Financial Risk Management
and Asset Pricing
Week 1
Philip Maymin,
[email protected]
Any Questions?
Blackboard
Intermission
Financial Risk Management and Asset
Pricing: What are we talking about?
What is finance?
What is risk?
What does it mean to manage r

Digital
Integrated
Jan M. Rabaey
Anantha Chandrakasan
Circuits
Borivoje Nikolic
A Design
Perspective
The Inverter
July 30, 2002
Digital Integrated Circuits2nd
Inverter
The CMOS Inverter: A
First Glance V
DD
V in
V out
CL
Digital Integrated Circuits2nd
I

Digital
Integrated
Jan M. Rabaey
Anantha Chandrakasan
Circuits
Borivoje Nikolic
A Design
Perspective
The Devices
July 30, 2002
Digital Integrated Circuits2nd
Devices
Goal of this chapter
Present intuitive understanding of device
operation
Introduction o

ECE 429/529 HW 1
Assigned Jan. 25, Due Feb. 4
Prob. 1: For each of the following systems, prove whether the system is (i) causal, (ii) linear,
(iii) time-invariant, (iv) stable.
(a) y[n] = 2en x[n]
(b) y[n] = x[2n 1]
n1
k=0 x[k], n 0
(c) y[n] =
0
n<0
Prob

MATH 410: THE PERMUTED LU FACTORIZATION ALGORITHM
AKA THE DOOLITTLE DECOMPOSITION WITH PARTIAL PIVOTING
1. Overview
In class we described the permuted LU factorization algorithm. These additional notes will rst
briey describe the history of this algorithm

7.2.24:
(a): Denote the basis vectors by e1 and e2 , respectively. We have
L(e1 ) = e1 2e2 ,
L(e2 ) = 4e1 + 3e2 .
1 4
.
2
3
(c): Denote the basis vectors by v1 and v2 respectively. We have
So the matrix of L is
L(v1 ) =
3
1
= v1 + 2v2 ,
L(v2 ) =
5
5
= 5v2

5.4.1(a):
t3 =
=
t3 , t2 1
t3 , t
t3 , 1
1+
t+ 2 1 2 3
1, 1
t, t
t 3, t
1 3
1 4
t dt
t
1
+ 1
1
1 2
dt
t
1
1
dt
t+
dt
t2
1
3
1
3
+
1
1
(t5 t3 /3) dt 2
(t6
1
(t 1/3)+ 1
1
1
(t2 1/3)2 dt
(t3
1
1
t3 , t3 3 t
5
3
t3 5 t, t3 3 t
5
3/5t4 ) dt
3/5t)2 dt
3
t3

HOMEWORK ASSIGNMENT 5 - MATH 410
DUE MONDAY, FEBRUARY 25 AT THE BEGINNING OF CLASS
Tip: Use a calculator or computer for row-reduction and integration unless instructed otherwise.
Note: Use the standard inner product on Rn unless instructed otherwise.
3.1

3.1.20(a): Let f (x) = x, g(x) = 1 + x2 . We compute f, g , f , and g using the L2 inner
product on [0, 1]:
1
x(1 + x2 ) dx =
f, g =
0
3
,
4
1
x2 dx =
f =
0
1
3
,
3
(1 + x2 )2 dx =
g =
0
2 105
.
15
3.2.1: We nd the angles between the two vectors:
(b): (1,

HOMEWORK ASSIGNMENT 4 - MATH 410
DUE MONDAY, FEBRUARY 18 AT THE BEGINNING OF CLASS
Tip: Use a calculator or computer to row-reduce matrices unless instructed otherwise.
2.4.1: Decide whether each of the following is a basis of R2 . Justify your answers!
1

2.4.1: We decide whether each of the following is a basis of R2 .
1
1
(b):
,
is not a basis:
1
1
We form the matrix of column vectors and row-reduce.
1
1
1
1
1
0
1
0
Since the matrix has rank 1, the two vectors do not form a basis.
An easier way to do thi

HOMEWORK ASSIGNMENT 3 - MATH 410
DUE MONDAY, FEBRUARY 11 AT THE BEGINNING OF CLASS
Note: If you cannot make it to class for some reason, you may email me your homework, but I
prefer a hard copy. I encourage you to work with others, but the work you submit

