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Ch12 - Wiener Processes and It's Lemma Chapter 12 1 Types...

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Wiener Processes and Itô’s Lemma Chapter 12 1

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Types of Stochastic Processes Discrete time; discrete variable Discrete time; continuous variable Continuous time; discrete variable Continuous time; continuous variable 2
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 3

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Markov Processes (See pages 259-60) 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 4
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 consistent with weak-form market efficiency 5

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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 normal probability distribution of with mean \$40 and standard deviation \$10 6
Questions What is the probability distribution of the stock price at the end of 2 years? ½ years? ¼ years? t years? Taking limits we have defined a continuous variable, continuous time process 7

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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 8
Variances & Standard Deviations (continued) In our example it is correct to say that the variance is 100 per year.

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