21-420 Continuous Time Finance
Homework 5 - due April 19, 2010
Problem 1. Exercise 1.10 from the course book.
Problem 2. Exercise 5.3 from the course book.
Problem 3. Exercise 5.4 from the course book
21-640, Continuous-Time Finance
Spring 2010 Homework 5 Solutions
Problem 1.
1. By denition of P,
1
P () =
1
Z ( ) dP( ) =
2 dx = 1.
1
2
0
Now let A1 , A2 , . . . be a sequence of disjoint measurable s
21-420 Continuous Time Finance
Homework 4 - due April 5, 2010
Problem 1. Multidimensional stochastic calculus. Assume we are given an mdimensional Brownian motion, i.e., W (t) is a vector (W1 (t), .,
21-420 Continuous-Time Finance, Spring 2010
Homework 4 Solutions
Problem 1.
1. Let f (t, x, y ) = x2 +y 2 so that Y (t) = f (t, W1 (t), W2 (t). Then f (0, 0, 0) = 0, ft (t, x, y ) =
0, fx (t, x, y ) =
21-420 Continuous Time Finance
Homework 3 - due March 29, 2010
Problems 1 - 2. Exercises 4.3 - 4.4 from the course book.
Problem 3. Let Wt , t 0 be a standard Brownian motion and (Ft )t0 be a correspo
21-640, Continuous-Time Finance Spring 2010
Homework 3 Solutions
Problem 1.
1. False. I (t) I (s) = (s) [W (t) W (s)] = W (s) [W (t) W (s)] is not independent of
Fs . If it were, we would have E (I (t
21-420 Continuous Time Finance
Homework 2 - due February 24, 2010
Problem 1. Consider the following tree as a model for stock evolution with equal branching
probabilities at each node:
1. Describe the
21-420 Homework 2 Solutions
Problem 1
1. First let's name the sample space. Let =
,
, where each denotes a possible path
= (100, 140, 150),
= (100, 140, 130),
= (100, 120, 100),
of the stock price. Sp
21-420 Continuous Time Finance
Homework 1 - due February 8, 2010
Problem 1. Prove the following properties of the covariance:
1. Cov(X, Y ) = Cov(Y, X ).
2. Cov(aX + bY, Z ) = aCov(X, Z ) + bCov(Y, Z
Problem 3.
Recall that since = ( ,
are independent if and only if
,
1.
,
=
,
= 0, so
2.
,
=
,
= 1, so
3.
,
=
,
= 0, so
4.
,
+3
2
) has a multivariate Gaussian distribution, if then
,
= 0.
and
are inde
Department of Mathematical Sciences
CARNEGIE MELLON UNIVERSITY
21-420 Continuous time nance
Spring 2010
Test 2 - April 21, 2010
Prove every statement that you make, except when you refer to theorems (
Department of Mathematical Sciences
CARNEGIE MELLON UNIVERSITY
21-420 Continuous time nance
Spring 2009
Test 2 - Solution
April 27, 2009
You are not allowed to consult any person nor material except o
Department of Mathematical Sciences
Spring 2010
CARNEGIE MELLON UNIVERSITY
21-420 Continuous time nance
Midterm
February 26, 2010
You are not allowed to consult any person nor material except one shee
Department of Mathematical Sciences
CARNEGIE MELLON UNIVERSITY
21-420 Continuous time nance
Spring 2009
Test 1
March 2, 2008
This is an individual exam. You are not allowed to consult any person nor m
Department of Mathematical Sciences
CARNEGIE MELLON UNIVERSITY
21-420 Continuous time nance
Spring 2009
Test 1 - Solution
March 2, 2008
This is an individual exam. You are not allowed to consult any p
Department of Mathematical Sciences
CARNEGIE MELLON UNIVERSITY
21-420 Continuous time nance
Spring 2008
Test 1
March 3, 2008
This is an individual exam. You are not allowed to consult any person nor m
21-420 Continuous Time Finance
Homework 3
Due Feb 27, 2009, before the class
Problem 1. Let Wt , t 0 be a Brownian motion and (Ft )t0 be a corresponding ltration.
Which of the following processes are
Department of Mathematical Sciences
Spring 2009
CARNEGIE MELLON UNIVERSITY
21-420 Continuous time nance
Final Exam
May 7, 2009
You are not allowed to consult any person nor material except one sheet o
Department of Mathematical Sciences
CARNEGIE MELLON UNIVERSITY
21-420 Continuous time nance
Spring 08
Final Exam
May 8, 2008
You are not allowed to consult any person nor material except one sheet of