MULTIPLE CHOICE QUESTIONS

Name: M375T=M396D Introduction to Actuarial Financial Mathematics Spring 2013 University of Texas at Austin Sample In-Term Exam Two: Pretest Instructor: Milica Čudina Notes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 75 minutes Standing assumptions: No-arbitrage All options are European in style MULTIPLE CHOICE QUESTIONS 1. (5 points) Assume that the current exchange rate is $1.3 per euro. The continuously compounded interest rate for the euro is 0.03, while continuously compounded interest rate for the USD is 0.04. Let the price of an at-the-money USD-denominated European call on on the euro with exercise date in 6 months be equal to 0.053 What is the price of an at-the-money Euro-denominated put on the USD with the exercise date in 6 months? (a) About 0.011. (b) About 0.031. (c) About 0.051 (d) About 0.071. (e) None of the above Solution: (b) Let x denote the exchange rate from euros to dollars. We are given that x(0) = 1.3. Using the put-call duality for options on currencies, we get Solution: (a) VP Euro (0, 1/x(0)) = (1/x(0) 2 VC USD (0, x(0)) 0.031. 2. (5 points)denote the continuously compounded interest rate by r. Let V CC (0) denote the price of a cash call on the asset S with strike K and exercise date T. Let V CP (0) denote the price of a cash put on the asset S with strike K and exercise date T. Then, (a) e rt (b) 1 (c) e rt (d) F P 0,T (S) (e) None of the above V CC (0) + V CP (0) =

2 3. (5 points) Denote the continuously compounded interest rate by r. Let V AC (0) denote the price of an asset call on t he asset S with strike K and exercise date T. Let V AP (0) denote the price of an asset put on the asset S with strike K and exercise date T. Then, (a) Ke rt (b) S(0) (c) F 0,T (S) (d) F P 0,T (S) (e) None of the above V AC (0) + V AP (0) = Solution: (d) Answer (b) was accepted as well, since I assumed that the students circling it made the nodividend assumption (since no dividends were explicitly mentioned in the problem statement). TRUE/FALSE QUESTIONS 1. (2 pts) Consider two exchange options, one that allows you to exchange a share of asset S for a share of asset Q, and another one that allows you to forfeit a share of asset Q and obtain a share of asset S in return. Then, the prepaid forward prices of the two assets are the same if and only if the two exchange options have the same price. Solution: TRUE 2. (2 pts) Consider a gap option whose trigger price is equal to its strike price. Then, the premium for this option is the same as that for an ordinary option with the same strike, the same exercise date and the same underlying asset. Solution: TRUE 3. (2 pts) Consider a European gap put option such that its trigger price exceeds its strike price. Then, the premium of this option is decreasing with respect to the trigger price. Solution: TRUE Let us look at the payoff of this option at time T ; I am adding the trigger price K 2 in the notation to emphasise that we are considering it to be the argument of the payoff function. V GP (T, K 2 ) = (K 1 S(T ))I [S(T )<K2 ]. Since we are given that K 1 < K 2, the above payoff is negative for all the values of S(T ) such that K 1 < S(T ) < K 2. Keeping all else fixed, and increasing the value of K 2, we see that the above region becomes wider-and-wider. It is evident that for all else kept intact, i.e., temporarily fixing K 1, S(T ), and T, the function V GP is decreasing as a function of K 2. We have to conclude that this effect is reflected in the initial premium as well. 4. (2 pts) Consider two European exchange options both with exercise date T, one that allows you to exchange a share of asset S for a share of asset Q, and another one that allows you to forfeit a share of asset Q and obtain a share of asset S in return.

