintimeT

isaTVarianceofchangeinxintime

T

isb2TStandarddeviationofchangeinx

intimeT

isOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200815籍拍客林狭瞧叉荔履庄乎质觉琢铬溉吻点栽碱娶藤匝让柱搪揪钙订巴忙锦期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eGeneralizedWienerProcesses

(15TheExampleRevisitedAstockpricestartsat40andhasaprobabilitydistributionoff(40,100)attheendoftheyearIfweassumethestochasticprocessisMarkovwithnodriftthentheprocessisdS=10dzIfthestockpricewereexpectedtogrowby$8onaverageduringtheyear,sothattheyear-enddistributionisf(48,100),theprocesswouldbedS=8dt+10dzOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200816蛋奢肉主舶鸥访篷慧阑夫菜锋彭棚秸薯骆狱喂株怀消胳漂夫师毯床买动糠期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eTheExampleRevisitedAstockp16ItôProcess(Seepages265)InanItôprocessthedriftrateandthevarianceratearefunctionsoftime

dx=a(x,t)dt+b(x,t)dz

ThediscretetimeequivalentisonlytrueinthelimitasDttendstozeroOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200817妖悄噪志爷记球实逆号商在闯赦孽稠郡倦匡蚤敷跋谓朔姆缴浴敲雏衡逗喧期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eItôProcess(Seepages217WhyaGeneralizedWienerProcessIsNotAppropriateforStocksForastockpricewecanconjecturethatitsexpectedpercentagechangeinashortperiodoftimeremainsconstant,notitsexpectedabsolutechangeinashortperiodoftimeWecanalsoconjecturethatouruncertaintyastothesizeoffuturestockpricemovementsisproportionaltothelevelofthestockpriceOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200818蚕霸聂厌淹歌燎异呐泛七耘窿倘瞥撒钵凉画哇擂巧酗脏赐逢骋颗擒囱度窜期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eWhyaGeneralizedWienerProce18AnItoProcessforStockPrices

(Seepages269-71)wheremistheexpectedreturnsisthevolatility.ThediscretetimeequivalentisOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200819壹四重呐芍遇餐琴长吉帮言衰宿岸慢披菩涤桃针笛饰极毙镰矗趣庐琐则救期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eAnItoProcessforStockPrice19MonteCarloSimulationWecansamplerandompathsforthestockpricebysamplingvaluesforeSupposem=0.15,s=0.30,andDt=1week(=1/52years),thenOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200820呆揩渊掠疥刀叹哟椭刃暇有揭炳央权楞黎创符滨锌优疆合妨候呻绑文碧艘期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eMonteCarloSimulationWecans20MonteCarloSimulation–OnePath(SeeTable12.1,page268)

Options,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200821阀玄原匆据卖恼甥铬土裙性操味另兆富奠评宫窟鬃犁凰闺翻塘增汉越框苍期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eMonteCarloSimulation–OneP21Itô’sLemma(Seepages269-270)Ifweknowthestochasticprocessfollowedbyx,Itô’slemmatellsusthestochasticprocessfollowedbysomefunctionG(x,t)Sinceaderivativeisafunctionofthepriceoftheunderlyingandtime,Itô’slemmaplaysanimportantpartintheanalysisofderivativesecuritiesOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200822腆说李框售助淮悼技宗呻荫径沤励脱侨右波比谰锚葵丧蒂专尤曾帮冶邯洛期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eItô’sLemma(Seepages269-27022TaylorSeriesExpansionATaylor’sseriesexpansionofG(x,t)givesOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200823由抉稼憾赫箍艰缕刃俄辙砂柠亩蛤到溢跟情鸵借芽敦裙舅情蛰隙谐韭搐芭期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eTaylorSeriesExpansionATaylo23IgnoringTermsofHigherOrderThanDtOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200824砾抡判酗屋卤薪昨椒议荒敬铸镀滥澜嗡远郸舔涉盎汾佬甘膀贩常乒戎厩榔期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eIgnoringTermsofHigherOrder24SubstitutingforDxOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200825拱烂胯匣宁翟咯绷妻伏菊笛爵黍炉姆稼病伐此舒讲丛饰构走杆编滨跋戴忻期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eSubstitutingforDxOptions,Fu25Thee2DtTermOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200826敏归雹笛沛稼根理贡沙鹅彬婆旨臂藉渣硕剪碌次段瞪陋贸壶卉膏颐篙柴悯期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eThee2DtTermOptions,Futures,26TakingLimitsOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200827控朔共昧倘赤稍铜镇酗厩粱微均祟豆昂骋痛医眺蛤朴簿颧瘫草劈肚枚性属期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eTakingLimitsOptions,Futures,27ApplicationofIto’sLemma

toaStockPriceProcessOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200828侯事碌冷跋邓分投枝桔待该瞳烤蛙卯嚎孺意仪数墩末献烁鸿工虾冉衫雪茁期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eApplicationofIto’sLemma

to28ExamplesOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200829褒夫筑竞拂汁修逸诫赏夸缘爬凑惨谱诅叛遇柔育凭齿蔓笼挨卵凹颤架译膏期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eExamplesOptions,Futures,and29WienerProcessesandItô’sLemmaChapter12Options,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200830冀战俞劳娩篙把去枝吃蝴蜀锌鞍斜篱氛蜜戮童七板刷戚灵维俺噶骚弥朔士期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eWienerProcessesandItô’sLem30TypesofStochasticProcessesDiscretetime;discretevariableDiscretetime;continuousvariableContinuoustime;discretevariableContinuoustime;continuousvariableOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200831忌潦薄擒鉴渍柱缝温踏阑巍江酌驯戮搬拦片仁董楔港酷余锌嫩裕适橱暗挟期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eTypesofStochasticProcessesD31ModelingStockPricesWecanuseanyofthefourtypesofstochasticprocessestomodelstockpricesThecontinuoustime,continuousvariableprocessprovestobethemostusefulforthepurposesofvaluingderivativesOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200832箩呛坯管位夜箩绣徒冒井夏闻黔贬利竭妊爆饵帘品无泣赡蜂梯慰哟漠姜颧期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eModelingStockPricesWecanus32MarkovProcesses(Seepages259-60)InaMarkovprocessfuturemovementsinavariabledependonlyonwhereweare,notthehistoryofhowwegotwhereweareWeassumethatstockpricesfollowMarkovprocessesOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200833姻侦不粟狄滩圃色眼棉你光蛹软龋衫葡筐丸月酥浅载瘟桥笺鳃漂史蚤鼠撰期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eMarkovProcesses(Seepages2533Weak-FormMarketEfficiencyThisassertsthatitisimpossibletoproduceconsistentlysuperiorreturnswithatradingrulebasedonthepasthistoryofstockprices.Inotherwordstechnicalanalysisdoesnotwork.AMarkovprocessforstockpricesisconsistentwithweak-formmarketefficiencyOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200834优颧蛾咖坍佐单谓呕苟贴哇辗废熏穗翘加纵讽胺咨棕佰沦鼓设砾绳跌囚关期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eWeak-FormMarketEfficiencyOpt34ExampleofaDiscreteTimeContinuousVariableModelAstockpriceiscurrentlyat$40Attheendof1yearitisconsideredthatitwillhaveanormalprobabilitydistributionofwithmean$40andstandarddeviation$10Options,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200835敛具投津橙竿只山钢案泳殿材育殴脯芭半毫佣好欢坯琢撅蹋引谁庇嚷搅硷期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eExampleofaDiscreteTimeCon35QuestionsWhatistheprobabilitydistributionofthestockpriceattheendof2years?½years? ¼years?

Dtyears?Takinglimitswehavedefinedacontinuousvariable,continuoustimeprocessOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200836釜轮吼彬孰钟摈划垦摄挪海梭五陀喻笔笆氯葱橡晶鞍传和诸腾虎焉遗辰淡期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eQuestionsWhatistheprobabili36Variances&StandardDeviationsInMarkovprocesseschangesinsuccessiveperiodsoftimeareindependentThismeansthatvariancesareadditiveStandarddeviationsarenotadditiveOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200837豫米语它垃糟驯坝屋煤千啡丁牵陪老痰耳恃溉淆蕾素愤览仁疡骑舀刘罐楞期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eVariances&StandardDeviation37Variances&StandardDeviations(continued)Inourexampleitiscorrecttosaythatthevarianceis100peryear.Itisstrictlyspeakingnotcorrecttosaythatthestandarddeviationis10peryear.Options,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200838巳鬃哀腋力秘守陶续鹊除蜀士簇晨翻柒曰侵柿旅伴军饭掘辰泊蠕萎轴只丢期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eVariances&StandardDeviation38AWienerProcess(Seepages261-63)WeconsideravariablezwhosevaluechangescontinuouslyDefinef(m,v)

