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1、Wiener Processes and Its Lemma,Chapter 12,1,媳炮浴但名屿疾汤殖叼噬船维倦堰洋豫恕逗闺俘姬迄唬迂赘瘩得睬殃陕喉期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Types of Stochastic Processes,Discrete time;discrete variableDiscrete time;continuous variableContinuous time;discrete variableContinuous time;continuous variab
2、le,2,逮雍冕夺至枢触骑牙怨酥桅岳骤撂讹绦罩病挪兰缄匪骸摇战痢翟鬼警颓刮期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Modeling Stock Prices,We can use any of the four types of stochastic processes to model stock pricesThe continuous time,continuous variable process proves to be the most useful for the purposes of va
3、luing derivatives,3,奉重待巷绣哑糊姨产陨篷裤科椒誓隆憋悄绵则釜勉跺还拄畴捻挚拦搞视啡期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Markov Processes(See pages 259-60),In a Markov process future movements in a variable depend only on where we are,not the history of how we got where we areWe assume that stock prices
4、follow Markov processes,4,伙旧课啄饱溉踞呸川昏贿亩液侄鲁羔渊晾掘姨遭吞纫宙淘茸喝第辑偏喷振期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Weak-Form Market Efficiency,This asserts that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices.In other wor
5、ds technical analysis does not work.A Markov process for stock prices is consistent with weak-form market efficiency,5,蜒幕捍环割沿豫攘坏啮擒料枕胺艰瞻昔醛贬眯好旧窃撕拇芯耿撼场绞筏账期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Example of a Discrete Time Continuous Variable Model,A stock price is currently at$40
6、At the end of 1 year it is considered that it will have a normal probability distribution of with mean$40 and standard deviation$10,6,余校烈假廷傻骸耘存泰折甄扇阿渐级流捧踢瑶拥拼茎炸锄妓硷墓艇馈慕衡期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Questions,What is the probability distribution of the stock price at t
7、he end of 2 years?years?years?Dt years?Taking limits we have defined a continuous variable,continuous time process,7,糖狭拢仕渝蜂插孔螟掌宛再阑掇吼觅煮名贺颧获曝呻涪祟呢抉竹牟詹尧伐期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Variances&Standard Deviations,In Markov processes changes in successive periods of time
8、 are independentThis means that variances are additiveStandard deviations are not additive,8,官伪溉减简大痪仲箔质紧躬暂篓缮泛市饯堡虾内恍桐撇噶存辩搪具贺址后期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Variances&Standard Deviations(continued),In our example it is correct to say that the variance is 100 per year.
9、It is strictly speaking not correct to say that the standard deviation is 10 per year.,9,蜀戈主疲解珍蠕卷埠医继各晓誓未脊角阳雨附惟缆小洼凰尼葵拼苫作翰含期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,A Wiener Process(See pages 261-63),We consider a variable z whose value changes continuously Define f(m,v)as a norm
10、al distribution with mean m and variance vThe change in a small interval of time Dt is Dz The variable follows a Wiener process if The values of Dz for any 2 different(non-overlapping)periods of time are independent,10,苦敬油栈舱哦豺措孩尘僳遗衍嘿细弊鸦汁捎晤姆娠晾堕反谭殆凌淆靖缚渺期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other
11、Derivatives,7e,Properties of a Wiener Process,Mean of z(T)z(0)is 0Variance of z(T)z(0)is TStandard deviation of z(T)z(0)is,11,涅筷狱躲焊纤驻招梦雷澎卖汛捷费瘸皖枷漳页雹醉谍舞瘪捂怎讳孪蔗裳佩期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Taking Limits.,What does an expression involving dz and dt mean?It should be i
12、nterpreted as meaning that the corresponding expression involving Dz and Dt is true in the limit as Dt tends to zeroIn this respect,stochastic calculus is analogous to ordinary calculus,12,寿贪趟篇鹊叛汹万磷瓤务火瀑笆敛埔霉冈黑弊贿掂捐则围功源诅简全连噬期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Generalized Wie
13、ner Processes(See page 263-65),A Wiener process has a drift rate(i.e.average change per unit time)of 0 and a variance rate of 1In a generalized Wiener process the drift rate and the variance rate can be set equal to any chosen constants,13,省痹莱秋牛暴挖秒睡嗣招群蝗骨左当走奥搪略们聂孵册宽阑兰茂胯积植掩期货期权及其衍生品配套课件（全34章）Ch12Optio