2.2.2: We decide whether each of the following is a subspace of R3 .
t
(b): The set of all vectors of the form t where t R is a subspace:
0
Suppose v1 and v2 are arbitrary vectors in the set, and suppose c is a real number.
Then
t1
t2
v1 = t1 for some t

HOMEWORK ASSIGNMENT 2 - MATH 410
DUE FRIDAY, JANUARY 25 AT THE BEGINNING OF CLASS
Note: Write up your solutions neatly. If you cannot make it to class for some reason, you may
email me your homework, but I prefer a hard copy. You are encouraged to work wi

HOMEWORK ASSIGNMENT 1 - MATH 410
DUE FRIDAY, JANUARY 18 AT THE BEGINNING OF CLASS
Note: Please write up your solutions neatly. You are welcome to type your solutions, but you do
not have to. Turn in your homework at the beginning of class. If you cannot m

1.2.7: Consider the matrices
1
3
4 2 ,
0
6
1
A = 1
3
6 0
3
,
4 2 1
2
3
C = 3 4 .
1
2
B=
(a) 3A B does not make sense.
(b) AB does not make sense.
(c)
BA =
3
1
6
4
0
2
(d) (A + B)C does not make sense.
(e) A + BC does not make sense.
(f)
1
A + 2CB = 3
7
11

FINAL EXAM TOPICS
MATH 410
The exam will cover the entire semesters material.
It will be helpful to have a calculator which can row-reduce a matrix.
You may use a 5x7 notecard during the exam (both sides). This notecard should only have
equations and form

HOMEWORK ASSIGNMENT 0 - MATH 410
(1) Complete the survey I handed out in class today. (If you were not in class or misplaced the
survey, email me and Ill send it to you.)
(2) Turn in the survey to me in my oce, no later than January 23. You can either
(a)

CONCEPTS TO KNOW FOR EXAM 3
MATH 410
The exam will cover chapters 7, 8, and the end of chapter 5. Note that we did not cover sections
5.7, 7.3, 7.4, 7.5, 8.1, or 8.5. So you should understand sections 5.3-5.6, 7.1-7.2, 8.2-8.4, and 8.6.
For the most part,

CONCEPTS TO KNOW FOR EXAM 2
MATH 410
The exam will cover chapters 2, 3, and the beginning of chapter 5. Note that we did not cover
section 3.6. So you should understand sections 2.1-2.5, 3.1-3.5, and 5.1-5.2.
For the most part, if you can do all the homew

(1) This is a Gram-Schmidt problem. First, nd a basis for the vector space of all polynomials
of degree less than or equal to 3. The usual basis for this vector space is 1, x, x2 , x3 . Lets
name them:
p1 = 1,
p2 = x,
p3 = x2 ,
p4 = x3 .
Now we perform th

HW 8: MAKING CORRECTIONS TO EXAM 3
MATH 410
How to write up your work.
For each problem you choose to re-work, do re-work the entire problem from beginning to end. Carefully show all your work; justify your reasoning with explanations and calculations as

CONCEPTS TO KNOW FOR EXAM 1
MATH 410
The exam will cover chapter 1. Note that we did not cover sections 1.6 or 1.7. (Well get to
section 1.6 later; its pretty short.) So you should understand sections 1.1-1.5 and 1.8-1.9.
For the most part, if you can do

1. We nd the rank of A and then decide how many solutions there are to the system Ax = b.
2 3 4
3
(a) A = 0 4 5 , so rank A = 3. b = 5 , so 1 solution.
0 0 6
7
0
3 4 5 6
0
0 5 7 9
(b) A =
0 0 7 8 , so rank A = 4. b = 0 , so 1 solution.
0
0 0 0 7
3

Financial Risk Management
and Asset Pricing
Week 1
Philip Maymin,
[email protected]
Any Questions?
NYU Classes
Intermission
Financial Risk Management and Asset
Pricing: What are we talking about?
What is finance?
What is risk?
What does it mean to manage

Financial Risk Management
and Asset Pricing
Week 11: Utility and CAPM
Options
Binomial trees
American vs European
Copyright 2012 Philip Maymin
When should American calls be
exercised early?
1. Never
2. Possibly the day before the ex
date of a large divi

Financial Risk Management
and Asset Pricing
Week 13: Alternative Portfolio
Construction, and Market Efficiency
Alternative Portfolio Construction
Mean-variance efficient frontier:
Weights of tangency portfolio
Black-Litterman:
Blend of MVE and Bayes