3 On the other hand, consider the maximum option with the payoff V max (T ) = max(s(t ), Q(T )), and the minimum option with the payoff V min (T ) = min(s(t ), Q(T )). (1) Then, in our usual notation, V EC (S(0), Q(0), 0) + V EC (Q(0), S(0), 0) = V max (0) + V min (0). Solution: FALSE If S(T ) Q(T ), the payoff of a long exchange option allowing you to give up a unit of Q and receive a unit of S is V EC (S(T ), Q(T ), T ) = (S(T ) Q(T )) + = 0, i.e., the option goes unexercised. On the other hand, the payoff of a long exchange option allowing you to give up a unit of S and receive a unit of Q is V EC (Q(T ), S(T ), T ) = (Q(T ) S(T )) + = Q(T ) S(T ). So, the payoff of the portfolio whose price is on the left-hand side of (1) is simply Q(T ) S(T ). The payoff of the portfolio whose initial cost is on the right-hand side of (1) is always S(T ) + Q(T ). So, it is impossible for the proposed equality in prices to always be true. 5. (2 points) Let the continuously compounded interest rate be denoted by r. Consider a futures contract for delivery at time T of a market index with the continuous dividend yield δ. As a function of time, the price of this contract at time t is denoted by F t,t. Denote the time t price of a European call on the futures contract with strike K and exercise date T < T by V C (t), and denote the time t price of a European put on the same futures contract with the same strike price and the same exercise date by V P (t). Then, the following equality is always true V C (t) V P (t) = F t,t e δ(t t) Ke rt. Solution: FALSE There are many things amiss with the right-hand side of the above expression. The correct put-call parity for options on futures reads as V C (t) V P (t) = e r(t t) (F t,t K). FREE RESPONSE PROBLEMS 1. (10 points) The price of a 6 month dollar denominated call option on the euro with a $0.90 strike is $0.0404. The price of an otherwise equivalent put option is $0.0141. Assume that for the dollar we have r = 5%. a. (5 pts) What is the 6 month dollar-euro forward price? b. (5 pts) If the euro-denominated annual continuously compounded interest rate is 3.5%, what is the spot exchange rate?

4 Solution: a. We can use put-call-parity to determine the forward price: V C (0) V P (0) = e rt F 0,T (x) Ke rt F 0,T (x) = e rt [ V C (0) V P (0) + Ke rt ] = e 0.05 0.5 [ $0.0404 $0.0141 + $0.9e 0.05 0.5] F 0,T (x) = $0.92697. b. Given the forward price from above and the pricing formula for the forward price, we can find the current spot rate: F 0,T (x) = x(0)e (r r f)t x 0 = F 0,T (x)e (r r f)t = $0.92697e (0.05 0.035)0.5 = $0.92. 2. (12 points) Suppose that the exchange rate is 0.95$/e, and that the euro-denominated continuously compounded interest rate is 4%, while the dollar-denominated continuously compounded interest rate is 6%. The price of a 1-year 0.93-strike European call on the euro is $0.0571. What is the price of the corresponding European put? Solution: Note: See Problem 9.4. in the textbook! We can make use of the put-call parity for currency options: V P (0) = e r f T x(0) + V C (0) + e rt K V P (0) = e 0.04 0.95 + 0.0571 + e 0.06 0.93 = 0.91275 + 0.0571 + 0.87584 = 0.0202. A $0.93 strike European put option has a value of $0.0202. 3. (5 points) An investor wants to hold 200 euros two years from today. The spot exchange rate is $1.31 per euro. If the euro denominated annual interest rate is 3.0% what is the price of a currency prepaid forward? Solution: F P 0,T (x) = 200e 0.03 2 1.31 = 246.67. 4. (20 points) The current price of a share of stock S is $100. The stock is assumed to be paying a continuous dividend with the dividend yield of 0.04. Assume that the continuously compounded interest rate equals 0.05 Consider the following European gap options with the same exercise date in one year and the same underlying asset S. I Gap call with strike price 100 and trigger price 100 II Gap put with strike price 100 and trigger price 100 III Gap call with strike price 100 and trigger price 110 IV Gap call with strike price 110 and trigger price 100 V Gap call with strike price 100 and trigger price 80. Which one of the above options has the highest price? Solution: Let us try compare the prices of options I and II, first. Since for the both of them the trigger and the strike prices are the same, we are in fact dealing with just plain vanilla options. The regular put-call parity applies, and in our usual notation, we have V I (0) V II (0) = F P 0,T (S) 100e rt = 100e 0.04 100e 0.05 = 100(e 0.04 e 0.05 ) > 0 Option III has a lower price than option I since the payoff curve for option I dominates the payoff of option III.

5 100 80 60 40 20 50 100 150 200 Using the same type of comparison, we see that the value of option I is greater than the value of option IV (again, the payoff curve for option I is always above or at the same level as the payoff curve for option IV. 100 80 60 40 20 50 100 150 Option I has the higher price than option V (again, its payoff curve is always above or at the same level as the payoff curve for option V). So, the price for option I is higher than the price of option V. 100 80 60 40 20 50 100 150 20 We conclude that the option with the highest price of the ones offered is option I.

