MATLAB: An Introduction with Applications, 6th Edition: An Introduction with Applications
6th Edition
ISBN: 9781119385134
Author: Amos Gilat
Publisher: WILEY
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Chapter 4, Problem 15P
A person at point A spots a child in trouble at point B across the river. The person can run at a speed of 8.6 ft/s and can swim at a speed of 3.9 ft/s. In order a to reach he child in the shortest time the person runs to point C and then swims to point B. as shown. Write a MATLAB program that determines the distance x w point C that minimizes the time the person can reach the child. In the program define a
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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 4 Solutions
MATLAB: An Introduction with Applications, 6th Edition: An Introduction with Applications
Ch. 4 - Prob. 1PCh. 4 - The altitude, h, as a function of air pressure can...Ch. 4 - Write a MATLAB program that determines the radius,...Ch. 4 - Radioactive decay can be modeled by the equation...Ch. 4 - A fuel tank is made of a half cylinder (r =14 in.)...Ch. 4 - A 300-lb garage door is being opened by pulling on...Ch. 4 - Write a MATLAB program in a script file that...Ch. 4 - The reduction of the amount of medication in the...Ch. 4 - The value of a saving account, V, after t years is...Ch. 4 - The electricity supply cables of the three houses...
Ch. 4 - Early explorers often estimated altitude by...Ch. 4 - An isosceles triangle sign is designed to have a...Ch. 4 - The angle ? at which a viewer sees the picture on...Ch. 4 - A 12-ft (144-in.) wire is cut into eight pieces...Ch. 4 - A person at point A spots a child in trouble at...Ch. 4 - The maximum stress smax at the edge of a hole...Ch. 4 - The airplane shown is flying at a constant speed...Ch. 4 - The intrinsic electrical conductivity a of a...Ch. 4 - The pressure drop ?p in pascals (Pa) for a fluid...Ch. 4 - The net heat exchange by radiation from plate 1...Ch. 4 - The equation of a circle in a plane with radius R...Ch. 4 - A truss is a structure made of members joined at...Ch. 4 - A truss is a Structure made of members joined at...Ch. 4 - The graph of the function f(x)=ax3+bx2+cx+d passes...Ch. 4 - The graph of the function f(x)=ax4+bx3+cx2+dx+e...Ch. 4 - The surface of many airfoils can be described with...Ch. 4 - During a golf match, a certain number of points...Ch. 4 - The dissolution of copper sulfide in aqueous...Ch. 4 - The heat index HI, calculated from the air...Ch. 4 - The stress intensity factor K at a crack is given...
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