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1、Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.1,Model of the Behaviorof Stock PricesChapter 10,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.2,Cat
2、egorization of Stochastic Processes,Discrete time; discrete variable Discrete time; continuous variable Continuous time; discrete variable Continuous time; continuous variable,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.3
3、,Modeling Stock Prices,We can use any of the four types of stochastic processes to model stock prices The continuous time, continuous variable process proves to be the most useful for the purposes of valuing derivative securities,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. H
4、ull Tang Yincai, 2003, Shanghai Normal University,10.4,Markov 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 are We will assume that stock prices follow Markov processes,Options, Futures, and Other Derivatives, 4th edi
5、tion 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.5,Weak-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 M
6、arkov process for stock prices is clearly consistent with weak-form market efficiency,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.6,Example of a Discrete Time Continuous Variable Model,A stock price is currently at $40 At
7、 the end of 1 year it is considered 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 University,10.7,Quest
8、ions,What is the probability 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 process,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Norma
9、l University,10.8,Variances & Standard Deviations,In Markov processes changes in successive periods of time are independent This means that variances are additive Standard deviations are not additive,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shangha
10、i Normal University,10.9,Variances & 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, Future
11、s, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.10,A 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 if 1. ,whe
12、re is a random drawing from (0,1). 2. The values of Dz for any 2 different (non- overlapping) periods of time are independent,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.11,Properties of a Wiener Process,Mean of z (T ) z
13、(0) is 0 Variance of z (T ) z (0) is T Standard deviation of z (T ) z (0) is,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.12,Taking Limits . . .,What does an expression involving dz and dt mean? It should be interpreted as
14、 meaning that the corresponding expression involving Dz and Dt is true in the limit as Dt tends to zero In this respect, stochastic calculus is analogous to ordinary calculus,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.13
15、,Generalized 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 variance rate can be set equal to any chosen constants,Options, Futures, and Other Derivatives, 4th edi
16、tion 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.14,Generalized 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 Normal University,10.15,Generalized Wiener Processes(continued),Mean change in x in time T is aT Variance of change in x in time T is b2T Standard deviat
