The First Midterm: A
(60) In a two step binomial tree model as given in Figure 1, assume that r = 0.25 and p = 0.9, q = 0.1.
Answer the following questions.
16
8
4
4
2
1
Figure 1: Binomial tree.
Using replication (not the risk-neutral expectations) to c
W4635 (Spring 2015)
Lecture 1
Hongzhong Zhang
Department of Statistics
Columbia University
January 20, 2015
1 / 11
Introduction
Purpose for mathematical/stochastic nance: pricing
nancial contracts. These contracts are used for hedging
risks associated wit
Homework # 1 for STAT W4635
Professor Protter
January 24, 2016
Due: This homework is due on Tuesday, February 2, in class.
Please do the following exercises, showing all of your work. For these exercises, we assume we
are given a space (, F, P, F), where
Homework # 3 for STAT W4635
Professor Protter
February 10, 2016
Due: This homework is due on Tuesday, February 16, in class.
Please do the following exercises, showing all of your work. For these exercises, we assume we
are given a space (, F, P, F), wher
Homework # 2 for STAT W4635
Professor Protter
January 30, 2016
Due: This homework is due on Tuesday, February 9, in class.
Please do the following exercises, showing all of your work. For these exercises, we assume we
are given a space (, F, P, F), where
Homework # 5 (Review Problems) for STAT
W4635
Professor Protter
February 28, 2016
Due: This homework consists of Review Problems for the Midterm, which is on Thursday,
March 10, in class. It is not to be turned in, and it will not be graded, but for pract
Homework # 4 for STAT W4635
Professor Protter
February 26, 2016
Due: This homework is due on Tuesday, March 1, in class.
Please do the following exercises, showing all of your work. For these exercises, we assume we
are given a space (, G, P, G), where G
Homework #9 for STAT W4635
Professor Protter
April 15, 2016
Special Note: This is the last homework for the semester. Since it is being assigned only on
Thursday, the due date will be in class on Tuesday, April 26.
We begin with some problems concerning m
Homework # 7 for STAT W4635
Professor Protter
March 27, 2016
Due: This homework is due on Tuesday, April 4, in class.
1.) Suppose a zero coupon bond with maturity 10 is issued by a risky firm at time 0. The default
(random) time has an arrival intensity w
W4635 (Spring 2015)
Lecture 5
Hongzhong Zhang
Department of Statistics
Columbia University
February 10, 2015
1 / 15
Pricing derivatives with risk-neutral measure
In the binomial tree model with no arbitrage, we can
always determine a unique probability me
W4635 (Spring 2015)
Lecture 4
Hongzhong Zhang
Department of Statistics
Columbia University
February 4, 2015
1 / 15
Formalization: Information&Filtration
A discrete-time stochastic process cfw_Sn n0 is a collection
of random variables. We assume that we re
W4635 (Spring 2015)
Lecture 3
Hongzhong Zhang
Department of Statistics
Columbia University
February 1, 2015
1 / 16
Pricing an option using replicating
Theorem
ln a N step binomial tree model, any option that pays at the
maturity N can be perfectly replica
W4635 (Spring 2015)
Lecture 11
Hongzhong Zhang
Department of Statistics
Columbia University
March 23, 2015
1 / 16
Delta hedge in a binomial tree
In a binomial tree, we select a strategy cfw_n 1nN to
replicate the payo of an European option. If we denote
b
W4635 (Spring 2015)
Lecture 12
Hongzhong Zhang
Department of Statistics
Columbia University
March 25, 2015
1 / 15
It integral of a Brownian motion
o
Recall that
t
1
1
Bs dBs = (Bt )2 t
2
2
0
Sanity check: both sides are martingales! And
E cfw_(Bt )2 = t.
W4635 (Spring 2015)
Lecture 8
Hongzhong Zhang
Department of Statistics
Columbia University
February 23, 2015
1 / 15
Hedging an American call option r = 5%,
= 15%
A possible trajectory of a winner stock
120
Stock price
115
110
105
100
0
0.05
0.1
0.15
0.2
0
W4635 (Spring 2015)
Lecture 13
Hongzhong Zhang
Department of Statistics
Columbia University
March 29, 2015
1 / 15
Its lemma in terms of a product table
o
Its lemma is essentially Taylors expansion. We need to
o
expand it to the second order because the ag
W4635 (Spring 2015)
Review for the Midterm
Hongzhong Zhang
Department of Statistics
Columbia University
March 4, 2015
1 / 14
Basic concepts in no-abitrage pricing
What is the risk-neutral measure? Is it unique?
Why use risk-neutral measure for pricing? Wh
W4635 (Spring 2015)
Lecture 10
Hongzhong Zhang
Department of Statistics
Columbia University
March 23, 2015
1 / 12
Reection principle I
For the Bachelier model, we have
m S0
Q(m T ) = Q(ST m) = 2Q(ST m) = 2[1N( )]
T
where m = infcfw_t 0 : St = m and ST =
W4635 (Spring 2015)
Lecture 7
Hongzhong Zhang
Department of Statistics
Columbia University
February 16, 2015
1 / 14
Review of stopping times
Are the following random variables stopping times?
(HH) = 2, (HT ) = 1, (TT ) = (TH) = 1.
(HH) = (TH) = 2, (TT )
W4635 (Spring 2012)
Lecture 6
Hongzhong Zhang
Department of Statistics
Columbia University
February 11, 2015
1 / 12
Exercising strategies
For an American option, we know that the optimal
exercise time is the rst time when the option price is
equal to the
Homework #8 for STAT W4635
Professor Protter
April 2, 2016
Due: This homework is due on Tuesday, April 12, in class.
1. Recall the framework from Exercise 2 of Homework 7: Suppose the default time comes
from a Cox process, where the state variable process