Math 489/889 Exam 1 Name:________________________________
Monday, October 26, 2009

1. (15 points) According to the article “Bullion bulls” on page 81 in the October 8, 2009 issue of The Economist, gold has risen from about $510 per ounce in January 2006 to about $1050 per ounce in October 2009, 46 months later.
   1. What is the continuously compounded annual rate of increase of the price of gold over this period?
   2. In October 2009, one can borrow or lend money at 5% interest, again assume it compounded continuously. In view of this, describe a strategy that will make a profit in October 2010, involving borrowing or lending money, assuming that the rate of increase in the price gold stays constant over this time.
   3. The article suggests that the rate of increase for gold will stay constant. In view of this, what do you expect to happen to interest rates and what principle allows you to conclude that?

2. (20 points) A long straddle option pays $|S-K|$ if it expires when the underlying stock value is $S$. The option is a portfolio composed of a call and a put on the same security with $K$ as the strike price for both. A stock currently has price $100 and goes up or down by 10% in each time period. What is the value of such a long straddle option with strike price $K=110$ at expiration 2 time units in the future? Assume a simple interest rate of 5% in each time period.

3. (20 points) A gambler plays a coin flipping game in which the probability of winning on a flip is $p=0.4$ and the probability of losing on a flip is $q=1-p=0.6$. The gambler wants to reach the victory level of $16 before being ruined with a fortune of $0. The gambler starts with $8, bets $2 on each flip when the fortune is $6,$8,$10 and bets $4 when the fortune is $4 or $12 Compute the probability of ruin in this game.

4. (20 points) Suppose that in a certain district, 40% of the registered voters prefer candidate A. A random sample of 50 registered voters is selected. Let $S_{50}$ denote the number in the sample who prefer A. Create a simple probability model for $S_{50}$. Find the approximate probability that $S_{50}$ is less than 19.