Introductory Statistics
1st Edition
ISBN: 9781938168208
Author: Barbara Illowsky, Susan Dean
Publisher: OpenStax College
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Chapter 11, Problem 70H
The standard deviation of the chi-square distribution is twice the
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Students have asked these similar questions
In this problem, we consider a Brownian motion (W+) t≥0. We consider a stock model (St)t>0
given (under the measure P) by
d.St 0.03 St dt + 0.2 St dwt,
with So 2. We assume that the interest rate is r = 0.06. The purpose of this problem is to
price an option on this stock (which we name cubic put). This option is European-type, with
maturity 3 months (i.e. T = 0.25 years), and payoff given by
F = (8-5)+
(a) Write the Stochastic Differential Equation satisfied by (St) under the risk-neutral measure
Q. (You don't need to prove it, simply give the answer.)
(b) Give the price of a regular European put on (St) with maturity 3 months and strike K = 2.
(c) Let X =
S. Find the Stochastic Differential Equation satisfied by the process (Xt)
under the measure Q.
(d) Find an explicit expression for X₁ = S3 under measure Q.
(e) Using the results above, find the price of the cubic put option mentioned above.
(f) Is the price in (e) the same as in question (b)? (Explain why.)
Problem 4. Margrabe formula and the Greeks (20 pts)
In the homework, we determined the Margrabe formula for the price of an option allowing you to
swap an x-stock for a y-stock at time T. For stocks with initial values xo, yo, common volatility
σ and correlation p, the formula was given by
Fo=yo (d+)-x0Þ(d_),
where
In (±²
Ꭲ
d+
õ√T
and
σ = σ√√√2(1 - p).
дго
(a) We want to determine a "Greek" for ỡ on the option: find a formula for
θα
(b) Is
дго
θα
positive or negative?
(c) We consider a situation in which the correlation p between the two stocks increases: what
can you say about the price Fo?
(d) Assume that yo< xo and p = 1. What is the price of the option?
We consider a 4-dimensional stock price model given (under P) by
dẴ₁ = µ· Xt dt + йt · ΣdŴt
where (W) is an n-dimensional Brownian motion,
π = (0.02, 0.01, -0.02, 0.05),
0.2
0
0
0
0.3
0.4
0
0
Σ=
-0.1
-4a За
0
0.2
0.4 -0.1 0.2)
and a E R. We assume that ☑0 = (1, 1, 1, 1) and that the interest rate on the market is r = 0.02.
(a) Give a condition on a that would make stock #3 be the one with largest volatility.
(b) Find the diversification coefficient for this portfolio as a function of a.
(c) Determine the maximum diversification coefficient d that you could reach by varying the
value of a?
2
Chapter 11 Solutions
Introductory Statistics
Ch. 11 - A factory manager needs to understand how many...Ch. 11 - Teachers want to know which night each week their...Ch. 11 - The expected percentage of the number of pets...Ch. 11 - Students in a social studies class hypothesize...Ch. 11 - A sample of 300 students is taken. Of the students...Ch. 11 - The Bureau of Labor Statistics gathers data about...Ch. 11 - Refer back to the information in Try It. How many...Ch. 11 - Do families and singles have the same distribution...Ch. 11 - Ivy League schools receive many applications, but...Ch. 11 - A SCUBA instructor wants to record the collective...
Ch. 11 - The FCC conducts broadband speed tests to measure...Ch. 11 - If the number of degrees of freedom for a...Ch. 11 - If df > 90, the distribution is _____________. If...Ch. 11 - When does the chi-square curve approximate a...Ch. 11 - Where is µ located on a chi-square curve?Ch. 11 - Is it more likely the df is 90, 20, or two in the...Ch. 11 - Determine the appropriate test to be used in the...Ch. 11 - Determine the appropriate test to be used in the...Ch. 11 - Determine the appropriate test to be used in the...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - x2test statistic = ______. Use the following...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Test of Independence Determine the appropriate...Ch. 11 - Test of Independence Determine the appropriate...Ch. 11 - Test of Independence Determine the appropriate...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - a = 0.05 a. Decision: ___________________ b....Ch. 11 - a = 0.01 a. Decision: ___________________ b....Ch. 11 - A math teacher wants to see if two of her classes...Ch. 11 - What are the null and alternative hypotheses for...Ch. 11 - A market researcher wants to see if two different...Ch. 11 - A meteorologist wants to know if East and West...Ch. 11 - What condition must be met to use the test for...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Which test do you use to decide whether an...Ch. 11 - What is the null hypothesis for the type of test...Ch. 11 - Which test would you use to decide whether two...Ch. 11 - Which test would you use to decide if two...Ch. 11 - How are tests of independence similar to tests for...Ch. 11 - How are tests of independence different from tests...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - As the number of degrees of freedom increases, the...Ch. 11 - The standard deviation of the chi-square...Ch. 11 - The mean and the median of the chi-square...Ch. 11 - A six-sided die is rolled 120 times. Fill in the...Ch. 11 - The marital status distribution of the U.S. male...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - The City of South Lake Tahoe, CA, has an Asian...Ch. 11 - Use the following information to answer the next...Ch. 11 - Use the following information to answer the next...Ch. 11 - In a goodness-of-fit test, the expected values are...Ch. 11 - In general, if the observed values and expected...Ch. 11 - Use a goodness-of-fit test to determine if high...Ch. 11 - The test to use to determine if a six-sided die is...Ch. 11 - In a goodness-of fit test, if the p-value is...Ch. 11 - A sample of 212 commercial businesses was surveyed...Ch. 11 - Table 11.42 contains information from a survey...Ch. 11 - A recent debate about where in the United States...Ch. 11 - Car manufacturers are interested in whether there...Ch. 11 - College students may be interested in whether or...Ch. 11 - Some travel agents claim that honeymoon hot spots...Ch. 11 - A manager of a sports club keeps information...Ch. 11 - A major food manufacturer is concerned that the...Ch. 11 - According to Dan Lenard, an independent insurance...Ch. 11 - Suppose that 600 thirty-year-olds were surveyed to...Ch. 11 - The number of degrees of freedom for a test of...Ch. 11 - The test for independence uses tables of observed...Ch. 11 - The test to use when determining if the college or...Ch. 11 - In a test of independence, the expected number is...Ch. 11 - An ice cream maker performs a nationwide survey...Ch. 11 - Table 11.52 provides a recent survey of the...Ch. 11 - A 2013 poll in California surveyed people about...Ch. 11 - A psychologist is interested in testing whether...Ch. 11 - Do men and women select different breakfasts? The...Ch. 11 - A fisherman is interested in whether the...Ch. 11 - In 2007, the United States had 1.5 million...Ch. 11 - When looking at energy consumption, we are often...Ch. 11 - The Insurance Institute for Highway Safety...Ch. 11 - Is there a difference between the distribution of...Ch. 11 - If df = 2, the chi-square distribution has a shape...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - Test of a Single Variance Use the following...Ch. 11 - A plant manager is concerned her equipment may...Ch. 11 - Consumers may be interested in whether the cost of...Ch. 11 - Isabella, an accomplished Bay to Breakers runner,...Ch. 11 - Airline companies are interested in the...Ch. 11 - The number of births per woman in China is 1.6...Ch. 11 - According to an avid aquarist, the average number...Ch. 11 - The manager of "Frenchies" is concerned that...Ch. 11 - You want to buy a specific computer. A sales...Ch. 11 - A company packages apples by weight. One of the...Ch. 11 - a. Explain why a goodness-of-fit test and a test...
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