Math 3589
Spring 2016
Introduction to Financial Mathematics
Homework Assignment #2 Due Tuesday January 26
1. A stock currently costs $4 per share. In each time period, the value of the
stock will either increase or decrease by 50%, and the risk-free inter

Name:
MATH 3589 INTRODUCTION TO FINANCIAL MATHEMATICS
SPRING 2016 MIDTERM 1
No books, notes or electronic devices.
Show your work and explain your answers.
1. (20 points)
(a) Dene a convex function (do not state Jensens inequality). Pictures will not
rece

Math 3589
Spring 2016
Introduction to Financial Mathematics
Homework Assignment #2 Solutions
1. A stock currently costs $4 per share. In each time period, the value of the
stock will either increase or decrease by 50%, and the risk-free interest rate
is 1

Math 3589
Spring 2016
Introduction to Financial Mathematics
Homework Assignment #4 Solutions
1. Consider the N-period binomial model. Let 0 , 1 , . . . , N 1 be any adapted
portfolio process. In other words, we start with X0 dollars, at time t = k
we purc

Math 3589
Spring 2016
Introduction to Financial Mathematics
Homework Assignment #3 February 2 Solutions
1. Let (, P) be a finite probability space. Recall that if A is an event,
then the probability of A is
P(A) =
X
P().
A
Let Ac be the compliment of A. S

Math 3589
Spring 2016
Introduction to Financial Mathematics
Homework Assignment #1 Due January 19 Solutions
1. Suppose S0 = 4, S1 (H) = 8, S1 (T ) = 2 and r = 0. The price of a European
Call option with strike price k = 10 is V0 = 2. Using a replicating p

Math 3589
Introduction to Financial Mathematics
Homework Assignment #1 Due September 6
1. Suppose S0 = 4, S1 (H) = 8, S1 (T ) = 2 and the risk-free interest rate
is r = 0. Someone is willing to buy or sell European Call options with
strike price k = 10 fo

Math 3589
Spring 2016
Introduction to Financial Mathematics
Homework Assignment #1 Due January 19
1. Suppose S0 = 4, S1 (H) = 8, S1 (T ) = 2 and r = 0. The price of a European
Call option with strike price k = 10 is V0 = 2. Using a replicating portfolio,

Math 3589
Spring 2016
Introduction to Financial Mathematics
Homework Assignment #3 Due Tuesday, February 2
1. Let (, P) be a nite probability space. Recall that if A is an event,
then the probability of A is
P(A) =
P().
A
Let Ac be the compliment of A. Sh

Math 3589
Spring 2016
Introduction to Financial Mathematics
Homework Assignment #4 Due Tuesday, February 16
1. Consider the N-period binomial model. Let 0 , 1 , . . . , N 1 be any adapted
portfolio process. In other words, we start with X0 dollars, at tim

Math 3589
Spring 2016
Introduction to Financial Mathematics
Homework Assignment #5 Due Tuesday, February 22
1. Consider the Binomial Asset pricing model as in Chapter 1, except that, after each movement in the stock price, a dividend is paid and the stock

Name:
MATH 3589
HOMEWORK
MIDTERM 2
No books, notes or electronic devices.
Show your work and explain your answers.
1. (20 points)
e are two positive probability measures on a finite sample space , define
(a) If P and P
e with respect to P.
the Radon-Nikod

Stochastic Calculus for Finance, Volume I and II
by Yan Zeng
Last updated: August 20, 2007
This is a solution manual for the two-volume textbook Stochastic calculus for nance, by Steven Shreve.
If you have any comments or nd any typos/errors, please email

Math 3589
Introduction to Financial Mathematics
Extra Credit Assignment Due September 1
Write 2+ pages, single spaced (size 12) on one of the following two topics. If
you cheat on margins, etc. no points will be awarded, however, you may include
up to hal

Math 3589
Introduction to Financial Mathematics
Homework Assignment #11
1. Prove that a symmetric random walk is a martingale.
Proof: Let cfw_Mn
n=0 be a symmetric random walk. Then
1
1
En [Mn+1 ] = (Mn + 1) + (Mn 1) = Mn .
2
2
2. Prove that a symmetric

Math 3589
Introduction to Financial Mathematics
Homework Assignment #9 Due November 8
1. Prove the Optimal Sampling theorem, Part I: A martingale stopped at
a stopping time is a martingale. A supermartingale (or submartingale)
stopped at a stoping time is

Name:
MATH 3589
INTRODUCTION TO FINANCIAL MATHEMATICS
1. Consider the binomial asset pricing world. You own three put options which expire
at time t = 1 on the stock with current price S0 = 4, the up factor u = 2, the down
factor d = 1/2, and the risk-fre

Math 3589
Introduction to Financial Mathematics
Homework Assignment #1 Due September 6
1. Suppose S0 = 4, S1 (H) = 8, S1 (T ) = 2 and the risk-free interest rate
is r = 0. Someone is willing to buy or sell European Call options with
strike price k = 10 fo

Math 3589
Introduction to Financial Mathematics
Homework Assignment #2 Due Tuesday September 13
1. (Put-call parity) A stock currently costs S0 per share. In each time period,
the value of the stock will either increase or decrease by u and d respectively

Math 3589
Introduction to Financial Mathematics
Homework Assignment #3 Solutions
1. Consider the N-period binomial model. Let 0 , 1 , . . . , N 1 be any adapted
portfolio process. In other words, we start with X0 dollars, at time t = k
we purchase (or sho

Math 3589
Introduction to Financial Mathematics
Homework Assignment #4 Solutions
1. Consider the Binomial Asset pricing model as in Chapter 1, except that, after each movement in the stock price, a dividend is paid and the stock price
is reduced by the am

Math 3589
Spring 2016
Introduction to Financial Mathematics
Homework Assignment #6 Due Tuesday, March 1
1. Under the conditions of Theorem 3.1.1, show the following analogues of
properties (1)(3):
1 > 0) = 1
(1) P(
Z
1] = 1
(2) E[
Z
Y ].
(3) for any ra

Math 3589
Spring 2016
Introduction to Financial Mathematics
Homework Assignment #11 Solutions
1. Consider the three period (N = 3) binomial model with S0 = 4, the up
factor u = 2, the down factor d = 1/2 and the risk-free interest rate is
r = 1/4. Draw th

Math 3589
Spring 2016
Introduction to Financial Mathematics
Homework Assignment #7 Solutions
1. Consider the N -period binomial model. Let P be the actual probability
e be the risk-neutral probability measure, and assume both
measure and P
measures are po

Math 3589
Introduction to Financial Mathematics
Homework Assignment #10 Solutions
1. Optimal Sampling theorem, Part I: A martingale stopped at a stopping
time is a martingale. A supermartingale (or submartingale) stopped at a
stoping time is a supermartin

Name:
MATH 3589
HOMEWORK
MIDTERM 2
No books, notes or electronic devices.
Show your work and explain your answers.
1. (20 points)
e are two positive probability measures on a finite sample space , define
(a) If P and P
e with respect to P.
the Radon-Nikod

Math 3589
Spring 2016
Introduction to Financial Mathematics
Homework Assignment #8 Solutions
Exercise 16. Consider the two period binomial model, with the stock price
at time t = 0, S0 = 4, the up factor u = 2, down factor d = 1/2, and
risk free interest