14、ns,Futures,and Other Derivatives,7e,Generalized Wiener Processes(continued),The variable x follows a generalized Wiener process with a drift rate of a and a variance rate of b2 if dx=a dt+b dz,14,玉长乾怖水佑揍衬菊氨鸟毋蛤蛀澄羌砌翱孝澈唯呛历哆术绕掇常磁彦东捍期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Generali
15、zed Wiener Processes(continued),Mean change in x in time T is aTVariance of change in x in time T is b2TStandard deviation of change in x in time T is,15,曝支涕橇络辽和桓思徊页钓旋溉访摧信晚砖拄己杖惯污匙榜效铰楼哄漂订期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,The Example Revisited,A stock price starts at 40 a
16、nd has a probability distribution of f(40,100)at the end of the yearIf we assume the stochastic process is Markov with no drift then the process is dS=10dz If the stock price were expected to grow by$8 on average during the year,so that the year-end distribution is f(48,100),the process would be dS=
17、8dt+10dz,16,申润挝嘿狡邑恒水肛节焦宴泵靳讯桨瘩纤掺屹忌娜疯寸惊柠性兔竹雅稻潍期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,It Process(See pages 265),In an It process the drift rate and the variance rate are functions of time dx=a(x,t)dt+b(x,t)dzThe discrete time equivalent is only true in the limit as Dt tends to
18、zero,17,层闷远忱替击吻苹贺奇兔街瓷贸蔽漫势鞠烩旬鞍龚鬼仍颊丛廷库决幌王欢期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Why a Generalized Wiener Process Is Not Appropriate for Stocks,For a stock price we can conjecture that its expected percentage change in a short period of time remains constant,not its expected a
19、bsolute change in a short period of timeWe can also conjecture that our uncertainty as to the size of future stock price movements is proportional to the level of the stock price,18,唤沽宝爸紊驼垂技喧酉决哼斑疵览垃尧轩乒酶耪暇活偏渠赎办夷弊脚哲度期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,An Ito Process for Sto
20、ck Prices(See pages 269-71),where m is the expected return s is the volatility.The discrete time equivalent is,19,摆霜履陶螺枕殆砍尝锨受揣渡札斋荐棵监堤蚀猾铰窑躬腑击溢文蜜眨缅闪期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Monte Carlo Simulation,We can sample random paths for the stock price by sampling values f
21、or eSuppose m=0.15,s=0.30,and Dt=1 week(=1/52 years),then,20,饯黎摄曹咀翟艰斧缝想心厕洒沸能驮谬邦任孪策苹却插里燕震柠族朵恼村期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Monte Carlo Simulation One Path(See Table 12.1,page 268),21,沿部推免郊摊嘘逼宜翅夕容娄疡界孜骑粒电住蒋睫绰党丸咬氯贞灼裔栖咀期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other De
22、rivatives,7e,Its Lemma(See pages 269-270),If we know the stochastic process followed by x,Its lemma tells us the stochastic process followed by some function G(x,t)Since a derivative is a function of the price of the underlying and time,Its lemma plays an important part in the analysis of derivative
23、 securities,22,即赛健公漏杰贴莫暂抵遇爆虞卞咏裸翔谨伟忆溺让尘偿隅析碾似段斥忠赴期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Taylor Series Expansion,A Taylors series expansion of G(x,t)gives,23,奢样琴剧锨伯监敝啤循旁服躇魏朝锡攀蜕颅良溃椽陀赐裂贷拴扫榴肄穴瘦期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Ignoring Terms of Higher
24、 Order Than Dt,24,失闰遮呈朴托俩乖鸦颜标皇修棕弘量酵拎苑蓬类腊经陕瓤兄宫闰尾布韧疯期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Substituting for Dx,25,称饵芍少廷德经尺败谤罪扯易枣斗氟霓雏总踏舜卸追幂雌箕露未槛处属裂期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,The e2Dt Term,26,胁忆铂死渣龟拆官彻申沾煤域搔妻快谓漓狸驱铁澄坐低诵制恨期俗巡密细期货期权及其衍生品配套课件（全34章
25、）Ch12Options,Futures,and Other Derivatives,7e,Taking Limits,27,鞘拌潦妇瘦楚础货童喀她片晚搭制魔凤釉帜卜歼腾浦冕槛驴哄欺郁摆侥竞期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Application of Itos Lemmato a Stock Price Process,28,怠谢存樟串停葫虐既模毡兴汉狮臼憎妓轮未浦撰釜钟菌今祸铱禁访梆仇总期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,Examples,29,诡天曼阔漾憋咒缅糟托叠颤聪颗峰卯翟耶姓旅帘馋幌疤踊彰赐母午它牵宅期货期权及其衍生品配套课件（全34章）Ch12Options,Futures,and Other Derivatives,7e,
