Model of the Behavior of stock Prices Chapter 11 Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
11.1 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Model of the Behavior of Stock Prices Chapter 11
11.2 Categorization of Stochastic Processes o Discrete time: discrete variable Discrete time continuous variable Continuous time discrete variable Continuous time continuous variable Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
11.2 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Categorization of Stochastic Processes • Discrete time; discrete variable • Discrete time; continuous variable • Continuous time; discrete variable • Continuous time; continuous variable
11.3 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 derivatives Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
11.3 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 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 derivatives
11.4 Markov processes (See pages 216-7 e 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 o We assume that stock prices follow Markov processes Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
11.4 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 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 are • We assume that stock prices follow Markov processes
11.5 Weak-Form market Efficiency e 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 e A Markov process for stock prices is clearly consistent with weak-form market efficiency Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
11.5 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Weak-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 Markov process for stock prices is clearly consistent with weak-form market efficiency
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)