Differential Equations with Boundary-Value Problems (MindTap Course List)
9th Edition
ISBN: 9781305965799
Author: Dennis G. Zill
Publisher: Cengage Learning
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Chapter 6.4, Problem 47E
To determine
The six Legendre polynomials after P1(x) using the recurrence relation (k+1)Pk+1(x)−(2k+1)xPk(x)+kPk−1(x)=0, where P0(x)=1, P1(x)=x.
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Construct a know-show table for each statement below that appears to be true.
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 6 Solutions
Differential Equations with Boundary-Value Problems (MindTap Course List)
Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - Prob. 3ECh. 6.1 - Prob. 4ECh. 6.1 - Prob. 5ECh. 6.1 - Prob. 6ECh. 6.1 - Prob. 7ECh. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - Prob. 10E
Ch. 6.1 - Prob. 11ECh. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - Prob. 13ECh. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - Prob. 16ECh. 6.1 - In Problems 17 and 18 use an appropriate series in...Ch. 6.1 - Prob. 18ECh. 6.1 - Prob. 19ECh. 6.1 - Prob. 20ECh. 6.1 - Prob. 21ECh. 6.1 - Prob. 22ECh. 6.1 - Prob. 23ECh. 6.1 - Prob. 24ECh. 6.1 - Prob. 25ECh. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - Prob. 28ECh. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - Prob. 34ECh. 6.1 - Prob. 35ECh. 6.1 - Prob. 36ECh. 6.1 - Prob. 37ECh. 6.1 - In Problems 3538 proceed as in Example 4 and find...Ch. 6.1 - Prob. 39ECh. 6.1 - Prob. 40ECh. 6.2 - In Problems 1 and 2 without actually solving the...Ch. 6.2 - Prob. 2ECh. 6.2 - In Problems 3–6 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - Prob. 7ECh. 6.2 - Prob. 8ECh. 6.2 - Prob. 9ECh. 6.2 - Prob. 10ECh. 6.2 - Prob. 11ECh. 6.2 - Prob. 12ECh. 6.2 - Prob. 13ECh. 6.2 - Prob. 14ECh. 6.2 - Prob. 15ECh. 6.2 - Prob. 16ECh. 6.2 - Prob. 17ECh. 6.2 - Prob. 18ECh. 6.2 - Prob. 19ECh. 6.2 - Prob. 20ECh. 6.2 - Prob. 21ECh. 6.2 - Prob. 22ECh. 6.2 - Prob. 23ECh. 6.2 - In Problems 23 and 24 use the procedure in Example...Ch. 6.2 - Without actually solving the differential equation...Ch. 6.2 - How can the power series method be used to solve...Ch. 6.2 - Prob. 27ECh. 6.2 - Prob. 28ECh. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 3ECh. 6.3 - Prob. 4ECh. 6.3 - Prob. 5ECh. 6.3 - Prob. 6ECh. 6.3 - Prob. 7ECh. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 9ECh. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 11ECh. 6.3 - Prob. 12ECh. 6.3 - In Problems 13 and 14, x = 0 is a regular singular...Ch. 6.3 - Prob. 14ECh. 6.3 - In Problems 1524, x = 0 is a regular singular...Ch. 6.3 - Prob. 16ECh. 6.3 - Prob. 17ECh. 6.3 - Prob. 18ECh. 6.3 - Prob. 19ECh. 6.3 - Prob. 20ECh. 6.3 - Prob. 21ECh. 6.3 - Prob. 22ECh. 6.3 - Prob. 23ECh. 6.3 - Prob. 24ECh. 6.3 - Prob. 25ECh. 6.3 - Prob. 26ECh. 6.3 - Prob. 27ECh. 6.3 - Prob. 28ECh. 6.3 - Prob. 29ECh. 6.3 - Prob. 30ECh. 6.3 - Prob. 31ECh. 6.3 - Prob. 32ECh. 6.3 - Prob. 33ECh. 6.3 - Prob. 35ECh. 6.3 - Prob. 36ECh. 6.3 - Prob. 37ECh. 6.4 - Prob. 1ECh. 6.4 - Prob. 2ECh. 6.4 - Prob. 3ECh. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Prob. 7ECh. 6.4 - Prob. 8ECh. 6.4 - Prob. 9ECh. 6.4 - Prob. 10ECh. 6.4 - In Problems 11 and 12 use the indicated change of...Ch. 6.4 - In Problems 11 and 12 use the indicated change of...Ch. 6.4 - Prob. 13ECh. 6.4 - Prob. 14ECh. 6.4 - Prob. 15ECh. 6.4 - Prob. 16ECh. 6.4 - Prob. 17ECh. 6.4 - Prob. 18ECh. 6.4 - Prob. 19ECh. 6.4 - In Problems 1320 use (20) to find the general...Ch. 6.4 - Prob. 21ECh. 6.4 - Assume that b in equation (20) can be pure...Ch. 6.4 - Prob. 23ECh. 6.4 - Prob. 24ECh. 6.4 - In Problems 2326 first use (20) to express the...Ch. 6.4 - In Problems 2326 first use (20) to express the...Ch. 6.4 - Prob. 27ECh. 6.4 - Prob. 28ECh. 6.4 - Prob. 29ECh. 6.4 - Prob. 30ECh. 6.4 - Prob. 31ECh. 6.4 - Use the recurrence relation in Problem 28 along...Ch. 6.4 - Prob. 33ECh. 6.4 - Prob. 34ECh. 6.4 - Use the change of variables s=2kmet/2 to show that...Ch. 6.4 - Prob. 36ECh. 6.4 - Use the result in parts (a) and (b) of Problem 36...Ch. 6.4 - Prob. 38ECh. 6.4 - Prob. 39ECh. 6.4 - (a) Use the explicit solutions y1(x) and y2(x) of...Ch. 6.4 - Prob. 47ECh. 6.4 - Prob. 48ECh. 6.4 - Find the first three positive values of for which...Ch. 6.4 - The differential equation y 2xy + 2y = 0 is known...Ch. 6.4 - (a) When = n is a nonnegative integer, Hermites...Ch. 6.4 - Prob. 55ECh. 6.4 - Prob. 56ECh. 6 - In Problems 1 and 2 answer true or false without...Ch. 6 - Prob. 2RECh. 6 - Prob. 3RECh. 6 - Prob. 4RECh. 6 - Prob. 5RECh. 6 - Prob. 6RECh. 6 - Prob. 7RECh. 6 - Prob. 8RECh. 6 - Prob. 9RECh. 6 - Prob. 10RECh. 6 - Prob. 11RECh. 6 - Prob. 12RECh. 6 - Prob. 13RECh. 6 - Prob. 14RECh. 6 - Prob. 15RECh. 6 - Prob. 16RECh. 6 - Prob. 17RECh. 6 - Prob. 18RECh. 6 - Prob. 19RECh. 6 - Prob. 20RECh. 6 - Prob. 21RECh. 6 - Prob. 22RECh. 6 - Prob. 23RECh. 6 - Prob. 24RECh. 6 - Prob. 25RECh. 6 - Express the general solution of the given...Ch. 6 - Prob. 27RECh. 6 - Prob. 28RE
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