Math 489/889 Exam 1 Name:________________________________
Friday, October 29, 2010

1. (20 points) You can enter into futures contract to buy a Treasury bond that in 6 months time will be worth $950. The current price of the Treasury bond is $930 and the current interest rate for borrowing or lending money is 6% per year continuously compounded. What is the value of the futures contract? What principle allowed you to conclude that price?

2. (20 points) A European cash-or-nothing binary option pays a fixed amount on the expiration date if the underlying stock value is above the strike price. The binary option pays nothing if it expires with the underlying stock value equal to or less than the strike price. A stock currently has price $100 and goes up or down by 20% in each time period. What is the value of such a cash-or-nothing binary option with payoff $20 at expiration 2 time units in the future and strike price $100? Assume a simple interest rate of 10% in each time period.

3. (20 points) A gambler plays a game in which the probability of winning $1 on a turn is $p=0.25$, the probability of losing on a turn is $q=0.25$ and the probability of staying the same is $r=0.5$. The gambler starts with $2. The gambler wants to reach the victory level of $4 before being ruined with a fortune of $0. Write and solve the equations for the expected duration of the game.

4. (20 points) An insurance company is concerned about health insurance claims. Through an extensive audit, the company has determined that overstatements (claims for more health insurance money than is justified by the medical procedures performed) vary randomly with an exponential distribution $X$ with a parameter $1∕100$ which implies that $E\left[X\right]=100$ and $Var\left[X\right]=100^2$. The company can afford some overstatements simply because it is cheaper to pay than it is to investigate and counter-claim to recover the overstatement. Given $100$ claims in a month, the company wants to know what amount of reserve will give $95$% certainty that the sum total of the overstatements for the month do not exceed the reserve. (All units are in dollars.) What assumptions are you using?