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73323260-12-Wiener-Process-and-Itos-Lemma - Wiener...

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· Wiener Processes and Ito's etntna Any variable whose value changes over time in an uncertain way is said to follow a stochastic process. Stochastic processes can be classified as discrete time or continuous time. A discrete-time stochastic process is one where the value of the variable can change only at certain fixed points in time, whereas a continuous-time stochastic process is one where changes can take place at any time. Stochastic processes can also be classified as continuous variable or discrete variable. In a continuous-variable process, the underlying variable can take any value within a certain range, whereas in a discrete- variable process, only certain discrete values are possible. This chapter develops a continuous-variable, continuous-time stochastic process for stock prices. Learning about this process is the first step to understanding the pricing of options and other more complicated derivatives. It should be noted that, in practice, we do not observe stock prices following continuous-variable, continuous- time processes. Stock prices are restricted to discrete values (e.g., multiples of a cent) and changes can be observed only .when the exchange is open. Nevertheless, the continuous-variable, continuous-time process proves to be a useful model for many purposes. Many people feel that continuous-time stochastic processes are so complicated that they should be left entirely to "rocket scientists". This is not so. The biggest hurdle to understanding these processes is the notation. Here we present a step-by-step approach aimed at getting the reader over this hurdle. We also explain an important result known as Ito's lemma that is central to the pricing of derivatives. 12.1 THE MARKOV PROPERTY A Markov process is a particular type of stochastic process where only the present value of a variable is relevant for predicting the future. The past history of the variable and the way that the present has emerged from the past are irrelevant. Stock prices are usually assumed to follow a Markov process. Suppose that the price of IBM stock is $100 now. If the stock price follows a Markov process, our predictions for the future should be unaffected by the price one week ago, one month 263
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264 CHAPTER 12 ago, or one year ago. The only relevant piece of information is that the price is now $100. 1 Predictions for the future are uncertain and must be expressed in terms of probability distributions. The Markov property implies that the probability distribu- tion of the price at any particular future time is not dependent on the particular path followed by the price in the past. The Markov property of stock prices is consistent with the weak form of market efficiency. This states that the present price of a stock impounds all the information contained in a record of past prices. If the weak form of market efficiency were not true, technical analysts could make above-average returns by interpreting charts of the past history of stock prices. There is very little evidence that they are in fact able to do this.
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