Name: T/F 2.13 M.C. Σ

Transcription

1 Name: M339D=M389D Introduction to Actuarial Financial Mathematics University of Texas at Austin In-Term Exam II Instructor: Milica Čudina Notes: This is a closed book and closed notes exam. The maximal score on this exam is 80 points. Time: 50 minutes TRUE/FALSE 2.1 (2) TRUE FALSE 2.2 (2) TRUE FALSE 2.3 (2) TRUE FALSE 2.4 (2) TRUE FALSE 2.5 (2) TRUE FALSE 2.6 (2) TRUE FALSE 2.7 (2) TRUE FALSE 2.8 (2) TRUE FALSE 2.9 (2) TRUE FALSE 2.10 (2) TRUE FALSE 2.11 (2) TRUE FALSE MULTIPLE CHOICE 2.14 (5) a b c d e 2.15 (5) a b c d e 2.16 (5) a b c d e 2.17 (5) a b c d e 2.18 (5) a b c d e 2.19 (5) a b c d e 2.20 (5) a b c d e 2.21 (5) a b c d e 2.22 (5) a b c d e 2.23 (5) a b c d e 2.12 (2) TRUE FALSE FOR GRADER S USE ONLY: T/F 2.13 M.C. Σ

2 TRUE/FALSE QUESTIONS. Please, circle the correct answer on the front page of this exam. Problem 2.1. (2 points) In the setting of the binomial asset-pricing model, let d and u denote the up and down factors, respectively. Moreover, let r denote the continuously compounded, risk-free interest rate. Let h denote the length of a single period in our model. Then, if, d < e rh < u then there is no possibility for arbitrage regardless of whether the stock pays dividends. True or false? FALSE Problem 2.2. (2 points) Let the current exchange rate of euros (e) to USD ($) be denoted by x(0), i.e., currently, 1 e = $X(0). Let r $ denote the continuously compounded, risk-free interest rate for the $, and let r e denote the continuously compounded, risk-free interest rate for the e. Denote the price of a $-denominated European call option with strike K and exercise date T by V C (0) and the price of an otherwise identical put option by V P (0). Then, True or false? FALSE The two interest rates have switched places. V C (0) V P (0) = x(0)e r $T Ke ret. Problem 2.3. If the strike is kept equal to the trigger of a gap call, then increasing the trigger reduces the premium of the gap call. True or false? TRUE If the trigger and strike prices are kept equal, the gap is closed and we are in fact dealing with a vanilla call. Call prices are decreasing with respect to the strike price. Problem 2.4. (2 points) Strangles are financial positions designed to hedge against decreasing prices of the underlying asset. True or false? FALSE Problem 2.5. (2 points) The payoff of a gap call option is always nonnegative regardless of the choice of the trigger and the strike. True or false? FALSE Problem 2.6. (2 points) In the replicating portfolio for a call option whose underlying asset s price is modeled using a binomial tree, the value of the is always nonnegative. True or false? TRUE

3 Problem 2.7. A European call option with strike K on a futures contract on a stock has the same value as the European call option with strike K on that same stock provided that the futures contract has the same expiration as the stock option. True or false? TRUE Problem 2.8. (2 points) The following is a replicating portfolio for a ratio spread: Long a two-year European call and write a three-year European call with the same strike price and the same underlying asset. True or false? FALSE 3 Problem 2.9. (2 points) An investor wants to speculate on low volatility combined with a higher likelihood of lower than higher prices. Then, he should long a ratio spread with fewer calls of the lower strike. True or false? TRUE Problem (2 points) In the binomial asset pricing model, the replicating portfolio for a put option has a bond investment which is equivalent to lending at the risk-free interest rate. True or false? TRUE Problem (2 points) A butterfly spread can be constructed in this way: Buy a 90 call, sell a 100 put, sell a 100 call, buy a 110 put. True or false? TRUE Problem The strike price at which the European call and the otherwise identical European put have the same premiums is the forward price for delivery of the underlying on the exercise date of the two options. True or false? TRUE 2.2. FREE-RESPONSE PROBLEMS. Problem (10 points) Let our market model include two continuous-dividend-paying stocks whose time t prices are denoted by S(t) and Q(t) for t 0. The current stock prices are S(0) = 160 and Q(0) = 80. The dividend yield for the stock S is δ S = 0.06 and the dividend yield for the stock Q is δ Q = The price of an exchange option giving its bearer the right to forfeit one share of Q for one share of S in one year is given to be $11. Find the price of a maximum option on the above two assets with exercise date in a year. Remember that the payoff of the maximum option is max(s(1), Q(1)).

