Fin 500J: Suggested Solutions to Homework 3
Yajun Wang
Olin Business School
Summer 2009
Problem 1. Consider the problem of maximizing f (x, y, z) = xyz + z, subject to the
constraints x2 + y 2 + z 6, x 0, y 0, z 0.
(1) Write out a complete set of rst orde
MATHEMATICAL FOUNDATIONS FOR FINANCE
Quadratic Taylor Series Approximation
Philip H. Dybvig
Washington University
Saint Louis, Missouri
Copyright c Philip H. Dybvig 2010
Background
It is often useful to approximate a function locally using derivatives. Fo
MATHEMATICAL FOUNDATIONS FOR FINANCE
Introduction to Probability and Statistics: Part 2
Philip H. Dybvig
Washington University
Saint Louis, Missouri
Copyright c Philip H. Dybvig 2011
Continuous random variables
Discrete random variables take on isolated v
Supplemental notes: Kuhn-Tucker rst-order conditions
P. Dybvig
Minimization problem (like in the slides):
Choose x N to
minimize f (x)
subject to (i E)gi (x) = 0, and
(i I)gi (x) 0.
x = (x1 , ., xN ) is a vector of choice variables.
f (x) is the scalar-va
MATHEMATICAL FOUNDATIONS FOR FINANCE
Eigenvalues and Eigenvectors
Philip H. Dybvig
Washington University
Saint Louis, Missouri
Copyright c Philip H. Dybvig 2010
Background
Many problems look simple in a univariate setting but complicated in a multivariate
Suggested Solutions to Homework 6
Yajun Wang
Olin Business School
Summer 2009
Problem 1. (i) Suppose that the p.d.f. of a certain random variable X has the following form:
f (x) =
cx 0 < x < 4
0
otherwise,
where c is a given constant. Determine the value
MATHEMATICAL FOUNDATIONS FOR FINANCE
Calculus Review
Philip H. Dybvig
Washington University
Saint Louis, Missouri
Copyright c Philip H. Dybvig 2011
Some underyling ideas
Functions: A lot of things in the world can be described by functions. For
example, t
Fin 500J Solutions to Homework 1
Yajun Wang
Olin Business School
Problem 1. For
A=
3 6
1
, B=
2 1
0
3
2 4 5
, C= 0 3 0
1 0 1
2
0 1 1 1
compute
(1)(A + AT )B
(2) Determinant of C
and verify your answers using Matlab.
Solution :
(1) (A + AT )B =
(2)|C| = 2
Problem Set 2: Probability using Calculus
FIN 500J Mathematical Foundations for Finance
P. Dybvig
Prepare a hard copy of your answers to submit in class. It is okay to work
with others on the problems, but do the write-up and any computer work
yourself. S
Problem Set 4, selected answers: Optimization and Kuhn-Tucker Conditions
FIN 500J Mathematical Foundations for Finance
P. Dybvig
1. Optimal investing: one period. An investor has $100,000 in wealth available for investment to retirement one year from now.
FIN 550J Exam Answers, October 21, 2010
Phil Dybvig
1. PROBABILITIES Let the stock price S take on values uniformly distributed on [40, 60]. A digital option on the stock has payo
(1)
D=
1 for S > 55
0 otherwise
a. What is the density (pdf) of the stock p
Problem Set 1: Probability
FIN 500J Mathematical Foundations for Finance
P. Dybvig
Here are some answers to help you see whether you are on the right track.
The answers to questions 2 and 3 are less detailed than what you need to
give in your write-ups.
1
FIN 550J Exam, October 20, 2011
Phil Dybvig
This is a closed-book examination. You may not use texts, notes, a crib sheet,
calculator, cell phone, listening device, or any other electronics. Answer all
questions as directed. Make sure each answer is clear
Problem Set 3, selected answers: Linear Algebra
FIN 500J Mathematical Foundations for Finance
P. Dybvig
1. Consider the matrices
B=
1 0
1 1
and C =
2 0 1
0 1 0
.
1. Compute the following.
A. B T (the transpose of B)
B. BC (the product of B and C)
BC =
2 0
Problem 4. For each of the following functions, nd the critical points and classify
these as local max, local min, saddle point or cant tell:
(1)xy 2 + x3 y xy,
(2)x2 + 6xy + y 2 3yz + 4z 2 + 6x + 17y 2z.
Solution : (1) First order conditions are:
(i)y 2