oftheBehaviorofStockPrices课件.pptx

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1、Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.1Model of the Behaviorof Stock PricesChapter 10Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.2Categorization of St

2、ochastic ProcessesDiscrete time;discrete variableDiscrete time;continuous variableContinuous time;discrete variableContinuous time;continuous variableOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.3Modeling Stock PricesWe can use a

3、ny 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 valuing derivative securitiesOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal Unive

4、rsity10.4Markov Processes 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 will assume that stock prices follow Markov processesOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai N

5、ormal University10.5Weak-Form Market EfficiencyThe assertion is that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices.In other words technical analysis does not work.A Markov process for stock prices is clearly consistent with we

6、ak-form market efficiencyOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.6Example of a Discrete Time Continuous Variable ModelA stock price is currently at$40At the end of 1 year it is considered that it will have a probability dist

7、ribution of(40,10),where(m,s)is a normal distribution with mean m and standard deviation s.Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.7QuestionsWhat is the probability distribution of the change in stock price over/during 2 yea

8、rs?years?years?Dt years?Taking limits we have defined a continuous variable,continuous time processOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.8Variances&Standard DeviationsIn Markov processes changes in successive periods of ti

9、me are independentThis means that variances are additiveStandard deviations are not additiveOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.9Variances&Standard Deviations(continued)In our example it is correct to say that the varian

10、ce is 100 per year.It is strictly speaking not correct to say that the standard deviation is 10 per year.(You can say that the STD is 10 per square root of years)Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.10A Wiener Process(See

11、 pages 220-1)We consider a variable z whose value changes continuously The change in a small interval of time Dt is Dz The variable follows a Wiener process if1.,where is a random drawing from(0,1).2.The values of Dz for any 2 different(non-overlapping)periods of time are independentOptions,Futures,

12、and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.11Properties of a Wiener ProcessMean of z(T)z(0)is 0Variance of z(T)z(0)is TStandard deviation of z(T)z(0)isOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shang

13、hai Normal University10.12Taking Limits.What does an expression involving dz and dt mean?It should be interpreted 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 calculusOptions,Futu

14、res,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.13Generalized Wiener Processes(See page 221-4)A Wiener process has a drift rate(ie average change per unit time)of 0 and a variance rate of 1 In a generalized Wiener process the drift rate&the var

15、iance rate can be set equal to any chosen constantsOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.14Generalized Wiener Processes(continued)The variable x follows a generalized Wiener process with a drift rate of a&a variance rate o

16、f b2 if dx=a dt+b dz Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.15Generalized 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 Opt

17、ions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.16The Example RevisitedA stock price starts at 40&has a probability distribution of(40,10)at the end of the yearIf we assume the stochastic process is Markov with no drift then the proces

18、s 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(48,10),the process is dS =8dt +10dzOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.17 Ito Process(See pages

19、224-5)In an Ito 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 zeroOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.18Why a

20、 Generalized Wiener Processis not Appropriate for StocksFor a stock price we can conjecture that its expected proportional change in a short period of time remains constant We can also conjecture that our uncertainty as to the size of future stock price movements is proportional to the level of the

21、stock priceOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.19An Ito Process for Stock Prices(See pages 225-6)where m is the expected return,s is the volatility.The discrete time equivalent isOptions,Futures,and Other Derivatives,4th

22、 edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.20Monte Carlo SimulationWe can sample random paths for the stock price by sampling values for Suppose m=0.14,s=0.20,and Dt=0.01,thenOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shangh

23、ai Normal University10.21Monte Carlo Simulation One Path(continued.See Table 10.1)Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.22Itos Lemma(See pages 229-231)If we know the stochastic process followed by x,Itos lemma tells us the

24、 stochastic process followed by some function G(x,t)Since a derivative security is a function of the price of the underlying&time,Itos lemma plays an important part in the analysis of derivative securitiesOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai

25、 Normal University10.23Taylor Series ExpansionA Taylors series expansion of G(x,t)givesOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.24Ignoring Terms of Higher Order Than DtOptions,Futures,and Other Derivatives,4th edition 2000 by

26、 John C.Hull Tang Yincai,2003,Shanghai Normal University10.25Substituting for DxOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.26The 2Dt TermOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Sha

27、nghai Normal University10.27Taking LimitsThis is Itos Lemma.Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University10.28Application of Itos Lemmato a Stock Price ProcessOptions,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tan

28、g Yincai,2003,Shanghai Normal University10.29Examples1、有时候读书是一种巧妙地避开思考的方法。3月-233月-23Wednesday,March 29,20232、阅读一切好书如同和过去最杰出的人谈话。16:26:1416:26:1416:263/29/2023 4:26:14 PM3、越是没有本领的就越加自命不凡。3月-2316:26:1416:26Mar-2329-Mar-234、越是无能的人，越喜欢挑剔别人的错儿。16:26:1416:26:1416:26Wednesday,March 29,20235、知人者智，自知者明。胜人者有力

29、，自胜者强。3月-233月-2316:26:1416:26:14March 29,20236、意志坚强的人能把世界放在手中像泥块一样任意揉捏。29 三月 20234:26:14 下午16:26:143月-237、最具挑战性的挑战莫过于提升自我。三月 234:26 下午3月-2316:26March 29,20238、业余生活要有意义，不要越轨。2023/3/29 16:26:1416:26:1429 March 20239、一个人即使已登上顶峰，也仍要自强不息。4:26:14 下午4:26 下午16:26:143月-2310、你要做多大的事情，就该承受多大的压力。3/29/2023 4:26:14 PM16:26:1429-3月-2311、自己要先看得起自己，别人才会看得起你。3/29/2023 4:26 PM3/29/2023 4:26 PM3月-233月-2312、这一秒不放弃，下一秒就会有希望。29-Mar-2329 March 20233月-2313、无论才能知识多么卓著，如果缺乏热情，则无异纸上画饼充饥，无补于事。Wednesday,March 29,202329-Mar-233月-2314、我只是自己不放过自己而已，现在我不会再逼自己眷恋了。3月-2316:26:1429 March 202316:26谢谢大家谢谢大家