Name: M339D=M389D Introduction to Actuarial Financial Mathematics University of Texas at Austin Sample In-Term Exam II Instructor: Milica Čudina Notes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 50 minutes MULTIPLE CHOICE 1 (5) a b c d e TRUE/FALSE 1 (2) TRUE FALSE 2 (2) TRUE FALSE 3 (2) TRUE FALSE 4 (2) TRUE FALSE 5 (2) TRUE FALSE 2 (5) a b c d e 3 (5) a b c d e 4 (5) a b c d e 5 (5) a b c d e 6 (5) a b c d e 7 (5) a b c d e 8 (5) a b c d e
2 FREE-RESPONSE PROBLEMS Problem 2.1. (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. Problem 2.2. Consider a two-period forward binomial tree, where the length of each period is 6 months. Assume the stock price is $50.00, σ = 0.20, r = 0.06 and the dividend yield δ = 0.035. What is the lowest strike price for which early exercise could occur after the first time step and with an American put option? Solution: With the given data, we get that approximately e σ h = 1.15 and e (r δ)h = 1.01. Hence, u = 1.16 and d = 0.88. S uu = 50 1.16 1.16 62.28; S ud = S du = 50 1.16 0.88 51.04; S dd = 50 0.88 0.88 38.72. Let us denote the strike by K. The payoffs, if there is no early exercise, as a function of K are V uu = (K 62.28) +, V ud = (K 51.04) +, and V dd = (K 38.72) +. The risk-neutral probability is p 1.01 0.88 = 1.16 0.88 = 0.46. So, the continuation values of the put after taking a single step back in the binomial tree are V cont u = e 0.03 [0.46 (K 62.28) + + 0.54 (K 51.04) + ] ; V cont d = e 0.03 [0.46 (K 51.04) + + 0.54 (K 38.72) + ]. On the other hand, the stock prices at these two nodes are So, the values of immediate/early exercise are S u = 50 1.16 58 and S d = 50 0.842 44. V imm u = (K 58) + and V imm d = (K 44) +. At any of these two nodes, early exercise will happen if V imm node > V node cont. Hence, we must have that K > 44 otherwise, no early exercise should happen at either of the two nodes. Then, for 51.04 K > 44, we have that V cont u = 0 and V cont d = 0.97 0.54 (K 38.72).
3 So, we need to figure out for which K the inequality K 44 > 0.97 0.54 (K 38.72) K(1 0.97 0.54) > 44 38.72 0.97 0.54 holds. We get K > 49.81.
4 MULTIPLE CHOICE QUESTIONS Please, circle the correct answer on the front page of this exam. Problem 2.3. The current price of a non-dividend-paying stock is $100 per share and its volatility is given to be 0.25. The continuously compounded, risk-free interest rate equals 0.06. Consider a $110-strike, one-year American put on the above stock. Use a two-period forward binomial stock-price tree to calculate the current price of the American put. (a) $20.03 (b) $15.41 (c) $13.38 (d) $11.11 Solution: (c) By the definition of the forward binomial tree, we obtain u = e (r δ)h+σ h = e 0.03+0.25 1 2 1.2297, d = e (r δ)h σ h = e 0.03 0.25 1 2 0.8635. in our usual notation. The binomial tree modeling the stock price is The risk-neutral probability of the stock price going up in a single period equals p = e(r δ)h d u d = e0.03 0.8635 1.2297 0.8635 = 0.4559. Should the American option not be exercised early the possible payoffs would be V uu = 0, V ud = 110 106.18 = 3.82, V dd = 110 74.56 = 35.44. It is not sensible to exercise the American put at the up node, so the value of the American put equals the continuation value at the up node. We get V A u = CV u = e 0.03 (1 0.4559) 3.82 = 2.017. At the down node, the value of immediate exercise is IE d = 110 86.35 = 23.65. On the other hand, the continuation value at the down node equals CV d = e 0.03 [0.4559 3.82 + (1 0.4559) 35.44] = 20.4031. We conclude that the American put s value at the down node equals the value of immediate exervise, i.e., V A d = 23.65.
