IEOR 4701
Assignment 4
2014 Fall
Due on next Wednesday, Oct. 22 in class
1. Let cfw_X(t) : t 0be a continuous time Markov chain with the following transition rate matrix
0
0
A=
1
2
3
4
2
1
1
2
3
1
2
1
5
1
1
3
2
1
1
0
1
2
For a set D dene TD = mincfw_t 0 :
Assignment 3
IEOR 4701 - Stochastic Models for Financial Engineering
Due October 1st, in class
1. Gamblers ruin problem.
We have two gamblers, A and B. We toss a coin: if it comes up H, then B pays 1 dollar
to A; otherwise (i.e. if T comes up), A pays 1 d
IEOR 4701 Lecture Notes - Lecture 26
Applications to the Optional Sampling Theorem
Optional sampling theorem (O.S.T.) states that, with regularity conditions, the
expectation of a martingale at a stopping time is equal to the expecation of its initial
val
IEOR 4701 Lecture Notes - Lecture 24
Further Topics in Martingale
The theory of martingale came into being with the aim of providing insight into the
apparent impossibility of making money by placing bets on fair games. This lecture is
intended to introdu
IEOR 4701 Lecture Notes - Lecture 22
Itos Lemma and applications
Itos Lemma is the key result in stochastic calculus. In this lecture we state and give an
outline proof of a general form of the result.
Suppose a process X(t) is given to us as a stochastic
IEOR 4701 Lecture Notes - Lecture 18
We rst show that Brownian motion is a Markov process. To see this is true, suppose cfw_X(t)
is a BM (, ). It is easy to see
Pcfw_X(t + s) [y, y + dy]|X(s) = x
=Pcfw_X(t + s) X(s) + x [y, y + dy]|X(s) = x
=Pcfw_X(t + s)
IEOR 4701 Lecture Notes - Lecture 17
1
Our goal is to compute V ar( 0 B(s)ds). Integrating by parts,
1
1
B(s)ds = B(1)
0
sdB(s).
0
Hence,
V ar(
1
B(s)ds) = V ar(B(1) + V ar(
1
sdB(s) 2Cov(B(1),
sdB(s).
0
0
0
1
The rst term on the r.h.s. is trivial and eq
IEOR 4701 Lecture Notes - Lecture 15
Recall in the last lecture we construct a sequence of binomial processes W (t) that converge to a geometric Brownian motion as 0. We start by introducing
1
1
w.p. 2 + /2
Jk () =
1 w.p. 1 /2.
2
Dene
W (t) =
Jk ().
t
k
IEOR 4701 Lecture Notes - Lecture 13
Recap - Continuous Time Markov Chain
1. We have constructed a Continuous Time Markov Chain using transition rate matrix
A.
2. Consider a call option with a payo function f (x) = (x K)+ where K is the strike
price and a
Assignment 7
IEOR 4701 - Stochastic Models for Financial Engineering
Due date: Monday, Dec. 8th 2014. (Question 6 is extra-credit. All questions are
equally weighted.)
Exercise 1: Consider a model of interest rate dynamics which follows the so-called
Cox-
Assignment 5
IEOR 4701 - Stochastic Models for Financial Engineering
Due November 5, in class
1. Recall that if X Gamma(, ) for , > 0, then
fX (x) =
ex x1
I(x > 0).
()
(a) Verify that if < , then
(
E exp(X) =
)
.
(b) let Z N (0, 1), and nd and such that
(