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1、 CHAPTER 13Wiener Processes and Its LemmaPractice QuestionsProblem 13.1.What would it mean to assert that the temperature at a certain place follows a Markov process? Do you think that temperatures do, in fact, follow a Markov process? Imagine that you have to forecast the future temperature from a)
2、 the current temperature, b) the history of the temperature in the last week, and c) a knowledge of seasonal averages and seasonal trends. If temperature followed a Markov process, the history of the temperature in the last week would be irrelevant. To answer the second part of the question you migh
3、t like to consider the following scenario for the first week in May: (i) Monday to Thursday are warm days; today, Friday, is a very cold day. (ii) Monday to Friday are all very cold days. What is your forecast for the weekend? If you are more pessimistic in the case of the second scenario, temperatu
4、res do not follow a Markov process. Problem 13.2.Can a trading rule based on the past history of a stocks price ever produce returns that are consistently above average? Discuss. The first point to make is that any trading strategy can, just because of good luck, produce above average returns. The k
5、ey question is whether a trading strategy consistently outperforms the market when adjustments are made for risk. It is certainly possible that a trading strategy could do this. However, when enough investors know about the strategy and trade on the basis of the strategy, the profit will disappear.
6、As an illustration of this, consider a phenomenon known as the small firm effect. Portfolios of stocks in small firms appear to have outperformed portfolios of stocks in large firms when appropriate adjustments are made for risk. Research was published about this in the early 1980s and mutual funds
7、were set up to take advantage of the phenomenon. There is some evidence that this has resulted in the phenomenon disappearing. Problem 13.3.A companys cash position, measured in millions of dollars, follows a generalized Wiener process with a drift rate of 0.5 per quarter and a variance rate of 4.0
8、per quarter. How high does the companys initial cash position have to be for the company to have a less than 5% chance of a negative cash position by the end of one year? Suppose that the companys initial cash position is. The probability distribution of the cash position at the end of one year is w
9、here is a normal probability distribution with mean and variance . The probability of a negative cash position at the end of one year is where is the cumulative probability that a standardized normal variable (with mean zero and standard deviation 1.0) is less than . From normal distribution tables
10、when: i.e., when. The initial cash position must therefore be $4.58 million. Problem 13.4.Variables and follow generalized Wiener processes with drift rates and and variances and . What process does follow if: (a) The changes in and in any short interval of time are uncorrelated?(b) There is a corre
11、lation between the changes in and in any short interval of time?(a) Suppose that X1 and X2 equal a1 and a2 initially. After a time period of length , X1 has the probability distribution and has a probability distribution From the property of sums of independent normally distributed variables, has th
12、e probability distribution i.e., This shows that follows a generalized Wiener process with drift rate and variance rate . (b) In this case the change in the value of in a short interval of time has the probability distribution: If , , , and are all constant, arguments similar to those in Section 13.
13、2 show that the change in a longer period of time is The variable, therefore follows a generalized Wiener process with drift rate and variance rate . Problem 13.5.Consider a variable, that follows the process For the first three years, and; for the next three years, and. If the initial value of the
14、variable is 5, what is the probability distribution of the value of the variable at the end of year six?The change in during the first three years has the probability distribution The change during the next three years has the probability distribution The change during the six years is the sum of a
15、variable with probability distribution and a variable with probability distribution. The probability distribution of the change is therefore Since the initial value of the variable is 5, the probability distribution of the value of the variable at the end of year six is Problem 13.6.Suppose that is a function of a stock price, and time. Suppose that and are the volatilities of and . Show that when the expected return of increases by , the growth rate of increases by , where is a constant. From Its lemma Also the dri
