ch11期權(quán)期貨與衍生證券（第五版）

1、Model of the Behaviorof Stock PricesChapter 11,Categorization of Stochastic Processes,Discrete time; discrete variableDiscrete time; continuous variableContinuous time; discrete variableContinuous time; continuous

2、variable,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 valuing deri

3、vatives,Markov Processes (See pages 216-7),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 follow Markov processes,W

4、eak-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 words technical analysis does not work.A Ma

5、rkov process for stock prices is clearly consistent with weak-form market efficiency,Example of a Discrete Time Continuous Variable Model,A stock price is currently at $40At the end of 1 year it is considered that it

6、will have a probability distribution of f(40,10) where f(m,s) is a normal distribution with mean m and standard deviation s.,Questions,What is the probability distribution of the stock price at the end of 2 years?½

7、; years?¼ years? dt years? Taking limits we have defined a continuous variable, continuous time process,Variances & Standard Deviations,In Markov processes changes in successive periods of time are

8、independentThis means that variances are additiveStandard deviations are not additive,Variances & Standard Deviations (continued),In our example it is correct to say that the variance is 100 per year.It is strict

9、ly speaking not correct to say that the standard deviation is 10 per year.,A Wiener Process (See pages 218),We consider a variable z whose value changes continuously The change in a small interval of time dt is dz

10、The variable follows a Wiener process if1. 2. The values of dz for any 2 different (non-overlapping) periods of time are independent,Properties of a Wiener Process,Mean of [z (T ) – z (0)] is 0Variance of

11、 [z (T ) – z (0)] is TStandard deviation of [z (T ) – z (0)] is,Taking Limits . . .,What does an expression involving dz and dt mean?It should be interpreted as meaning that the corresponding expression involving d

12、z and dt is true in the limit as dt tends to zeroIn this respect, stochastic calculus is analogous to ordinary calculus,Generalized Wiener Processes(See page 220-2),A Wiener process has a drift rate (i.e. average c

13、hange 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,Generalized Wiener Processes(continued),The variable x fo

14、llows a generalized Wiener process with a drift rate of a and a variance rate of b2 if dx=adt+bdz,Generalized Wiener Processes(continued),Mean change in x in time T is aTVariance of change in x in time T is b2TStan

15、dard deviation of change in x in time T is,The Example Revisited,A stock price starts at 40 and has a probability distribution of f(40,10) at the end of the yearIf we assume the stochastic process is Markov with no dri

16、ft 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(48,10), the process is dS = 8dt + 10dz,Ito Process (See pages 222)

17、,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 zero,Why a Generalized Wiener Process

18、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 absolute change in a short period of timeWe can also con

19、jecture that our uncertainty as to the size of future stock price movements is proportional to the level of the stock price,An Ito Process for Stock Prices(See pages 222-3),where m is the expected return s is the volati

20、lity. The discrete time equivalent is,Monte Carlo Simulation,We can sample random paths for the stock price by sampling values for eSuppose m= 0.14, s= 0.20, and dt = 0.01, then,Monte Carlo Simulation – One Path (See

21、Table 11.1),,Ito’s Lemma (See pages 226-227),If we know the stochastic process followed by x, Ito’s lemma tells us the stochastic process followed by some function G (x, t )Since a derivative security is a function of t

22、he price of the underlying and time, Ito’s lemma plays an important part in the analysis of derivative securities,Taylor Series Expansion,A Taylor’s series expansion of G(x, t) gives,Ignoring Terms of Higher Order Than d