Should the option be exercised at time 0, the payoff would be 10. The continuation value at the root node is CV 0 = e 0.03 [0.4559 2.017 + (1 0.4559) 23.65] = 13.38. So, the price we were looking for is $13.38. Problem 2.4. The current price of a continuous-dividend-paying stock is $100 per share. Its volatility is given to be 0.2 and its dividend yield is 0.03. The continuously compounded risk-free interest rate equals 0.06. Consider a $95-strike European put option on the above stock with nine months to expiration. Using a three-period forward binomial tree, find the price of this put option. (a) $2.97 (b) $3.06 (c) $3.59 (d) $3.70 Solution: (c) The up and down factors in the above model are u = e 0.03 0.25+0.2 0.25 = 1.1135, d = e 0.03 0.25 0.2 0.25 = 0.9116. The relevant possible stock prices at the leaves of the binomial tree are S ddd = d 3 S(0) = 100(0.9116) 3 = 75.7553, S ddu = d 2 us(0) = 92.5335. The remaining two final states of the world result in the put option being out-of-the-money at expiration. The risk-neutral probability of the stock price moving up in a single period is p 1 = 1 + e = 0.475. 0.2 0.25 So, the price of the European put option equals V P (0) = e 0.06(3/4) [ (95 75.7553)(1 0.475) 3 + (95 92.5335)(3)(1 0.475) 2 (0.475) ] = 3.5884. Problem 2.5. (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. 5
6 (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 symmetry/duality for options on currencies, we get VP Euro (0, 1/x(0)) = (1/x(0) 2 VC USD (0, x(0)) 0.031. Problem 2.6. The following two one-year European put options on the same asset are available in the market: a $50-strike put with the premium of $5, a $55-strike put with the premium of $10. The continuously compounded, risk-free interest rate is 0.04. Which of the following positions certainly exploits the arbitrage opportunity caused by the above put premia? (a) Put bull spread. (b) Put bear spread. (c) Both of the above positions. (d) There is no arbitrage opportunity. Solution: (a) Problem 2.7. You are given that the price of: a $50-strike, one-year European call equals $8, a $65-strike, one-year European call equals $2. Both options have the same underlying asset. What is the maximal price of a $56-strike, one-year European call such that there is no arbitrage in our market model? (a) $4.40 (b) $5 (c) $5.60 (d) $6.02 Solution: (c) Using the convexity of call price with respect to the strike, we get the following answer: 3 5 8 + 2 5 2 = 24 + 4 5 = 5.60.
Problem 2.8. (5 points) Consider a non-dividend-paying stock with the initial price of S(0) = 100. Assume that the annual risk-free continuously compounded interest rate equals r = 0.05. Let the annualized standard deviation of the sontinuously compounded stock return, i.e., the volatility be σ = 0.25. Using a one-period forward binomial tree, calculate the price of a one-year at-the-money European call on this underlying asset. (a) $11.07 (b) $12.46 (c) $13.38 (d) $14.58 Solution: (d) By the definition of the forward binomial tree, with the given data, u = e (r δ)h+σ h = e 0.05+0.25 = e 0.3 1.35, d = e (r δ)h σ h = e 0.2 < 1. We do not care about the actual value of d since the option is at-the-money and we get the payoff of 0 at the lower node regardless of the actual value. Also, we do not need d explicitly to calculate the risk-neutral probability in this model, since Finally, by risk-neutral pricing p = 1 1 + e = 1 = 0.4378. σ h 1 + e0.25 V C (0) = e 0.05 (135 100) 0.4378 14.58. 7 2.1. TRUE/FALSE QUESTIONS. Problem 2.9. (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).
8 Problem 2.10. (2 points) It is never optimal to exercise an American call option on a nondividend paying stock early. Solution: TRUE Problem 2.11. (2 points) In the setting of the one-period binomial model, denote by i the effective interest rate per period. Let u denote the up factor and let d denote the down factor in the stock-price model. If d < u 1 + i then there certainly is no possibility for arbitrage. Solution: FALSE Problem 2.12. (2 points) You are using a binomial asset-pricing model to model the evolution of the price of a particular stock. Then, the in the replicating portfolio of a single call option on that stock never exceeds 1. Solution: TRUE The call s will always be between 0 and 1. Problem 2.13. The expiration date of a futures option cannot exceed the delivery date of the underlying futures contract. Solution: TRUE Problem 2.14. (2 points) The price of a European call option on a non-dividend-paying stock is equal to the price of an otherwise identical American call option. Solution: TRUE