文档介绍：Model of the Behavior of Stock Prices Chapter 11
Categorization of Stochastic Processes
Discrete time; discrete variable
Discrete time; continuous variable
Continuous time; discrete variable
Continuous time; continuous variable
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
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
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
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.
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
Variances & Standard Deviations
In Markov processes changes in essive periods of time are independent
This means that variances are additive
Standard deviations are not additive
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.
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