EBK MATLAB: AN INTRODUCTION WITH APPLIC
6th Edition
ISBN: 8220103113663
Author: GILAT
Publisher: YUZU
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Chapter 5, Problem 7P
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In a network with 12 links, one of the links has failed. The failed link is randomlylocated. An electrical engineer tests the links one by one until the failed link is found.a. What is the probability that the engineer will find the failed link in the first test?b. What is the probability that the engineer will find the failed link in five tests?Note: You should assume that for Part b, the five tests are done consecutively.
Problem 3. Pricing a multi-stock option the Margrabe formula
The purpose of this problem is to price a swap option in a 2-stock model, similarly as
what we did in the example in the lectures. We consider a two-dimensional Brownian
motion given by W₁ = (W(¹), W(2)) on a probability space (Q, F,P). Two stock prices
are modeled by the following equations:
dX
=
dY₁ =
X₁ (rdt+
rdt+0₁dW!)
(²)),
Y₁ (rdt+dW+0zdW!"),
with Xo
xo and Yo =yo. This corresponds to the multi-stock model studied in class,
but with notation (X+, Y₁) instead of (S(1), S(2)). Given the model above, the measure
P is already the risk-neutral measure (Both stocks have rate of return r). We write
σ = 0₁+0%. We consider a swap option, which gives you the right, at time T, to
exchange one share of X for one share of Y. That is, the option has payoff
F=(Yr-XT).
(a) We first assume that r = 0 (for questions (a)-(f)). Write an explicit expression for
the process Xt.
Reminder before proceeding to question (b): Girsanov's theorem…
Problem 1. Multi-stock model
We consider a 2-stock model similar to the one studied in class. Namely, we consider
=
S(1)
S(2)
=
S(¹) exp (σ1B(1) + (M1 - 0/1 )
S(²) exp (02B(2) + (H₂-
M2
where (B(¹) ) +20 and (B(2) ) +≥o are two Brownian motions, with
t≥0
Cov (B(¹), B(2)) = p min{t, s}.
"
The purpose of this problem is to prove that there indeed exists a 2-dimensional Brownian
motion (W+)+20 (W(1), W(2))+20 such that
=
S(1)
S(2)
=
=
S(¹) exp (011W(¹) + (μ₁ - 01/1) t)
롱)
S(²) exp (021W (1) + 022W(2) + (112 - 03/01/12) t).
where σ11, 21, 22 are constants to be determined (as functions of σ1, σ2, p).
Hint: The constants will follow the formulas developed in the lectures.
(a) To show existence of (Ŵ+), first write the expression for both W. (¹) and W (2)
functions of (B(1), B(²)).
as
(b) Using the formulas obtained in (a), show that the process (WA) is actually a 2-
dimensional standard Brownian motion (i.e. show that each component is normal,
with mean 0, variance t, and that their…
Chapter 5 Solutions
EBK MATLAB: AN INTRODUCTION WITH APPLIC
Ch. 5 - Prob. 1PCh. 5 - Use the plot command to plot the function...Ch. 5 - Use the plot command to plot the function...Ch. 5 - Plot the function (x)=x2exand its derivative for...Ch. 5 - Make two separate plots of the function...Ch. 5 - Use the fplot command to plot the funtion...Ch. 5 - Plot the function f(x) = sin(2x) cos2(0.5x) and...Ch. 5 - The orbit of the planet Mercury around the sun can...Ch. 5 - Prob. 9PCh. 5 - The butterfly curve (Fay. T. H. “The Butterfly...
Ch. 5 - A plot of an astroid is shown in the figure on the...Ch. 5 - Make the plot of the astroid that is shown in the...Ch. 5 - Plot the function f(x)=x26x+7x38 in the domain 0x4...Ch. 5 - Plot the function f(x)=x+1x21 for 4x4 . Notice...Ch. 5 - Prob. 15PCh. 5 - The shape of the pretzel shown is given by the...Ch. 5 - Prob. 17PCh. 5 - Make a polar of the function: =n135.7 r=n for n=1,...Ch. 5 - Prob. 19PCh. 5 - Plot two ellipses is one figure (shown). The...Ch. 5 - The following data gives the height (in inches) of...Ch. 5 - The volage Vc t second after closing the switch in...Ch. 5 - The force F (in N) acting between a particle with...Ch. 5 - The curvilinear motion of a particle is defined by...Ch. 5 - The height and speed of a projectile shoot at a...Ch. 5 - The position x as a function of time of a particle...Ch. 5 - According to Planck’s law of black- body...Ch. 5 - A high-pass filter passes signals with frequencies...Ch. 5 - A resistor, R = 4O, and an inductor, L = 1.3 H,...Ch. 5 - In a typical tension test a dog-bone shaped...Ch. 5 - According to special relativity, a rod of length L...Ch. 5 - A railroad bumper is designed to slow down a...Ch. 5 - The ideal gas equation states that PVRT=n, where P...Ch. 5 - Two sound waves of slightly different frequencies...Ch. 5 - Consider the diode circuit shown in the figure....Ch. 5 - A monochromatic light that passes through a double...Ch. 5 - A simply supported beam is loaded as shown. The...Ch. 5 - Biological oxygen demand (BOD) is a measure of the...Ch. 5 - The shape of a asymmetric four-digitseries NACA...Ch. 5 - The Taylor series expansion for sin2x is:...
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