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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Textbook Question
Chapter 19.3, Problem 1JC
Why do we need the control group?
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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 19 Solutions
Intro Stats, Books a la Carte Edition (5th Edition)
Ch. 19.3 - Why do we need the control group?Ch. 19.3 - Prob. 2JCCh. 19.3 - Prob. 3JCCh. 19.3 - Prob. 4JCCh. 19.3 - Prob. 5JCCh. 19.3 - Prob. 6JCCh. 19.4 - Prob. 7JCCh. 19.4 - Prob. 8JCCh. 19.4 - Prob. 9JCCh. 19 - Human births If there is no seasonal effect on...
Ch. 19 - Bank cards At a major credit card bank, the...Ch. 19 - Prob. 3ECh. 19 - Prob. 4ECh. 19 - Customer ages An analyst at a local bank wonders...Ch. 19 - Bank cards, once more A market researcher working...Ch. 19 - Human births, last time For the data in Exercise...Ch. 19 - Prob. 8ECh. 19 - Iliad injuries 800 BCE Homers Iliad is an epic...Ch. 19 - Prob. 10ECh. 19 - Prob. 11ECh. 19 - Prob. 12ECh. 19 - Dice After getting trounced by your little brother...Ch. 19 - MMs As noted in an earlier chapter, Mars Inc. says...Ch. 19 - Nuts A company says its premium mixture of nuts...Ch. 19 - Prob. 16ECh. 19 - NYPD and race Census data for New York City...Ch. 19 - Violence against women In its study When Men...Ch. 19 - Fruit flies Offspring of certain fruit flies may...Ch. 19 - Prob. 20ECh. 19 - Prob. 21ECh. 19 - Lottery numbers The fairness of the South African...Ch. 19 - Prob. 23ECh. 19 - Prob. 24ECh. 19 - Childbirth, part 2 In Exercise 23, the table shows...Ch. 19 - Prob. 26ECh. 19 - Prob. 27ECh. 19 - Prob. 28ECh. 19 - Prob. 29ECh. 19 - Does your doctor know? (part 4) In Exercises 24,...Ch. 19 - Prob. 31ECh. 19 - Prob. 32ECh. 19 - Prob. 33ECh. 19 - NYPD The table below shows the rank attained by...Ch. 19 - Prob. 35ECh. 19 - NYPD again Examine and comment on this table of...Ch. 19 - Cranberry juice Its common folk wisdom that...Ch. 19 - Prob. 38ECh. 19 - Montana A poll conducted by the University of...Ch. 19 - Fish diet Medical researchers followed 6272...Ch. 19 - Prob. 41ECh. 19 - Working parents In April 2009, Gallup published...Ch. 19 - Grades Two different professors teach an...Ch. 19 - Full moon Some people believe that a full moon...Ch. 19 - Grades, again In some situations where the...Ch. 19 - Full moon, next phase In Exercise 44, you found...Ch. 19 - Racial steering A subtle form of racial...Ch. 19 - Survival on the Titanic Newspaper headlines at the...Ch. 19 - Prob. 49ECh. 19 - Education by age Use the survey results in the...
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