11.6 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 o (40, 10) Whereψ(μ,o) is a normal distribution with mean u and standard deviationσ. Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
11.6 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 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
117 Ouestions e What is the probability distribution of the stock price at the end of 2 years? 佐 years? 74 years? ° St years? Taking limits we have defined a continuous variable. continuous time process Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
11.7 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Questions • What is the probability distribution of the stock price at the end of 2 years? • ½ years? • ¼ years? • dt years? Taking limits we have defined a continuous variable, continuous time process
11.8 Variances standard Deviations o In Markov processes changes in successive periods of time are independent e This means that variances are additive e Standard deviations are not additive Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
11.8 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 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
11.9 Variances Standard Deviations(continued) e In our example it is correct to say that the variance is 100 per year o It is strictly speaking not correct to say that the standard deviation is 10 per year Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
11.9 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 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
11.10 A Wiener Process(See pages 218) e We consider a variable z whose value changes continuously o the change in a small interval of time ft is δz The variable follows a Wiener process if 1.8zEvSt where e is a random drawing from (0, 1 2. The values of Sz for any 2 different(non- overlapping) periods of time are independent Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
11.10 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 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 • The variable follows a Wiener process if 1. 2. The values of dz for any 2 different (nonoverlapping) periods of time are independent dz = dt where i s a random drawing from f(0,1)