MSF 501Math with Fin Applications
Exam 2, Fall 2011 KEY
Use your notes and text. Use your calculator.
Do NOT use a laptop.
Good luck!
Name: _
Multiple Choice (5 points each). CIRCLE the letter of the BEST answer.
1(10). What is the integrating factor you
MSF 501 Math with Financial Apps
Lecture 9
Getting to Know and Love the Black Scholes Model
More on Put-Call Parity
I.
No-Arbitrage Pricing with Put-Call Parity
Several students have asked for clarification of my in-class comments on
put-call parity. Firs
MSF 501Math with Fin Apps
Exam 2, Fall 2009
!
KEY
1(6): (10 points) Consider the following differential equation: dg t rX 1 g t dt .
2
a.
Use the appropriate integrating factor to solve the differential equation for g T , given an initial (t = 0)
value of
MSF 501Math with Fin Apps
Exam 2, Fall 2010
Use your notes and text. Use your calculator.
Do NOT use a laptop.
Good luck!
Solutions
Multiple Choice (5 points each). CIRCLE the letter of the BEST answer.
1.
An investment manager has $100 million under mana
Differential Equations INTEGRATING FACTOR METHOD
Graham S McDonald A Tutorial Module for learning to solve 1st order linear differential equations
q Table of contents q Begin Tutorial
c 2004 g.s.mcdonald@salford.ac.uk
Table of contents
1. 2. 3. 4. 5. 6. T
MSF 502 Lecture 3
Fall 2012
What this course is about:
Estimation and Hypothesis Testing
Estimation Theory
I.
Point Estimation
A point estimate of a population parameter is a single value of a statistic of that
parameter. Examples of point estimate includ
MSF 502 Lecture 4
Fall 2012
Introducing the Simple Linear Regression Model:
Dont worry, things will get weirder!
Introduction to Econometrics
I.
What is Econometrics?
The literal meaning of econometrics is measurement in economics. But from a
Finance pers
MSF 502 Lecture 5
Fall 2012
Brooks Chapter 3, with a Little Chapter 4:
OLS Hypothesis Testing and Model Diagnostics
Testing Simultaneous Hypotheses:
The Wald F-Test
I.
Joint (Linear) Hypotheses
In Lecture 4 we examined the t-test for individual regression
MSF 502 Lecture 6
Fall 2012
Brooks Chapter 4:
Difficulties with X-Variables
Real-World X-Variable Hassles
I.
Digression Estimating Covariance Matrices
2
Sample variance is s1
x1,t x1 2
T 1
x T x
2
2
1,t
1
T 1
.
Proof for numerator:
x 2
x1,t x1 2 x12,
MSF 502 Lecture 7
Fall 2012
Brooks Chapter 4:
The Problems and Remedies for Unruly Residual Errors
Testing for Normality of Residuals
I.
Residuals Gone Wild!
A Reminder of Classical OLS Assumptions:
1. The true, or population regression equation is linear
MSF 501 Math with Financial Applications
Introduction to Derivatives
Basics of Options
Fun with Multivariate Its Lemma
I.
Multiplication Rule
Suppose we have two Geometric Brownian Motions that are multiplied
together. The stochastic process of this produ
MSF 501 Lecture 6, page 1
Weeks 12 & 13:
Introduction to Continuous-Time Finance
Its Lemma
I.
Most Securities are Derivatives
The value of any security ultimately comes down to present value
problem. We must answer, What is the present value of future cas
MSF 501 Lecture 5, page 1
Week 10
Some Numerical Methods & Fun with Integration
Numerical Differentiation
I.
Motivation
Sometimes we have a function that is just too complex to find a closedform solution for its derivative, or if we do, it is impossibly u
The CAPM Model
A. Portfolio Consisting of A riskfree asset and a risky asset Let Let Let Let Let = the rate of return on the portfolio. = the rate of return on the risk free asset. = the rate of return on the risky asset. = the proportion of the portfolio
8.4. CHOLESKY FACTORIZATION
31
8.4
Cholesky Factorization
Let A be Hermitian, positive denite (A = A and x Ax 0, x), then by exploring symmetry GE 1 costs 3 n3 and there is no need for pivoting. A = R R R = A A + A = R R A = R R
A
1 2
(8.70) (8.71)
= O()
Linear Algebra Notes
Chapter 8
EIGENVALUES AND EIGENVECTORS
In previous chapters, we have seen a few uses for the eigenvalues and eigenvectors
of a matrix, but I have not explained how to nd the eigenvectors. In this chapter,
we will learn a general metho
Chapter 10
Eigenvalues and Singular
Values
This chapter is about eigenvalues and singular values of matrices. Computational
algorithms and sensitivity to perturbations are both discussed.
10.1
Eigenvalue and Singular Value Decompositions
An eigenvalue and
Math with Fin Apps (MSF 501)
Exam 2 Practice Problems (Solutions)
1.
A covariance matrix of 3 stock portfolios is shown below.
0.04 0.048 0.04
0.048 .09 0.072
0.04 0.072 0.16
The eigenvalues and eigenvectors of the matrix are
1
0.224
3
0.011
E1
0.31
Math with Financial Applications (MSF 501
Final Exam Practice Set
1.
Intel, Corp. is selling at $20 per share and pays no dividends. It historical standard deviation of returns is
40%. The risk-free T-bills have a continuously compounded rate of 2%. Use t
Please read:
a personal appeal from
Wikipedia founder Jimmy Wales
Read now
Integration by parts
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Topics in calculus
Fundamental theorem
Limits of functions
Continuity
Mean value theorem
Roll
MSF 501 Week 5, page 1
Math with Financial Applications
Optimization with Matrix Calculus
Portfolio Management and the CAPM
Second Order Conditions for
Constrained Optimization
Problems
I.
Necessary and Sufficient Conditions
Necessary Conditiona prerequi
MSF 501 Lecture 4, page 1
Week 8:
Eigen Values and Principle Components Analysis
A Matrix Algebra Digression:
Whats up with Eigenvalues and
Eigenvectors?
I.
Why Do Finance People Care about Eigenvalues/vectors?
We are going to cover eigenvalues and eigenv
Foundations of Finance: The Capital Asset Pricing Model (CAPM)
Prof. Alex Shapiro
Lecture Notes 9
The Capital Asset Pricing Model (CAPM)
I. II. III. IV. V. VI. Readings and Suggested Practice Problems Introduction: from Assumptions to Implications The Mar