MØA 155 PROBLEM SET: Options Exercise 1. Arbitrage [2] In the discussions of some of the models in this course, we relied on the following type of argument: If two investment strategies have the same payoff tomorrow, they must have the same price today. (Which we also called an no arbitrage argument.) Which of the following theories relied explicitly on these kinds of no-arbitrage arguments? (a) (b) (c) The Put Call parity of option pricing. The Miller Modigliani theorems for capital structure. The CAPM. Exercise 2. Mutual Fund Manager [2] A mutual fund announces that the salaries of its fund managers will depend on the performance of the fund. If the fund loses money, the salaries will be zero. If the fund makes a profit, the salaries will be proportional to the profit. 1. Describe the salary of a fund manager as an option. 2. How is a fund manager motivated to behave with this type of renumeration package? Exercise 3. Pricing factors [1] Which of the following are always positively related to the price of a European call option on a stock? (Hint: There is more than one) 1. The stock price. 2. The strike price. 3. The time to expiration. 4. The volatility 5. The risk-free rate 6. The magnitude of dividends anticipated during the life of the option. Exercise 4. Product [3] A bank has sold a product that offers investors the total return (excluding dividends) on the Oslo OBX index over a one year period. The return is capped at 2%. If the index goes down the original investment of the investor is returned. 1. What option position is equivalent to the product? Exercise 5. Put [2] Suppose that a European put option to sell a share for $6 costs $4 and is held until maturity. 1. Under what circumstances will the seller of the option(i. e. the party with the short position) make a profit? 2. Under what circumstances will the option be exercised? 1
3. Draw a diagram illustrating how the profit from a short position in the option depends on the stock price at the maturity of the option. Exercise 6. Straddles and Butterflies [3] Option traders often refer to straddles and butterflies. Here is an example of each: Straddle Buy call with exercise price of $1 and simultaneously buy put with exercise price of $1. Butterfly Simultaneously buy one call with exercise price of $1, sell two calls with exercise price of $11, and buy one call with exercise price of $12. 1. Draw position diagrams for the straddle and butterfly, showing the payoffs from the investor s net position. 2. Each strategy is a bet on variability. Explain briefly the nature of this bet. Exercise 7. Bank [3] You are working for an investment bank in South East Asia, and have sold a large number of straddles on the Japanese stock market index, with exercise price 18,5. 1. Give two ways to achieve this straddle position. 2. What does the profit structure of your position look like? 3. What are the conditions that make this a profitable transaction? 4. How much can you lose? Exercise 8. Breakeven [1] A one year call option on a stock with a strike price of $3 costs $3. A one year put option on the stock with a strike price of $3 costs $4. Suppose that a trader buys two call options and one put option. 1. What is the breakeven stock price above which the traders makes a profit? 2. What is the breakeven stock price below which the traders makes a profit? Exercise 9. [2] A default-free zero-coupon bond costs $91 and will pay $1 at maturity in 1 year. What is the effective annual interest rate? What is the payoff diagram for the bond? What is the profit diagram for the bond? Exercise 1. Put Call Parity [3] You can currently purchase a share of stock for $25, purchase a put option on the stock for $3, and write a call option against the stock for $4. Suppose that holding these three positions guarantees a payoff of $3 one year from today. 1. If the risk free rate (with discrete, annual compounding), is 2%, does put-call parity hold? If not, then what new price of the put option would allow put-call parity to hold? Exercise 11. Arbitrage? [4] For the following cases, determine whether these data define an arbitrage opportunity. If it does involve an arbitrage opportunity, describe the transactions that can be used to exploit it. The price of an ABC call with exercise price 4, expiring at the end of six months, is currently 1.5. ABC stock is at 4, and the annual rate on a risk free security is 6%. 2
A September ABC 4 European call is trading at $3 and a September ABC 5 European call is trading at $4. Exercise 12. [2] A Canadian company wants to purchase GBP 6 months from now. The current forward rate is F = 1.6. They enter into a range forward contract, which can simplest be described in terms of the payoffs: Spot rate Contract rate S > 1.61 Purchase GBP at 1.61 1.61 > S > 1.59 Purchase GBP at S 1.59 > S Purchase GBP at 1.59 or as graph illustrating the contract price as a function of the exchange rate. 1.61 1.59 contract rate 1.59 1.61 spot 1. Discuss how such a contract can be constructed from more simple building blocks, specifically put and call options. Exercise 13. [2] A call option is at expiration with an exercise price of 15 on a stock trading for 1. What is the value of the option? 1.. 2. 2.5 3. 5. 4. Impossible to determine without knowing the standard deviation of the stock. 5. I choose not to answer. Exercise 14. A stock is selling for $31. There is a call option on the stock with an exercise price of $27. Which of the below is an appropriate minimum value of the call option today? (a) $ (b) $4 (c) $27 (d) Cannot be determined without knowing the time to expiration. (e) I choose not to answer 3
