Download Fall 10 489f10t1.pdf

Math 489/889
Friday, October 29, 2010
Exam 1
Name:
Problem 1 2 3 4 Total
Possible 20 20 20 20
80
Points
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?
1
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.
2
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.
3
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 [X] = 100 and
Var [X] = 1002 . 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?
4