Homework 2 - Due 9/30/2014
Math 485
October 9, 2014
1
Textbook - For your reference, not required for
submission
Section 2.2, Page 29: 1,2,3,4.
Section 2.4 Page 35: 1,2,5,6.
2
Homework problems
1. (a) Suppose Apple is currently trading at $500.00. Alice b
Homework 2 - Solution
Math 485
October 3, 2014
1. (a) Suppose Apple is currently trading at $500.00. Alice believes Apple will go
up in a major way and wants to bet $60,000 on this speculation. Right now, suppose
a January call at strike $600 costs $6.00.
The multi-period binomial model
Math 485
October 29, 2014
1
Description of the multi-period binomial model
To make our model richer, well transition from the 1-period model to the multi-period
binomial model. Specically well have:
1.1
Notations
The presen
The stock model for continuous time - Pricing Euro
style derivative
Math 485
December 3, 2014
1
1.1
Geometric Brownian motion
Denition
As motivated in Lecture notes 5a, a natural model for us to use for the underlying
process St , as the limit of the mult
Math 485
Fall 2015
Final exam
12/15/2015
Name (Print):
This exam contains 4 pages (including this cover page) and 7 problems. Check to see if any pages
are missing. Enter all requested information on the top of this page, and put your initials on the
top
The American options
Math 485
October 14, 2014
1
1.1
Some preliminaries
Denition
An American call option with strike price K and expiration T on an underlying S
gives the holder the right to choose a time between 0 and T to buy 1 share of S with
price K.
From discrete to continuous time model
Math 485
October 31, 2014
1
Introduction
We have seen that the discrete time binomial model provides some rich features to
our modeling of a nancial asset. In particular, it allows us to discuss the pricing
path depe
Brownian motion and Ito calculus
Math 485
November 6, 2014
1
1.1
Brownian motion and Ito integral
Denition
A stochastic process Wt is a Brownian motion (abbreviated BM) if
a. W0 = 0
b. Wt Ws is independent of Wr , 0 r s.
c. Wt Ws has Normal(0, t s) distri
The multi-period binomial model (Cont)
Math 485
September 19, 2014
1
1.1
Conditional expectation in the multi-period model
The value of a forward contract in the future
Suppose were in the multi-period model with the present being k = 0. Consider the
forw
A review of basic probability theory
Math 485
December 3, 2014
1
Probability and Events
1.1
Events and their properties
Consider an experiment where we toss a coin twice. All the possible outcomes are
cfw_T T , cfw_HH, cfw_T H, cfw_HT .
We call these (ele
Introduction to basic nancial products and pricing
(Cont)
Math 485
October 9, 2014
1
1.1
Intro to the 1 period model
Value of a European option
Consider a European call option on an asset S with expiration T and strike price
K. From now on, we will denote
Homework 1 - Solution
Math 485
September 14, 2015
1. John, your friend, tells you that he has two daughters. You also know that he
has three children in total. What, then, is the probability that his youngest child is
a girl? (Assuming that a boy and a gi
Math 485
Fall 2016
Midterm exam 1
10/13/16
Name (Print):
This exam contains 5 pages (including this cover page) and 6 problems. Check to see if any pages
are missing. Enter all requested information on the top of this page, and put your initials on the
to
Math 485
Fall 2014
Final exam
12/16/2014
Name (Print):
This exam contains 8 pages (including this cover page) and 8 problems. Check to see if any pages
are missing. Enter all requested information on the top of this page, and put your initials on the
top
Math 485
Fall 2014
Midterm 2
11/20/2014
Name (Print):
This exam contains 6 pages (including this cover page) and 5 problems. Check to see if any pages
are missing. Enter all requested information on the top of this page, and put your initials on the
top o
Math 485
Fall 2014
Midterm 1
10/09/2014
Name (Print):
This exam contains 7 pages (including this cover page) and 6 problems. Check to see if any pages
are missing. Enter all requested information on the top of this page, and put your initials on the
top o
Fundamental theorems of asset pricing - Discrete
models
Math 485
December 15, 2014
1
Introduction
In this note we discuss the two fundamental theorems of asset pricing. The mathematical tool for discussion is martingale theory in discrete time. We will de
Introduction to basic nancial products and pricing
Math 485
September 21, 2014
1
1.1
The nancial products
Forward contract
Suppose you run a factory, and you know that in November you will need a large
amount of oil (say 10,000 barrels). Suppose the price
A review of basic probability theory (Cont)
Math 485
December 3, 2014
1
1.1
Conditional expectation
Conditional distribution, conditional density
We have discussed conditional probability P (A|B), which is the probability that A
happened given the knowled
The Greeks
Math 485
November 19, 2014
1
Motivation:
Suppose an option seller sells a Euro-style derivative that pays VT = (ST ) at time
T . We already learned that he should charge V0 = E ( erT (ST ) for the option at
time 0.
Now the question is what shou
Fundamental theorems of asset pricing - Continuous
models
Math 485
November 30, 2014
1
Introduction
In this note we discuss the two fundamental theorems of asset pricing for the continuous time model, in particular the Black-Scholes model. The mathematica
Homework 4 - Due 10/06/2016
Math 485
September 30, 2016
1. Show that for any random variable X and any constant c
E(X E(X)2 E(X c)2 .
2. Let X, Y be independent random variables, Y having Normal(0,1) distribution
and X has distribution P (X = 1.5) = P (X
Homework 4
Math 485
October 24, 2014
A. Section 3.3, Page 54: 1,2,3,4 (for 1-3, only do case a - c from the table of
Exercise 1).
B. For Exercise 3, case b of Exercise 1, nd the optimal exercise time and verify
that
V0 = E Q er (K S )+ .
C. Section 3.4: 1
Weekly Homework 5
Math 485
November 6, 2014
1. Compute X (t) := E exp(itX) , where exp(x) := ex , i is the imaginary number:
i2 = 1 and X has N (, 2 ) distribution. Recall that the density of N (, 2 ) is
f (x) =
1
2 2
exp
(x )2
2 2
.
(t) is called the c
Homework 3 - Selected solution
Math 485
November 6, 2014
1
Textbook (Stampi and Goodman)
A. Section 3.1, Page 49: 1,2,3.
Answer:
1. $ 2.66. The two Tuesday nodes are $ 2.18 and $ 3.63.
2. pu + (1 p)d = 1.04. A formula on page 48 of the textbook states tha
Homework 3 - Due 10/16/2014
Math 485
October 9, 2014
Read Section 3.1, 3.2 in the Textbook.
I. Textbook Section 3.1, Page 49: 1,2,3.
II. Textbook Section 3.2 Page 51: 1,2,3,4.
III. Repeat problem 1 of Section 3.2 and nd the replicating portfolio at time
t
Weekly Homework 5
Math 485
November 14, 2014
1. Compute X (t) := E exp(itX) , where exp(x) := ex , i is the imaginary number:
i2 = 1 and X has N (, 2 ) distribution. Recall that the density of N (, 2 ) is
f (x) =
1
2 2
exp
(x )2
2 2
.
(t) is called the
Homework1 - Due 9/18/2014
Math 485
October 9, 2014
1. You visit your friends family, who has 3 children. Upon arriving at the house,
you see two girls playing at the front yard. John, your friend, tells you that those are
his two daughters. What, then, is