Intro Stats, Books a la Carte Edition (5th Edition)
5th Edition
ISBN: 9780134210285
Author: Richard D. De Veaux, Paul Velleman, David E. Bock
Publisher: PEARSON
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Chapter 15.6, Problem 6JC
To determine
Define the parameter.
State null and alternative hypothesis.
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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 15 Solutions
Intro Stats, Books a la Carte Edition (5th Edition)
Ch. 15.2 - A research team wants to know if aspirin helps to...Ch. 15.2 - An allergy drug has been tested and found to give...Ch. 15.2 - The new drug is tested and the P-value is 0.0001....Ch. 15.4 - The research team that wants to know if aspirin...Ch. 15.4 - Prob. 5JCCh. 15.6 - Prob. 6JCCh. 15.6 - Prob. 7JCCh. 15.6 - Prob. 8JCCh. 15 - Better than aspirin? A very large study showed...Ch. 15 - Psychic A friend of yours claims to be psychic....
Ch. 15 - Parameters and hypotheses For each of the...Ch. 15 - Hypotheses and parameters As in Exercise 3, for...Ch. 15 - SECTION 15.2 5. Better than aspirin again?...Ch. 15 - GRE performance A test preparation company claims...Ch. 15 - SECTION 15.3 7. Hispanic origin According to the...Ch. 15 - Empty houses According to the 2010 Census, 11.4%...Ch. 15 - Prob. 9ECh. 15 - Marriage In 1960, census results indicated that...Ch. 15 - Prob. 11ECh. 15 - Golf balls The United States Golf Association...Ch. 15 - Prob. 13ECh. 15 - Prob. 14ECh. 15 - CHAPTER EXERCISES 15. Hypotheses Write the null...Ch. 15 - More hypotheses Write the null and alternative...Ch. 15 - Negatives After the political ad campaign...Ch. 15 - Dice The seller of a loaded die claims that it...Ch. 15 - Prob. 19ECh. 15 - Cars A survey investigating whether the proportion...Ch. 15 - Prob. 21ECh. 15 - Prob. 22ECh. 15 - Prob. 23ECh. 15 - Prob. 24ECh. 15 - Dowsing In a rural area, only about 30% of the...Ch. 15 - Abnormalities In the 1980s, it was generally...Ch. 15 - Prob. 27ECh. 15 - Educated mothers The National Center for Education...Ch. 15 - Contributions, please II We learned in Chapter 13,...Ch. 15 - Prob. 30ECh. 15 - Prob. 31ECh. 15 - Scratch and dent An appliance manufacturer...Ch. 15 - Prob. 33ECh. 15 - Football 2016 During the first 15 weeks of the...Ch. 15 - WebZine A magazine is considering the launch of an...Ch. 15 - Seeds A garden center wants to store leftover...Ch. 15 - Prob. 37ECh. 15 - Prob. 38ECh. 15 - Prob. 39ECh. 15 - Acid rain A study of the effects of acid rain on...Ch. 15 - Prob. 41ECh. 15 - TV ads A start-up company is about to market a new...Ch. 15 - Prob. 43ECh. 15 - Prob. 44ECh. 15 - Normal temperature again From the measurements of...Ch. 15 - Hot dogs again The nutrition lab in Chapter 14,...Ch. 15 - Prob. 47ECh. 15 - Computer lab fees again The technology committee...Ch. 15 - More Ruffles Recall from Chapter 14, Exercise 55...Ch. 15 - More Doritos We saw in Chapter 14, Exercise 56...Ch. 15 - Prob. 51ECh. 15 - Prob. 52ECh. 15 - Prob. 53ECh. 15 - Prob. 54ECh. 15 - Maze Here are the data from the researcher...Ch. 15 - Facebook friends According to...Ch. 15 - Maze revisited A student resampled the Maze times...Ch. 15 - Prob. 58ECh. 15 - Prob. 59ECh. 15 - Prob. 60E
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- Question 1. Your manager asks you to explain why the Black-Scholes model may be inappro- priate for pricing options in practice. Give one reason that would substantiate this claim? Question 2. We consider stock #1 and stock #2 in the model of Problem 2. Your manager asks you to pick only one of them to invest in based on the model provided. Which one do you choose and why ? Question 3. Let (St) to be an asset modeled by the Black-Scholes SDE. Let Ft be the price at time t of a European put with maturity T and strike price K. Then, the discounted option price process (ert Ft) t20 is a martingale. True or False? (Explain your answer.) Question 4. You are considering pricing an American put option using a Black-Scholes model for the underlying stock. An explicit formula for the price doesn't exist. In just a few words (no more than 2 sentences), explain how you would proceed to price it. Question 5. We model a short rate with a Ho-Lee model drt = ln(1+t) dt +2dWt. Then the interest rate…arrow_forwardIn 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.)arrow_forwardThe managing director of a consulting group has the accompanying monthly data on total overhead costs and professional labor hours to bill to clients. Complete parts a through c. Question content area bottom Part 1 a. Develop a simple linear regression model between billable hours and overhead costs. Overhead Costsequals=212495.2212495.2plus+left parenthesis 42.4857 right parenthesis42.485742.4857times×Billable Hours (Round the constant to one decimal place as needed. Round the coefficient to four decimal places as needed. Do not include the $ symbol in your answers.) Part 2 b. Interpret the coefficients of your regression model. Specifically, what does the fixed component of the model mean to the consulting firm? Interpret the fixed term, b 0b0, if appropriate. Choose the correct answer below. A. The value of b 0b0 is the predicted billable hours for an overhead cost of 0 dollars. B. It is not appropriate to interpret b 0b0, because its value…arrow_forward
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