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Problem Set 3

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UNIVERSITY OF TORONTO AT SCARBOROUGH DEPARTMENT OF MANAGEMENT MGTC71: Introduction to Derivatives Markets Problem SET 3 1.The following are European call and put prices for a stock on November 9 (today): Last transaction Prices ($) Calls Puts Strike Price Dec Mar June Dec Mar June 65 3 4 5/8 -- 2 -- 4 The maturity dates of the options are on Fridays preceding the third Saturday in each month which is 37 days from now (for the December options). You may assume that the annual risk free interest rate is 7%. The stock closed at 65 1/4 on November 9. a) is there a violation of the put-call parity condition. If so indicate clearly which securities you would buy and sell and indicate the amount of your risk-free arbitrage profit. 2. Suppose a two-month European put option on a non-dividend-paying stock is currently selling for $2 and that the stock price is $47, the strike price is $50, and the risk-free interest rate is 6% (C.C.). a) Is there an opportunity for arbitrage? Design a strategy to exploit this arbitrage? b) If the stock pays a dividend of $2/per share in 1 month, would the above market prices still provide an arbitrage opportunity? 3. You own a call option on Microsoft with an expiration date of 6 months from now. Microsoft is going to start paying a large dividend every quarter to shareholders. (Investors are indifferent to Microsoft paying the dividend and retaining the cash in the company - thus there is no stock movement at the time of the announcement.) a) What do you expect to happen to the price of the call option you own? What about a put option if you owned that? (Assume that volatility, exercise price and interest rates remain constant). b) Assume you are selling Microsoft put options to investors. You would like to sell or write a put option with a strike price of $65 with 4 months to maturity. How can you use a combination of Microsofts stock, risk-free bonds and Microsoft call options to hedge the risk you would have from selling put options? Be precise: (Drawing some graphs may help you solve this problem - but still decide what securities you should buy or sell.) Assume that risk free interest rate is 6%, and a call with a strike price of $65.00 sells for $2.75. The current stock price is $62.00. c) What is the minimum amount you should charge for this put option that you will sell? 4. A stock is currently selling for $100. The current price of a 6-month call option on the stock with an exercise price of $105 is selling for $12. The risk-free interest rate is 5%, compounded continuously. a) Assuming there is no dividend paid on the stock, compute the price of a 6-month put option on the stock with exercise price equal to $105. b) Imagine that the put is selling in the market for $13. Is there an arbitrage opportunity? If so, execute a strategy to take advantage of the arbitrage. Clearly state your positions and cash flows in an arbitrage table. c) Ignore parts (a) and (b). Imagine that you are unable to short-sell a particular stock. Using put-call parity, replicate a short position in the stock, assuming that the stock pays no dividends, there is a put and a call option, both of which have the same exercise price, X, and the same time to expiration, T. You are able to borrow and lend at the continuously compounded risk-free rate, r. Use an arbitrage table to support your argument. 5. XYZ is trading at $90 per share, and an American call option on XYZ with strike price $80 and 6 months to maturity is trading at $11. The risk-free interest rate is 8% (C.C). XYZ will make a dividend payment in exactly 1 month and will not make any other dividend payment within the next 6 months. The amount of XYZ dividend in 1 month is not known yet and could be anywhere between $1 per share to $5 per share. a) Do the above market prices provide riskless arbitrage opportunity? b) If XYZ announces that the dividend in 1 month will be $1 per share and the above prices stay the same, would there be an arbitrage opportunity? Use a table to show how you can exploit it. 6. Consider a forward contract, with delivery price K and maturity T, on a stock that pays a continuous dividend yield of q. (at time t, the stock pays dividend q. St per unit time.) a) Use the Put-Call Parity (for options on a stock with a continuous dividend yield) to derive the formula for the value of the forward contract. (Note: think of the forward contract as a portfolio composed of call and put options.) b) Use the derived formula to calculate the market value of a forward contract to purchase 1 million DM in 6 months at the price of $.65/DM. Assume that the US and German interest rates are, respectively, 5% and 7% (C.C.). The current Dollar-DM exchange rate is $.66/DM 7. On Friday November 17, 2005 XYZs stock price closed at 25. The following option prices were quoted as: Strike Expiration Call Price Put Price 22.50 Dec 3.30 0.90 25 Dec 1.85 1.80 27.50 Dec 0.85 3.20 30 Dec 0.35 5.40 a) Plot the profit diagram for a short position in a December straddle with a strike price of 25 as a function of the stock price at expiration and fill in the requested information. b) Plot the profit diagram (at expiration) for a bull spread based on the December call options with strike prices 25 and 30. c) Show the profit (at expiration) from an investment that buys the stock and sells the December call option with a strike price of 27.50. d) Plot the profit diagram (at expiration) for a butterfly spread based on the call options with strike prices of 22.50, 25 and 27.50. e) Plot the profit diagram (at expiration) for a collar based on a short call option with strike prices of 27.50, and a long put option with a strike price of 22.50, and a long position in the stock. 8. Suppose only securities available for trading are calls and puts (with any strike but same maturity T) on a stock. Answer the following three questions: a) Show in a table and then graph the payoff at maturity of a portfolio consisting of short 1 put option with strike price of $40 and long 2 call options with strike price of $70.b) Show in table and graph a the payoff at maturity of a portfolio consisting of long 2 call options with strike price of $30 and short 4 call options with strike price of $50, and long 2 call options with a strike price of $70. Briefly explain the bet behind such a portfolio. c) Suppose you are interested in replicating the following payoff at time T. What portfolio of Calls (same maturity any strike) and/or bonds (paying $1 dollar at maturity) will give you the desired payoff? Payoff 50 30 ST 30 50 100 9. Consider an American call w
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