CHAPTER 13
Wiener Processes and It’s Lemma
Practice Questions
Problem .
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) 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 might 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, temperatures do not follow a Markov process.
Problem .
Can a trading rule based on the past history of a stock’s 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 key 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.
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 were set up to take advantage of the phenomenon. There is some evidence that this has resulted in the phenomenon disappearing.
Problem .
A company’s cash position, measured in millions of dollars, follows a generalized Wiener process with a drift rate of per quarter and a variance rate of per quarter. How high does the company’s 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 company’s initial cash position isx. The probability distribution of the cash position at the end of one year is ?(x?4?0?5?4?4)??(x?2?0?16)
where ?(m?v) is a normal probability distribution with mean m and variance v. The probability of a negative cash position at the end of one year is
where N(x) is the cumulative probability that a standardized normal variable (with mean zero and standard deviation is less than x. From normal distribution tables
?x?2?0?N?? ??0?05
4??when:
x?2?0 ???1?64494
., whenx?4?5796. The initial cash position must therefore be $ million.
Problem .
Variables X1 and X2 follow generalized Wiener processes with drift rates ?1 and
?x?2?0?N???4? ??2 and variances ?12 and ?22. What process does X1?X2 follow if:
(a) The changes in X1 and X2 in any short interval of time are uncorrelated (b) There is a correlation ? between the changes in X1 and X2 in any short interval of time
(a) Suppose that X1 and X2 equal a1 and a2 initially. After a time period of length T, X1 has the probability distribution
?(a1??1T??12T)
and X2 has a probability distribution
2?(a2??2T??2T)
From the property of sums of independent normally distributed variables, X1?X2 has the probability distribution
2??a1??1T?a2??2T??12T??2T?
.,
22???a1?a2?(?1??2)T?(?1??2)T??
This shows that X1?X2 follows a generalized Wiener process with drift rate ?1??2
2and variance rate ?12??2.
(b) In this case the change in the value of X1?X2 in a short interval of time ?t has the probability distribution:
22 ??(???)?t?(????2??1?2)?t?1212??
If ?1, ?2, ?1, ?2 and ? are all constant, arguments similar to those in Section show that the change in a longer period of time T is
22??(???)T?(????2??1?2)T?1212??
The variable,X1?X2, therefore follows a generalized Wiener process with drift rate
2?1??2 and variance rate ?12??2?2??1?2.
Problem .
Consider a variable,S, that follows the process dS??dt??dz
For the first three years, ??2 and??3; for the next three years, ??3 and??4. If the initial value of the variable is 5, what is the probability distribution of the value of the variable at the end of year six
The change in S during the first three years has the probability distribution ?(2?3?9?3)??(6?27)
The change during the next three years has the probability distribution ?(3?3?16?3)??(9?48)
The change during the six years is the sum of a variable with probability distribution
?(6?27) and a variable with probability distribution?(9?48). The probability distribution of the change is therefore ?(6?9?27?48) ??(15?75)
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 ?(20?75)
Problem .
Suppose that G is a function of a stock price, S and time. Suppose that ?S and ?G are the volatilities of S and G. Show that when the expected return of S increases by ??S, the growth rate of G increases by ??G, where ? is a constant.
From It’s lemma
?G ?GG??SS
?SAlso the drift of G is
JohnHull《期货、期权和衍生证券》13章习题解答