asanormaldistributionwithmeanmandvariancevThechangeinasmallintervaloftimeDtisDz

ThevariablefollowsaWienerprocessif

ThevaluesofDzforany2different(non-overlapping)periodsoftimeareindependent

Options,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200839勇胞耸址还武瞩冕叼沟跪陈渤盘赛勘个英盏砌燥彪脉宛屎努产尧贿酉孕冉期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eAWienerProcess(Seepages2639PropertiesofaWienerProcessMeanof[z(T)–z(0)]is0Varianceof[z(T)–z(0)]isTStandarddeviationof[z(T)–z(0)]isOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200840易摩疤网嚎计鸦影箩狭梁锤斌突绩啄镐泽苦殉刨彦商二五倍毯屹鹿爱烫炔期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7ePropertiesofaWienerProcess40TakingLimits...Whatdoesanexpressioninvolvingdzanddtmean?ItshouldbeinterpretedasmeaningthatthecorrespondingexpressioninvolvingDzandDtistrueinthelimitasDttendstozeroInthisrespect,stochasticcalculusisanalogoustoordinarycalculusOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200841仅番劲伍奄酗榔戎嚏呢辫所揖镰梳莽总垮沪如趾揩拢乏仕咙闸抵阜害擒疮期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eTakingLimits...Whatdoesa41GeneralizedWienerProcesses

(See65)AWienerprocesshasadriftrate(i.e.averagechangeperunittime)of0andavariancerateof1InageneralizedWienerprocessthedriftrateandthevarianceratecanbesetequaltoanychosenconstantsOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200842寸宴绑晦磋申孔蔑茨么貉袋彭磨现独抖邹飞遍峨驯柯份蜕盏鄂再审赋痉艺期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eGeneralizedWienerProcesses

(42GeneralizedWienerProcesses

(continued)ThevariablexfollowsageneralizedWienerprocesswithadriftrateofaandavariancerateof

b2ifdx=adt+bdz

Options,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200843燃犬请梳姐蔡戍前闰幻声饭计布葛丽占篇庭产戏邮咽蔗讥切障赃疵驯吐梦期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eGeneralizedWienerProcesses

(43GeneralizedWienerProcesses

(continued)MeanchangeinxintimeT

isaTVarianceofchangeinxintime

T

isb2TStandarddeviationofchangeinx

intimeT

isOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200844籍拍客林狭瞧叉荔履庄乎质觉琢铬溉吻点栽碱娶藤匝让柱搪揪钙订巴忙锦期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eGeneralizedWienerProcesses

(44TheExampleRevisitedAstockpricestartsat40andhasaprobabilitydistributionoff(40,100)attheendoftheyearIfweassumethestochasticprocessisMarkovwithnodriftthentheprocessisdS=10dzIfthestockpricewereexpectedtogrowby$8onaverageduringtheyear,sothattheyear-enddistributionisf(48,100),theprocesswouldbedS=8dt+10dzOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200845蛋奢肉主舶鸥访篷慧阑夫菜锋彭棚秸薯骆狱喂株怀消胳漂夫师毯床买动糠期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eTheExampleRevisitedAstockp45ItôProcess(Seepages265)InanItôprocessthedriftrateandthevarianceratearefunctionsoftime

dx=a(x,t)dt+b(x,t)dz

ThediscretetimeequivalentisonlytrueinthelimitasDttendstozeroOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200846妖悄噪志爷记球实逆号商在闯赦孽稠郡倦匡蚤敷跋谓朔姆缴浴敲雏衡逗喧期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eItôProcess(Seepages246WhyaGeneralizedWienerProcessIsNotAppropriateforStocksForastockpricewecanconjecturethatitsexpectedpercentagechangeinashortperiodoftimeremainsconstant,notitsexpectedabsolutechangeinashortperiodoftimeWecanalsoconjecturethatouruncertaintyastothesizeoffuturestockpricemovementsisproportionaltothelevelofthestockpriceOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200847蚕霸聂厌淹歌燎异呐泛七耘窿倘瞥撒钵凉画哇擂巧酗脏赐逢骋颗擒囱度窜期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eWhyaGeneralizedWienerProce47AnItoProcessforStockPrices

(Seepages269-71)wheremistheexpectedreturnsisthevolatility.ThediscretetimeequivalentisOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200848壹四重呐芍遇餐琴长吉帮言衰宿岸慢披菩涤桃针笛饰极毙镰矗趣庐琐则救期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eAnItoProcessforStockPrice48MonteCarloSimulationWecansamplerandompathsforthestockpricebysamplingvaluesforeSupposem=0.15,s=0.30,andDt=1week(=1/52years),thenOptions,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200849呆揩渊掠疥刀叹哟椭刃暇有揭炳央权楞黎创符滨锌优疆合妨候呻绑文碧艘期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eMonteCarloSimulationWecans49MonteCarloSimulation–OnePath(SeeTable12.1,page268)

Options,Futures,andOtherDerivatives,7thInternationalEdition,Copyright©JohnC.Hull200850阀玄原匆据卖恼甥铬土裙性操味另兆富奠评宫窟鬃犁凰闺翻塘增汉越框苍期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,andOtherDerivatives,7eMonteCarloSimulation–OneP50Itô’sLemma(Seepages269-270)Ifwe