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\title{Math 489/889 \\
Stochastic Processes and\\
Advanced Mathematical Finance}
\author{Steve Dunbar}
\date{August 23, 2010}
\begin{document}
\maketitle
This background knowledge probe is not for a grade. It is intended for
diagnostic and instructional preparation purposes only. I will credit
you with 50 homework points automatically if you hand this in on Friday,
August 27, 2010 after making an honest effort at working the problems.
You may use any reference, including asking me for hints but work the
problems completely yourself.
\begin{enumerate}
\item
% Present value calculations
%adapted from Problem 6, page 41, {\it Principles of Corporate
%Finance}, Brealy and Myers, 1984, McGraw-Hill
Suppose that you win \$10,000 in the Nebraska Lottery Mega
Millions game (as did an Omaha woman on August 12, 2010). Your
financial adviser gives you a choice of the following payouts
for the prize:
\begin{enumerate}
\item
\$10,000 right now
\item
\$10,300 1 year from now
\item
\$245 each year forever, starting now
\item
\$1,025 for each of ten years, starting now
\end{enumerate}
Which is the most valuable payout in terms of present value?
Assume the interest rate is 3.0\% and is compounded continuously
(roughly the best available interest rate at the time of writing
this probe.)
\item
%Interest calculations
%adapted from Problem 6, page 41, {\it Principles of Corporate
%Finance}, Brealy and Myers, 1984, McGraw-Hill
A investment of \$232 will be worth \$312.18 in 2 years. What
is the effective annual interest rate assuming quarterly
compounding? Assuming continuously compounded interest?
\item
%Calculating binomial probabilities
% Adapted from Problem 3.6.2, page 145, {\it An Introduction to
% Probability Theory and Its Applications}, Larsen and Marx,
% Prentice-Hall, 1985,
Each day a stock price moves up one point or down one point with
probabilities $ 1/3 $ and $ 2/3 $ respectively. What is the
probability that after 4 days, the stock will have returned to
its original price? Assume the daily price fluctuations are
independent events.
\item
% Calculating binomial and geometric probabilities
% Adapted from Problem 31, page 118, {\it A First Course in
% Probability} by S Ross, Macmillan, 1976.
Consider a roulette wheel consisting of 38 numbers, $ 1 $
through $ 36 $, $ 0 $ and double $ 0 $. If Bond always bets
that the outcome will be one of the numbers $ 1 $ through $ 12 $,
what is the probability that Bond will lose his first $ 5 $
bets? What is the probability that his first win will occur on
his fifth bet?
\item
% Calculating normal probabilities
% Adapted from Problem 10, page 146, {\it A First Course in
% Probability} by S Ross, Macmillan, 1976.
If $ X $ is a normal random variable with parameters $ \mu = 10 $
and $ \sigma^2 = 36 $, compute
\begin{enumerate}
\item
$ \Prob{ X > 5} $
\item
$ \Prob{4 < X < 16} $
\item
$ \Prob{X < 8} $
\item
$ \Prob{X < 20} $
\item
$ \Prob{X > 16} $
\end{enumerate}
\item
% Using normal approximation to the binomial
% Adapted from Problem 10, page 146, {\it A First Course in
% Probability} by S Ross, Macmillan, 1976.
In $ 10,000 $ independent tosses of a coin, the coin landed
heads $ 5432 $ times. Is it reasonable to assume the coin is
fair? Explain.
\item
Suppose that $ X $ is a random variable having the probability
density function
$$
f(x) =
\begin{cases}
R x^{R-1} & \text{ for $0 \le x \le 1$ } \\
0 & \text{ elsewhere }
\end{cases}
$$
\begin{enumerate}
\item
Determine the mean $ \E{X} $.
\item
Determine the variance $ \Var{X} $
\item
Determine the standard deviation.
\end{enumerate}
\item
%Calculating moments and variances.
%Adapted from Problem 40, page 78 of S. Ross, {\it Introduction
% to Probability Models}, S. Ross, Academic Press, 1985.
If $ X $ is a uniformly distributed random variable over $ (0,1)
$, then calculate $ \E{X^n} $ and $ \Var{X^n} $ for $ n=1,2,3,\dots
$
%% \item % Conditional probabilities
%% % Adapted from Problem 10, page 79, {\it A First Course in
%% % Probability} by S Ross, Macmillan, 1976.
%% Three cards are randomly selected, without replacement, from an
%% ordinary deck of 52 playing cards. Compute the conditional
%% probability that the first card selected is a spade, given the
%% second and third cards are spades.
%% \item % Conditional probabilities
%% % Adapted from Problem 10, page 79, {\it A First Course in
%% % Probability} by S Ross, Macmillan, 1976.
%% If two fair dice are rolled, what is the conditional probability that
%% the first one lands on $6$, given that the sum of the dice is
%% $k$, $k=2,3,\dots 12$. Compute for all values of $k$. What is
%% the probability that at least one of the fair dices lands on $6$
%% given that the sum of the dice is $k$?
\end{enumerate}
\end{document}
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