oftheBehaviorofStockPrices(金融工程学,华东

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 University10.2Categori

2、zation of Stochastic 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 St

3、ock PricesWe 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 valuing derivative securitiesOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai

4、, 2003, Shanghai Normal University10.4Markov 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

5、. Hull Tang Yincai, 2003, Shanghai Normal University10.5Weak-Form Market Efficiency The 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 sto

6、ck prices is clearly consistent with weak-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 c

7、onsidered that it will have a probability distribution of f(40,10), where f(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 University10.7Questions What is the probability

8、 distribution of the change in stock price over/during 2 years? 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 &am

9、p; Standard DeviationsIn Markov processes changes in successive periods of time 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 University10.9Variance

10、s & Standard Deviations (continued)In our example it is correct to say that the variance 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 e

11、dition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University10.10A Wiener Process (See 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 f(0,1).

12、 2. The values of Dz for any 2 different (non- overlapping) periods of time are independent tzDDOptions, Futures, 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)

13、is TStandard deviation of z (T ) z (0) isTOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University10.12Taking Limits . . .What does an expression involving dz and dt mean?It should be interpreted as meaning that the corresponding express

14、ion involving Dz and Dt is true in the limit as Dt tends to zeroIn this respect, stochastic calculus is analogous to ordinary calculusOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University10.13Generalized Wiener Processes(See page 221-

15、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 variance rate can be set equal to any chosen constantsOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 20

16、03, Shanghai Normal University10.14Generalized Wiener Processes(continued) The variable x follows a generalized Wiener process with a drift rate of a & a variance rate of b2 if dx = a dt + b dz Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai

17、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 DDDxatbt b TOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal

18、 University10.16The Example RevisitedA stock price starts at 40 & has a probability distribution of f(40,10) 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,

19、so that the year-end distribution is f(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 224-5)In an Ito process the drift rate and the variance rate are functions

20、of time dx=a(x,t) dt + b(x,t) dzThe discrete time equivalent is only true in the limit as Dt tends to zeroDDDxa x ttb x tt( , )( , )Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University10.18Why a Generalized Wiener Processis not Appro

21、priate 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 stock priceOptions, Futures, and Other

22、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 isdSSdtSdzms DDDSS tStms Options, Futures, and Other Derivatives, 4th edi

23、tion 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, thenSSS02.00014.0 DOptions, Futures, and Other Derivatives, 4th edition 2000 by John C. Hul

24、l Tang Yincai, 2003, Shanghai Normal University10.21Monte Carlo Simulation One Path (continued. See Table 10.1) PeriodStock Price atStart of PeriodRandomSample for Change in StockPrice, DS020.0000.520.236120.2361.440.611220.847-0.86-0.329320.5181.460.628421.146-0.69-0.262Options, Futures, and Other

25、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 stochastic process followed by some function G (x, t )Since a derivative security is a function of th

26、e 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 Normal University10.23Taylor Series ExpansionA Taylors series expansion of G (x , t )

27、 givesDDDDDDDGGxxGttGxxGx txtGtt 2222222Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University10.24Ignoring Terms of Higher Order Than DtIn ordinary calculus we getstic calculus we getbecause has a component which is of order In stocha

28、 DDDDDDDDDGGxxGttGGxxGttGxxxt222Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University10.25Substituting for DxSuppose ( , )( , )so that = + Then ignoring terms of higher order than dxa x t dtb x t dzxatbttGGxxGttGxbtDDDDDDDD2222Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yinc