Exercise 15. Options [4] A put is worth $1 and matures in one year. A call on the same stock is worth $15 and matures in one year also. Both options are European. The put and call have the same exercise price of $4. The stock price is $5. The current price of a (risk free) discount bond (zero coupon bond) paying $1 that matures in one year is $.9. How do you make risk free profits given these prices? Exercise 16. Put Upper bound [5] Show that the following is an upper bound for the price of a put option P t (1 + r) (T t) where P t is the current put price, is the exercise price, r is the risk free interest rate and (T t) is the time to maturity of the option. 4
Empirical Solutions MØA 155 PROBLEM SET: Options Exercise 1. Arbitrage [2] (a) and (b). Exercise 2. Mutual Fund Manager [2] 1. Salary behaves like a call option. 2. Go for broke (high volatility) Exercise 3. Pricing factors [1] 1, 4 og 5: S, volatility, risk free rate Exercise 4. Product [3] 1. Suppose that S is invested in the product where S is the index level today. The value of the investment in one year is S plus the payoff from a bull spread. The bull spread is created from a long call option with strike price S and a short call option with strike price 1.2S. The payoffs (in terms of returns) from the product can be summarized as 2% Percentage return S 1.2S S T Exercise 5. Put [2] Short put Exercised if S T <. Profit if or p = 4 = 6 Payoff = max(, S T ) Profit = p max(, S T ) p ( S T ) > S T > p 5
Profit p = 4 = 6 S T Exercise 6. Straddles and Butterflies [3] Profit Buying a straddle. 1 S T 6
Profit Buying a butterfly. 11 S T A straddle is a gamble that the stock price will have a large movement either up or down. A butterfly is a gamble on the price staying close to the exercise price. Exercise 7. Bank [3] 1. Some ways to get a short straddle: Sell a put and a call. Buy the underlying, and sell two call options. Short the underlying, sell two put options. 2. Profit structure Profit Selling a straddle. S T 3. If the price stays around 18,5 you make a profit. However, you risk large losses if the price moves far away from 18,5. 4. The bank. 7
Exercise 8. Breakeven [1] +2 call +1 put Cost: 2 3 + 4 = 1. Payoff: 2 max(, S T 3) + max(, 3 S T ). 1. $35 2. $2 Exercise 9. [2] With continous compounding: 91 = e r 1 r = ln(91/1) = 9.43%. With discrete, annual compounding 91 = 1 1+r r = 91 1 1 = 9.89%. Payoff 91 Profit 9 Exercise 1. Put Call Parity [3] This can be solved several ways. One is to observe that for put call parity to hold, arbitrage opportunities must not be present. The return on a risk free payoff should equal the risk free rate. Investment is $25 + $3 - $4 = $24, Guaranteed payoff = $3, Guaranteed return = 3 24 1 = 25%. Put call parity does not hold, since the annually compounded risk free rate is 2%. Alternatively we can use the put call relation directly. C P = S 1 + r 8
Then have to argue that the exercise price equals 3, from the fact that this portfolio is known to equal 3 at maturity: 3 = S T max(, S T ) + max(, S T ) = Put call parity does not hold. Exercise 11. Arbitrage? [4] Both cases involve arbitrage opportunity C P = 4 3 = 1 S 3 = 25 1 + r 1 +.2 = 1. The first violates the inequality c max(, S e r(t t) ). To exploit this arbitrage opportunity, can for example use the following strategy. time t time T S 4 S > 4 Buy Call 1.5 S 4 Short stock 4 S S Invest risk free 4 1.5 = 38.95 38.95e.6 1 2 = 4.136 4.136 Sum 4.136 S >.136 > 2. Here there is a mispricing, the option with the higher exercise price should have a lower price. Buy the 5 option, sell the 4 option, receive 4 3 = 1 today, and max(, S 5) max(, S 4) > at option expiry. Exercise 12. [2] The following figure translates this into payoff space. payoff 1.59 1.61 spot From which we realize that it is really a combination of two options: payoff 1.59 1.61 spot Sell 1 put with = 1.59, and buy 1 call with = 1.61. Exercise 13. [2] 9
Th option is not in the money, the value is zero, (a) is correct. Exercise 14. (b) is correct Exercise 15. Options [4] Put call parity is violated. C > S P V (X) S X = 31 27 = 4 C P = 15 1 = 5 S P V (X) = 5 P V (4) 1 No need to calculate PV(4), since it must be less than or equal 4, but can do it using To make money, buy sheep and sell deer. Exercise 16. Put Upper bound [5] r = 1.9 1 = 11.11% S P V (X) = 5 The most you can get at exercise is (if stock price is zero). The value today is less than the present value of this, or 1 p (1 + r) (T t) 4 1 +.1111 = 14 Put 1 (1+r)(T t) 1 (1+r)(T t) S T 1