Introduction To Probability And Statistics
15th Edition
ISBN: 9781337554428
Author: Mendenhall, William.
Publisher: Cengage Learning,
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Chapter 2.3, Problem 3E
To determine
To approximate the value of s using 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 2 Solutions
Introduction To Probability And Statistics
Ch. 2.1 - Measures of CenterFor the data sets in Exercises...Ch. 2.1 - Prob. 2ECh. 2.1 - Prob. 3ECh. 2.1 - Prob. 4ECh. 2.1 - Prob. 5ECh. 2.1 - Prob. 6ECh. 2.1 - Prob. 7ECh. 2.1 - Prob. 8ECh. 2.1 - Prob. 9ECh. 2.1 - Prob. 10E
Ch. 2.1 - Prob. 11ECh. 2.1 - Prob. 12ECh. 2.1 - Prob. 13ECh. 2.1 - Prob. 14ECh. 2.1 - Prob. 15ECh. 2.1 - Prob. 16ECh. 2.1 - Prob. 17ECh. 2.1 - Fortune 500 Revenues Ten of the...Ch. 2.1 - Prob. 19ECh. 2.1 - Prob. 20ECh. 2.1 - Prob. 21ECh. 2.1 - Prob. 22ECh. 2.2 - Calculating the Standard Deviation I For the data...Ch. 2.2 - Calculating the Standard Deviation I For the data...Ch. 2.2 - Calculating the Standard Deviation I For the data...Ch. 2.2 - Prob. 4ECh. 2.2 - Prob. 5ECh. 2.2 - Prob. 6ECh. 2.2 - Prob. 7ECh. 2.2 - Prob. 8ECh. 2.2 - For the data sets in Exercises 79, find the range,...Ch. 2.2 - Prob. 10ECh. 2.2 - Prob. 11ECh. 2.2 - Sleep and the College Student A group of 10...Ch. 2.2 - Gas Mileage The miles per gallon (mpg) foreach of...Ch. 2.2 - Polluted Seawater Petroleum pollution in seas...Ch. 2.2 - Prob. 15ECh. 2.3 - Approximating the Standard Deviation For the data...Ch. 2.3 - Approximating the Standard Deviation For the data...Ch. 2.3 - Prob. 3ECh. 2.3 - Prob. 4ECh. 2.3 - Prob. 5ECh. 2.3 - Data Set I A distribution of measurements is...Ch. 2.3 - Prob. 7ECh. 2.3 - Data Set I A distribution of measurements is...Ch. 2.3 - Prob. 9ECh. 2.3 - Prob. 10ECh. 2.3 - Prob. 11ECh. 2.3 - Prob. 12ECh. 2.3 - Data Set II A distribution of measurements has a...Ch. 2.3 - Prob. 14ECh. 2.3 - Prob. 15ECh. 2.3 - Prob. 16ECh. 2.3 - Prob. 17ECh. 2.3 - Prob. 18ECh. 2.3 - Prob. 19ECh. 2.3 - Breathing Rates Breathing rates for humans canbe...Ch. 2.3 - Prob. 21ECh. 2.3 - Prob. 22ECh. 2.3 - Prob. 23ECh. 2.3 - Prob. 24ECh. 2.3 - Prob. 25ECh. 2.3 - Prob. 26ECh. 2.3 - Prob. 27ECh. 2.3 - Prob. 28ECh. 2.3 - Prob. 29ECh. 2.3 - Prob. 30ECh. 2.3 - Prob. 31ECh. 2.3 - TV Commercials The mean duration of...Ch. 2.4 - z-ScoresFor the data sets in Exercises 13, find...Ch. 2.4 - z-ScoresFor the data sets in Exercises 13, find...Ch. 2.4 - z-ScoresFor the data sets in Exercises 13, find...Ch. 2.4 - Median and Quartiles For the data in Exercises 46,...Ch. 2.4 - Median and Quartiles For the data in Exercises 46,...Ch. 2.4 - Median and Quartiles For the data in Exercises 46,...Ch. 2.4 - -Number Summary and the Box Plot For each of the...Ch. 2.4 - Five-Number Summary and the Box Plot For each of...Ch. 2.4 - Five-Number Summary and the Box Plot For each of...Ch. 2.4 - Prob. 10ECh. 2.4 - z-Scores vs. Box Plots For the data sets in...Ch. 2.4 - Percentiles What are the percentiles in Exercises...Ch. 2.4 - Percentiles What are the percentiles in Exercises...Ch. 2.4 - Percentiles What are the percentiles in Exercises...Ch. 2.4 - Percentiles What are the percentiles in Exercises...Ch. 2.4 - For the data in Exercises 1617, find the sample...Ch. 2.4 - For the data in Exercises 1617, find the sample...Ch. 2.4 - €For the data in Exercises 1819, find the...Ch. 2.4 - For the data in Exercises 1819, find the...Ch. 2.4 - Prob. 20ECh. 2.4 - Prob. 21ECh. 2.4 - Presidential Vetoes The number of vetoesused by...Ch. 2.4 - Prob. 23ECh. 2.4 - Prob. 24ECh. 2.4 - Ages of Pennies Here are the ages of50 pennies,...Ch. 2.4 - Prob. 26ECh. 2 - Raisins The number of raisins in each of...Ch. 2 - Raisins, continued Refer to Exercise 1. a. Find...Ch. 2 - Prob. 3RWYLCh. 2 - Prob. 4RWYLCh. 2 - Prob. 5RWYLCh. 2 - Prob. 6RWYLCh. 2 - Prob. 7RWYLCh. 2 - Prob. 8RWYLCh. 2 - Prob. 9RWYLCh. 2 - Prob. 10RWYLCh. 2 - Prob. 11RWYLCh. 2 - Prob. 12RWYLCh. 2 - Prob. 13RWYLCh. 2 - Prob. 14RWYLCh. 2 - Prob. 15RWYLCh. 2 - Prob. 16RWYLCh. 2 - Prob. 17RWYLCh. 2 - Prob. 18RWYLCh. 2 - Prob. 19RWYLCh. 2 - Prob. 20RWYLCh. 2 - Prob. 21RWYLCh. 2 - Prob. 22RWYLCh. 2 - Prob. 23RWYLCh. 2 - Prob. 24RWYLCh. 2 - Prob. 25RWYL
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