Download backknowprobe10.pdf

Math 489/889
Stochastic Processes and
Advanced Mathematical Finance
Steve Dunbar
August 23, 2010
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.
1. 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:
(a) $10,000 right now
(b) $10,300 1 year from now
(c) $245 each year forever, starting now
(d) $1,025 for each of ten years, starting now
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.)
2. 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?
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3. 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.
4. 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?
5. If X is a normal random variable with parameters µ = 10 and σ 2 = 36,
compute
(a) P [X > 5]
(b) P [4 < X < 16]
(c) P [X < 8]
(d) P [X < 20]
(e) P [X > 16]
6. In 10, 000 independent tosses of a coin, the coin landed heads 5432
times. Is it reasonable to assume the coin is fair? Explain.
7. Suppose that X is a random variable having the probability density
function
(
RxR−1 for 0 ≤ x ≤ 1
f (x) =
0
elsewhere
(a) Determine the mean E [X].
(b) Determine the variance Var [X]
(c) Determine the standard deviation.
8. 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, . . .
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