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 (nonoverlapping) periods of time are independent Dz = Dt