sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Math489/889
Stochastic Processes and
Advanced Mathematical Finance
Homework 7
Steve Dunbar
Due Monday, November 19, 2007
1. Let W1 (t) be a Brownian motion and W2 (t) be another independent
Brownian motion, and is a constant between 1 and 1. Then consider
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of NebraskaLincoln
Lincoln, NE 685880130
http:/www.math.unl.edu
Voice: 4024723731
Fax: 4024728466
Stochastic Processes and
Advanced Mathematical Finance
Stochastic Processes
Rating
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of NebraskaLincoln
Lincoln, NE 685880130
http:/www.math.unl.edu
Voice: 4024723731
Fax: 4024728466
Stochastic Processes and
Advanced Mathematical Finance
Speculation and Hedging
Rat
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of NebraskaLincoln
Lincoln, NE 685880130
http:/www.math.unl.edu
Voice: 4024723731
Fax: 4024728466
Stochastic Processes and
Advanced Mathematical Finance
Randomness
Rating
Student:
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of NebraskaLincoln
Lincoln, NE 685880130
http:/www.math.unl.edu
Voice: 4024723731
Fax: 4024728466
Stochastic Processes and
Advanced Mathematical Finance
Options and Derivatives
Rat
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of NebraskaLincoln
Lincoln, NE 685880130
http:/www.math.unl.edu
Voice: 4024723731
Fax: 4024728466
Stochastic Processes and
Advanced Mathematical Finance
Mathematical Modeling
Ratin
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Math489/889
Stochastic Processes and
Advanced Mathematical Finance
Homework 10
Steve Dunbar
Due Friday, December 14, 2007
1. (a) What is the price of a European call option on a nondividendpaying stock when the stock price is $52, the strike price is $50
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Math489/889
Stochastic Processes and
Advanced Mathematical Finance
Homework 9
Steve Dunbar
Due Monday, December 5, 2007
1. Find the mode (the value of the independent variable with the highest probability) of the lognormal probability density function. (U
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Math489/889
Stochastic Processes and
Advanced Mathematical Finance
Homework 8
Steve Dunbar
Due Wed, December 5, 2007
1. (a) Dierentiate the c.d.f. of Ta to obtain the expression for the p.d.f
of Ta .
Solution: The c.d.f. is:
2
FTa (t) =
2
exp(y 2 /2) dy
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Math489/889
Stochastic Processes and
Advanced Mathematical Finance
Homework 6
Steve Dunbar
Due Friday, November 9, 2007
1. Let W (t) be standard Brownian motion.
(a) Find the probability that 0 < W (1) < 1.
(b) Find the probability that 0 < W (1) < 1 and
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Math489/889
Stochastic Processes and
Advanced Mathematical Finance
Homework 5
Steve Dunbar
Due Fri, October 19, 2007
1. Calculate the m.g.f. of the random variable with uniform distribution
on [0, 1] and then obtain E[X ] and Var[X ].
Solution:
1
exp t 1
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Math489/889
Stochastic Processes and
Advanced Mathematical Finance
Homework 4
Steve Dunbar
Due Wednesday, October 10, 2007
1. Suppose X is an exponentially distributed random variable with mean
E[X ] = 1. For x = 0.5, 1, and 2, compare P[X x] with the Mar
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Math489/889
Stochastic Processes and
Advanced Mathematical Finance
Homework 3
Steve Dunbar
Due Wed, October 3, 2007
1. In a random walk starting at the origin nd the probability that the
point a > 0 will be reached before the point b < 0.
Solution: Let th
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Math 489/889
Stochastic Processes and
Advanced Mathematical Finance
Homework 2
Steve Dunbar
Due Sept 26, 2007
1. Consider a stock whose price today is $50. Suppose that over the next
year, the stock price can either go up by 10%, or down by 3%, so
the sto
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Math489/889
Stochastic Processes and
Advanced Mathematical Finance
Homework 1
Steve Dunbar
Friday, September 14, 2007
1. Puts and calls are not the only option contracts available, just the most
fundamental and the simplest. Puts and calls are designed to
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of NebraskaLincoln
Lincoln, NE 685880130
http:/www.math.unl.edu
Voice: 4024723731
Fax: 4024728466
Stochastic Processes and
Advanced Mathematical Finance
A Binomial Model of Mortgag
sTOCHASTIC PROCESSES AND ADVANCED MATHEMATICAL FINANCE
MATH 489

Fall 2009
Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of NebraskaLincoln
Lincoln, NE 685880130
http:/www.math.unl.edu
Voice: 4024723731
Fax: 4024728466
Stochastic Processes and
Advanced Mathematical Finance
Arbitrage
Rating
Student: c
Math 489/889 Stochastic Processes and Advanced Mathematical Finance Homework 2
Steve Dunbar Due Monday, Sept 13, 2010
1. Consider the hypothetical country of Elbonia, where the government has declared a currency band policy. This means exchange rate betwe