EBK MATLAB: AN INTRODUCTION WITH APPLIC
5th Edition
ISBN: 8220102007642
Author: GILAT
Publisher: YUZU
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Chapter 1, Problem 36P
Newton’s law of cooling gives the temperature T(t) of an object at time tin terms of T0, its temperature at t = 0, and the temperature of the surroundings.
A police officer arrives at a crime scene in a hotel room at 9:18 PM. where he finds a dead body. He immediately measures the body’s temperature and finds it to be 79.5°F. Exactly one hour later he measures the temperature again and finds it to be 78.0°F. Determine the time of death, assuming that victim body temperature was normal (98.6°F) prior to death and that the room temperature was constant at 69°F.
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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 1 Solutions
EBK MATLAB: AN INTRODUCTION WITH APPLIC
Ch. 1 - Prob. 1PCh. 1 - Calculate: 4125.22e5100.53 1323+ln5008Ch. 1 - Calculate: (a) 14.836.3213+52 (b)...Ch. 1 - Calculate: 24.5+64/3.52+8.312.5376.428/15...Ch. 1 - Calculate: a. cos7x9+tan715sin15 b....Ch. 1 - Define the variable xas x = 6.7, then evaluate: a....Ch. 1 - Define the variable tas t = 3.2, then evaluate:...Ch. 1 - Define the variables x and y as x = 5.1 and y =...Ch. 1 - Define the variables a, b, c, and d as:...Ch. 1 - Prob. 10P
Ch. 1 - Prob. 11PCh. 1 - Two trigonometric identities are given by:...Ch. 1 - Two trigonometric identities are given by:...Ch. 1 - Define two variables: alpha=/6 , and beta=3/8 ....Ch. 1 - Given: xsinaxdx=sinaxa2xcosaxa . Use MATLAB to...Ch. 1 - In the triangle shown a=5.3in.,=42,andb=6 in....Ch. 1 - In the triangle shown a = 5 in., b = 7 in., and =...Ch. 1 - In the ice cream cone shown, L = 4 in. and =35 ....Ch. 1 - Prob. 19PCh. 1 - The parametric equations of a line in space are:...Ch. 1 - The circumference of an ellipse can be...Ch. 1 - 315 people have to be transported using buses that...Ch. 1 - 739 apples are to be packed and shipped such that...Ch. 1 - Assign the number 316,501.673 to a variable, and...Ch. 1 - Prob. 25PCh. 1 - Prob. 26PCh. 1 - Prob. 27PCh. 1 - Prob. 28PCh. 1 - Prob. 29PCh. 1 - Prob. 30PCh. 1 - Radioactive decay of carbon-14 is used for...Ch. 1 - The greatest common divisor is the largest...Ch. 1 - The Moment Magnitude Scale (MMS), denoted Mw,...Ch. 1 - Prob. 34PCh. 1 - Prob. 35PCh. 1 - Newton’s law of cooling gives the temperature T(t)...Ch. 1 - The stress intensity factor K predicts the stress...Ch. 1 - Prob. 38PCh. 1 - Use the Help Window to find a display format that...Ch. 1 - Prob. 40P
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