华东师范大学：《金融工程》英文版 Chapter 10 Model of the Behavior of Stock prices

10.1 Model of the Behavior of Stock prices Chapter 10 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 10.1 Model of the Behavior of Stock Prices Chapter 10

10.2 Categorization 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, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 10.2 Categorization of Stochastic Processes ➢Discrete time; discrete variable ➢Discrete time; continuous variable ➢Continuous time; discrete variable ➢Continuous time; continuous variable

10.3 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. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 10.3 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

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 edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull 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

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 Markov process for stock prices is clearly consistent with weak-form market efficlency Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 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 Markov process for stock prices is clearly consistent with weak-form market efficiency

10.6 Example of a Discrete Time Continuous variable model )A stock price is currently at $40 At the end of 1 year it is considered that it will have a probability distribution of d (40, 10), where p(u, o)is a normal distribution with mean u and standard deviationσ. Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

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 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

10.7 uestions What is the probability distribution of the change in stock price over/during 2 years? years? 14 years? △ 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, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 10.7 Questions • What is the probability distribution of the change in stock price over/during ➢ 2 years? ➢ ½ years? ➢ ¼ years? ➢ Dt years? Taking limits we have defined a continuous variable, continuous time process

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, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal 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

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 SID is 10 per square root of years) Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai 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)

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 At is az The variable follows a Wiener process if 1.△=E√At, where s is a random drawing from o(0, 1) 2. The values of Az 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, C 2003, Shanghai Normal University

Options, Futures, 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. ,where  is a random drawing from f(0,1). 2. The values of Dz for any 2 different (non￾overlapping) periods of time are independent Dz =  Dt