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?

   Solution: $8.08

   Find the price by the principle of no-arbitrage. That is, simultaneously buying and selling two assets cannot provide a riskless profit. Let $X$ be the current price or value of the futures contract. Then assume you buy the futures contract for the bond and simultaneously short or sell the bond. This gives a total of $930-X$. Invest or loan that amount of money for $1∕2$ year at $6\%$ compounded continuously. Then you have $\left(930-X\right)exp\left(0.06\ast \left(1∕2\right)\right)=958.32-Xexp\left(0.03\right)$ at the end of the 6 months. Execute the futures contract to buy a bond at 950. The net is $8.32-Xexp\left(0.03\right)$. If this were positve, you would have a strategy for a riskless profit. If this were negative, then reverse your strategy to buy a bond and short the futures contract, which would reverse the sign, again yielding a riskless profit. Hence $8.32-Xexp\left(0.03\right)=0$, or $X=8.32exp\left(-0.03\right)=8.08$.

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.

   Solution: The recombinant binomial tree is:

    
                                  144  payoff = 20  
                                 /  
                                /  
                             120  
                            /  \     -- Strike = 100  
                           /    \  
                       100       96    payoff = 0  
                          \    /  
                           \  /  
                            80  
                              \  
                               \  
                                 64    payoff = 0  
   
   $$\begin{array}{llll}\hfill \pi & =\frac{1.10-0.80}{1.20-0.80}=0.75& \hfill & \\ \hfill 1-\pi & =\frac{1.20-1.10}{1.20-0.80}=0.25& \hfill & \\ \hfill & & \hfill \end{array}$$

   The value of the option at the first time period is either $V_{1,1}=\left(1∕1.1\right)\left[0.75\cdot 20+0.25\cdot 0\right]=13.6363...$ or $V_{1,0}=\left(1∕1.1\right)\left[0.75\cdot 0+0.25\cdot 0\right]=0$. Now the value of the option at time 0 is $V_{0,0}=1∕\left(1.1\right)\left[0.75\cdot 13.6363...+0.25\cdot 0\right]=9.2975\dots$.

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.

   Using our standard notation of $D_n$ for the duration of the game until either victory or ruin from a fortune of $n$ the first step analysis yields the equations working up from the ruin level:

   $$\begin{array}{llll}\hfill D_0& =0& \hfill & \\ \hfill D_1& =0.25\cdot D_0+0.5\cdot D_1+0.25D_2+1& \hfill & \\ \hfill D_2& =0.25\cdot D_1+0.5\cdot D_2+0.25D_3+1& \hfill & \\ \hfill D_3& =0.25\cdot D_2+0.5\cdot D_3+0.25D_4+1& \hfill & \\ \hfill D_4& =0& \hfill & \\ \hfill & & \hfill \end{array}$$

   The solution is $D_0=6,D_2=8,D_3=6$.

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?

   Solution: Let $X_i$ be the size of overstatement i in that month. Assume that the $X_i$ are independent and identically distributed, so that the Central Limit Theorem applies. Then we seek a value $S$ such that

   $$Pr\left[\sum _{i=1}^{100}{X}_i>S\right]\le 0.95$$

   From the Central Limit Theorem we can say that this will be approximately the same as

   $$Pr\left[Z>\frac{S-100\cdot 100}{100\cdot \sqrt{100}}>S\right]\le 0.95$$

   To assure this, we require

   $$\frac{S-100\cdot 100}{100\cdot \sqrt{100}}>1.65$$

   or $S>11,650$