6 40 30 20 10 20 40 60 80 100 120 140 10 20 5. (20 points) Consider a two-period binomial model for the stock price with both periods of length one year. Let the initial stock price be S(0) = 100 and assume that the stock pays no dividends. Let the up and down factors be u = 1.25 and d = 0.75, respectively. Let the continuously compounded interest rate be r = 0.05 per annum. Roger is interested in purchasing a chooser option with the provision that he can choose if the option is a put or a call after one year. The strike for this option is $100 and the expiry date is two years. Using the above binomial tree, find the price of the chooser option. Solution: With the fiven u and d, we get the following tree modeling the stock price 156.25 125 100 93.75 75 56.25 The risk-neutral probability of the stock price going up is p = e0.05 0.75 1.25 0.75 = 2(e0.05 0.75) 0.6025. We can price the chooser option in question in two ways.

Method I. One way is to consider all of the possible payoffs for both the put and the call and see which one the rational investor would choose at time one depending on whether he is in the up or the down node. In the up node, the value of the call is In the same node, the value of the put is e 0.05 [0.6025 56.25] = 32.24. e 0.05 [(1 0.6025) 6.25] = 2.36. So, a prudent investor would choose for his option to become a call if he/she is in the up node and the value of his chooser option at this node is 32.24. Similarly, at the down node, the value of the call is zero, while the value of the put is e 0.05 [0.6025 6.25 + (1 0.6025) 43.75] = 20.12. So, at this node, the rational investor chooses for the option to become a put and, thus, chooser option is worth 20.12. Finally, the time-0 value of the chooser option is e 0.05 [0.6025 32.24 + (1 0.6025) 20.12] = 26.08. Method II. The alternative method involves the pricing formula for chooser options we developed and used in class. Here, the time-0 price of a chooser option can be written as V C (0, T, K) + V P (0, t, Ke r(t t ) ) where V C (0, T, K) stands for the time-0 price of a European call where K denotes the strike and T is the expiration date and where V P (0, t, Ke r(t t ) ) stands for the time-0 price of a European put where Ke r(t t ) is the strike and t is the expiration date. In the current problem, and V C (0, 2, 100) = e 0.05 2 56.26 0.6025 2 = 18.48, V P (0, 1, 95.12) = e 0.05 20.12 (1 0.6025) = 7.61. So, the price of the chooser option is 18.48 + 7.61 = 26.09. The difference in cents between these two answers is due to rounding errors. 6. (25 points) Consider a two-period binomial model for a non-dividend paying asset S with S(0) = 50 and u = 1/d = 2. Let i = 0.25 denote the effective interest rate per period. You need to price a European put option on S which expires at the end of the two periods and has the strike K = 70. (i) (10 pts) Find the values of the given option at all the nodes in the binomial tree. In particular, find the fair price at time 0 of this option. (ii) (10 pts) Find the number of shares one needs to invest in at every node in the tree in order to replicate the option. (iii) (5 pts) If the option were American, would there be early exercise? 7

8 Solution: Is is easy to construct the binomial tree with the given parameters and to get the risk-neutral probability p = 1/2. Since the option is European, we could evaluate its price directly as V P (0) = 1 1.25 2 1 ( (K Suu ) + + 2(K S ud ) + + (K S dd ) +) 4 = 1 1.25 2 1 ( (70 200) + + 2(70 50) + + (70 12.5) +) 4 = 4 97.5 = 15.6. 25 However, we need to find the value of the put at the other nodes of the tree as well. Evidently, the values at the leaves (notes uu, ud and dd) are precisely the payoffs of the put depending on the stock price, i.e., V uu = 0, V ud = 20 and V dd = 57.5. At node u, we have At node d, the value is As for the s, V u = 1 1.25 1 (0 + 20) = 8. 2 V d = 1 1.25 1 (20 + 57.5) = 31. 2 u = 0 20 3 2 100 = 2 15 20 57.5 d = 3 2 25 = 1 0 = 8 31 = 23 50 75 As for early exercise, at node d, the payoff of immediate exercise would be 3 2 (70 25) + = 45 > V d. So, one would create a higher profit by exercising early.