Financial Derivatives Spring 2016 page 1 of 3 Problem Set 3: Option Pricing and Hedging 1. The following prices are observed. Formulate an arbitrage strategy to profit from the situation (ignoring transactions costs). • Digital Datawhack stock is selling for $95 per share. • Call options on Digital Datawhack at 90, with August 2016 expiration, are selling for $12 per share. • Put options on Digital Datawhack at 90, with August 2016 expiration, are selling for $2.50 per share. • At the current T-Bill rate, $86 invested today would grow to $90 at the expiration date of the options. 2. The following prices are observed. Formulate an arbitrage strategy to profit from the situation (ignoring transactions costs). • Arcott Labs stock is selling for $50 per share. • Call options on Arcott Labs at 45, with expiration in nine months, are selling for $11 per share. • Call options on Arcott Labs at 50, with expiration in nine months, are selling for $2 per share. • Put options on Arcott Labs at 45, with expiration in nine months, are selling for $3 per share. • Put options on Arcott Labs at 50, with expiration in nine months, are selling for $5 per share. • At the current T-Bill rate, $42.70 invested today would grow to $45 (and $47.44 would grow to $50) at the expiration date of the options. 3. The following prices are observed. Show how to borrow money at the risk-free rate, using only option contracts. • Morgenstern Labs stock is selling for $91.50 per share. • Call options on Morgenstern Labs at 85, with expiration in nine months, are selling for $14.50 per share. • Call options on Morgenstern Labs at 90, with expiration in nine months, are selling for $11.875 per share. • Put options on Morgenstern Labs at 85, with expiration in nine months, are selling for $3.75 per share. • Put options on Morgenstern Labs at 90, with expiration in nine months, are selling for $5.875 per share. • At the current T-Bill rate, $85.50 invested today would grow to $90 (and $80.75 would grow to $85) at the expiration date of the options. Financial Derivatives Spring 2016 page 2 of 3 4. Digital Datawhack stock rose by $2, from $60 to $62. “Crunch” Murdoch owns call options on Datawhack, which he bought two months ago for $5 per share. They expire in seven months, and have an exercise price of $55. Which of the following is a theoretically possible response of the option value (on a per-share basis), to the recent move of the stock? a. rise by $2.10 b. rise by $1.80 c. fall by $2.05 d. fall by $1.75 e. None of the above 5. Option A is a call option written on Digital Datawhack stock, with expiration in one month. Option B is a call on Datawhack with expiration in six months. Both options have an exercise price of $50 per share. Which of the following is a theoretically correct relationship? a. Option A is worth more than Option B. b. Option B is worth more than Option A. c. Option B is worth the same as Option A. 6. Option A is a call option written on Digital Datawhack, with exercise price of $50. Option B is also a call option on Datawhack, with exercise price of $55. Both options expire six months from now. Which of the following is a theoretically correct relationship? a. Option A is worth more than Option B. b. Option B is worth more than Option A. c. Option B is worth the same as Option A. 7. Suppose Treasury Bill rates have risen, but nothing else has changed. What would be the theoretical response of call option values (for calls written on U.S. stocks)? a. rise b. fall c. remain unchanged 8. Suppose the exercise price of an option is $50, the Treasury Bill rate is 5%, and there are 179 days remaining until expiration. Discounted continuously, what is the present value of the exercise price? 9. Suppose the stock price is $56 and the exercise price of a call option is $50. Volatility of the stock is a standard deviation of 20%. The information in problem 8 holds for this call. Do the following calculations: a. Find d1 and d2 b. Using the Normal Distribution table, estimate N(d1) and N(d2) c. Use these estimates to calculate the value of the call option d. Use the option calculator to find the value of this call. Explain why the computer’s answer differs from your hand calculations. e. What is Delta for this option? f. Find the value of a put with the same exercise price and expiration. Financial Derivatives Spring 2016 page 3 of 3 10. Use the option calculator spreadsheet to find Gamma for the option in problem 9. Suppose there is also an option with the same expiration and an exercise price of $55. Formulate a Gamma hedge for the option in problem 9. (See the discussion of Gamma hedging in the text.) 11. Suppose the option in problem 9 has a market premium of $822.80 for a 100-share contract ($8.228 per share). Use the option calculator spreadsheet or one of the programs provided with the text to find the implied volatility by trial and error. 12. Explain each of the following concepts as they relate to call options. What roles do they play in hedges? a. Delta b. Gamma c. Rho d. Vega e. Theta 13. Explain what we mean when we say that the binomial model is a discrete time model and the Black-Scholes model is a continuous time model. 14. Be prepared to explain Brownian Motion in class, giving examples. What makes us think stock price changes follow this pattern? 15. Suppose you subscribe to a service that gives you estimates of the theoretically correct volatilities of stocks. You note that the implied volatility of a particular option is substantially higher than the reported volatility. Assuming you trust the accuracy of the estimates you receive, what action would you consider? 16. Answer the following questions as they relate to implied volatilities. a. Can implied volatilities be expected to vary for options on the same stock with the same exercise prices but different expirations? b. Can implied volatilities be expected to vary for options on the same stock with the same expiration but different exercise prices? 17. Is it practical to expect a delta hedge to be risk free? Explain. 18. Is it practical to expect a gamma hedge to be risk free? Explain. Financial Derivatives 1. 2. 3. Solutions: Problem Set 3 This is a violation of put-call parity, which offers a nice opportunity for arbitrage. To take advantage of it, buy stock and puts for $97.50 per share. Sell calls and bonds (that is, write a call and borrow the present value of the exercise price) and receive $98.00 per share. Having bought a put and written a call with exercise price of $90 per share, you are committed to sell your stock for $90, which will be just enough money to pay off the loan. Your future obligations are covered, and you can still pocket a profit of $.50 per share. If you can do this in a large volume, say 100 contracts at 100 shares per contract, you'll make an arbitrage profit of $5,000. This is a double violation of put-call parity, which offers an opportunity to take advantage of a box spread. At the $45 exercise price, you would buy stock and puts while selling calls and bonds. At the $50 exercise price, you would sell stock and puts while buying calls and bonds. The stock transactions cancel each other, and the bond transactions leave you with a net purchase of $5 face value. You are thus committed to buy a share of stock for $50 and sell it for $45 (having bought a call and written a put with exercise price of $50 per share, and also having bought a put and written a call with exercise price of $45)—so you break even, after you collect the $5 from the bond investment. Your future obligations are completely covered, and you can still pocket a profit of $6.26 per share. Contract to buy stock at $90 per share and sell it at $85, thus creating an obligation to pay $5 on the expiration date (this is the loan repayment). In order to do this, create a box spread in which you buy a call and write a put at the $90 exercise price, while selling a Spring 2016 call and buying a put at the $85 exercise price. 4. B. 5. B. Option B has longer to expiration. 6. The lower the exercise price, the higher the value of the call option. Prior to expiration, A is worth more than B. After expiration, both are worthless. 7. A 8. Enter .05 * 179/365. Change the sign to negative, then press the ex button (exp on some calculators). Multiply the result times $50. The answer is $48.79. Save this in the calculator’s memory so you have the full precision for use in later problems. 9. d1 is 1.0543 and d2 is 0.9142. Delta is 0.8541, and the model premium is $7.84. Remaining questions are for class discussion. 10. For class discussion 11. For class discussion 12. For class discussion 13. For class discussion 14. For class discussion 15. For class discussion 16. For class discussion 17. For class discussion 18. For class discussion