4 4 As we showed in class V max (0) = F0,1(Q) P + V EC (0, S, Q) = Q(0)e δ Q + V EC (0, S, Q) = 80e =

5 2.3. MULTIPLE CHOICE QUESTIONS. Please, circle the correct answer on the front page of this exam. Problem (5 points) Let K 1 = 50, K 2 = 55 and K 3 = 65 be the strikes of three European call options on the same underlying asset and with the same exipration date. Let C i (0) denote the price at time 0 of the option with strike K i for i = 1, 2, 3. We are given that C 1 (0) = 16 and C 3 (0) = 1. What is the maximum possible value of C 2 (0) which still does not violate the convexity property of option prices? (a) C 2 (0) = 11 (b) C 2 (0) = 12 (c) C 2 (0) = 13 (d) C 2 (0) = 14 (a) The convexity requirement for call-option prices is 16 C C So, 10(16 C) 5(C 1) 33 3C 11 C. 5 Problem We are given the following European-call prices for options on the same underlying asset: $50-strike $10 $55-strike $6 $60-strike $4 Assume that the continuously-compounded, risk-free interest rate is strictly positive. Which of the following portfolios would exploit an arbitrage opportunity stemming from the above stock prices? (a) The call bear spread only. (b) The call bull spread only. (c) Both the call bull and the call bear spread. (d) Neither the call bull or call bear spread, but there is an arbitrage opportunity. (e) There is not enough information provided. (e) Problem (5 points) The current exchange rate is 0.90 euros per US dollar. A European eurodenominated call on the US dollar has the strike price of 0.80 euros and a premium of 0.09 euros. The call expires in 6 months. Calculate the value of a European US-dollar-denominated put on the euro that has the strike price of $1.25 and also expires in 6 months. (a) (b) (c) 0.145

6 6 (d) 0.15 (a) Let x be the exchange rate of euros per USD and let K = 0.80 euros. By the put-call symmetry of currency-option prices, we have V P (0, 1/x, 1/K) = V C(0, x, K) x(0)k = = 1 8 =

7 Problem (5 points) An investor acquires a call bull spread consisting of the call with strike K 1 = 100 and K 2 = 110 and with expiration in one year. The initial price of the 100 strike call option equals $11.34, while the price of the 110 strike option equals $7.74. At expiration, it turns out that the stock price equals $105. Given a continuously compouned annual interest rate of 5.0%, what is the profit to the investor? (a) $3.78 loss (b) $1.22 loss (c) $1.22 gain (d) $5 gain (c) The total initial cost of establishing the investor s position is = The future value of this amount at expiration is 3.60e 0.05 = The payoff at expiration is So, the profit is = (S(T ) 100) + (S(T ) 110) + = ( ) + ( ) + = 5. Problem An investor buys an $850-strike, two-year straddle on gold. The price of gold two years from now is modeled using the following distribution: What is the investor s expected payoff? (a) About $11.25 (b) About $23.00 (c) About $23.75 (d) About $36.25 (d) $800, with probability 0.35, $850, with probability 0.4, $925, with probability = Problem The current stock price is 20 per share. The price at the end of a four-month period is modeled with a one-period binomial tree so that the stock price can either increase by $5, or decrease by $5. The stock pays dividends continuously with the dividend yield The continuously-compounded, risk-free interest rate is What is the stock investment in a replicating portfolio for four-month, $20-strike European call option on the above stock? (a) Long shares (b) Long shares (c) Long 0.5 shares (d) Short 0.5 shares

8 8 (b) = e δh V u V d S u S d = e 0.04/

9 Problem The current price of a continuous-dividend-paying stock is $65 per share. Its dividend yield is We model the stock price at the end of two years using a binomial tree. It is assumed that the stock price can either go up, or go down by 30%. The continuously-compounded, risk-free interest rate equals Consider a two-year, $70-strike European call option on the above stock. What is the risk-free component of the replicating portfolio for this option? (a) Borrow $ (b) Lend $ (c) Borrow $ (d) Lend $ (a) The two possible stock prices at the end of the two years are S u = 84.5 and S d = So, the two possible call payoffs are V u = 14.5 and V d = 0. The risk-free component of the replicating porfolio is This means borrowing $ (2) 0.7(14.5) B = e = Problem Consider a continuous-dividend-paying stock whose current price is $50 and whose dividend yield is The continuously-compounded, risk-free interest rate is Consider a portfolio consisting of: (1) a (45, 60) call bull spread, and (2) a (45, 60) put bear spread. All the options are European with exercise date in one year. What is the price of the above portfolio? (a) $13.97 (b) $14.13 (c) $14.27 (d) $14.41 (c) This is a box spread. So, the price is (60 15)e 0.05 = Alternatively, using put-call parity, we have that the portfolio s price is V C (45) V C (60) V P (45) + V P (60) = (V C (45) V P (45)) (V C (60) V P (60)) = F 0,1 (S) 45e r (F 0,1 (S) 60e r ) = 15e 0.05 = Problem Consider a continuous-dividend-paying stock with the current price of $45 and dividend yield The continuously-compounded, risk-free interest rate is Consider a pair of six-month, $50-strike, $45-trigger gap options. The gap call sells for $1.70. What is the price of the gap put? (a) $5.17 (b) $6.16

10 10 (c) $7.27 (d) $7.41 (b) By put-call parity for gap options, we have V GP (0) = e e 0.01 = Problem The current futures price is given to be $80. The continuously-compounded, risk-free interest rate is The price of a 1-year, 85-strike European call option on the futures contract is $3.78. What is the price of the otherwise identical put option? (a) $7.34 (b) $7.74 (c) $8.14 (d) $8.54 (d) By put-call parity for futures options, we have V P (0) = V C (0) e rt (F 0,TF K) = 3.78 e 0.05 (80 85) = e 0.05 = 8.54.