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.
Problem 4. Consider a model with one stock (with mean
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 subsets of = [0, 1]. Then, by
linearity of integrals, we
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), ., Wm (t). Consider a ddimensional vector process X (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 ) = 2x, fy (t, x, y ) = 2y , fxy (t, x, y ) = 0, and fxx (
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 corresponding
ltration. Which of the following processes are ma
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) I (s)2 |Fs = E (I (t) I (s)2 . However,
E (I (t) I (s
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 -algebras (S0 ), (S0 , S1 ), (S0 , S1 , S2 ) and (S0 ,
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. Specifically, let
= (100, 80, 100), and
= (100, 80, 90).
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 ).
3. Var(X + Y ) = Var(X ) + Var(Y ) + 2Cov(X, Y ).
Pr
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 independent.
and
and
are not independent.
are independent.
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 (or major results). Write clearly.
Problem 1 (100 points
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 one sheet of paper which
you were allowed to prepare (wr
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 sheet of paper which
you were allowed to prepare.
The maxim
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 material
except one sheet of paper which you were allowe
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 person nor material
except one sheet of paper which you
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 material except
one sheet of paper which you were allowe
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 martingales with respect to (Ft ):
a) Wt 1
b) Wt3
c) Wt
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 of paper which
you were allowed to prepare.
Problem 1 (1
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 paper which
you were allowed to prepare (writing on one