Options Ans

Options 1. The common stock of PUTT Corporation has been trading in a narrow price range for the past month and you are convinced it is going to break far out of that range in the next three months. You do not know whether it will go up or down. The current price of the stock is 100 per share and the price of the three month call option at an exercise price of 100 is 10. a. If the risk free interest rate is 10% p.a., what must be the price of a three month put at an exercise price of 100 ( Stock pays no dividends) b. What would be a simple option strategy to exploit your conviction about the stock price’s future movements? How far would it have to move in either direction for you to make a profit on your initial investment? a. From put-call parity, P = C – S0 + X/(1 + rf)T P = 10 – 100 + 100/(1.10)1/4 = 7.645 b. Purchase a straddle, i.e., both a put and a call on the stock. The total cost of the straddle would be 10 + 7.645 = 17.645, and this is the amount by which the stock would have to move in either direction for the profit on the call or put to cover the investment cost (not including time value of money considerations). Accounting for time value, the 1/4 stock price would need to swing in either direction by 17.645 χ(1.10)  = 18.07. 2. The common stock of Call Corporation has been trading in a narrow price range around 50 per share for months and you are convinced it is going to stay within that range for the next three months. The price of the three month put option at an exercise price of 50 is 4. a. If the risk free interest rate is 10% p.a., what must be the price of a three month call at an exercise price of 50 if it is at the money? ( Stock pays no dividends) b. What would be a simple option strategy using a put and a call to exploit your conviction about the stock price’s future movements? What is the most money you can make on this position? How far can the stock price move in either direction before you lose money? c. How can you create a position involving a put, a call and risk less lending that would have the same payoff structure as the stock at expiration? What is the net cost of establishing that position now? a. From put-call parity, C = P + S0 – X/(l + rf)T C = 4 + 50 – 50/(1.10)1/4 = 5.18 b. Sell a straddle, i.e., sell a call and a put to realize premium income of 4 + 5.18 = 9.18. If the stock ends up at 50, both the options will be worthless and your profit will be 9.18. This is your maximum possible profit since at any other stock price, you will need to pay off on either the call or the put. The stock price can move by 9.18 in either direction before your profits become negative. c. Buy the call, sell (write) the put, lend 50/(1.10)1/4. The payoff is as follows: Position

Call (long) Put (short)

Immediate CF

C=

5.18

– P = – 4.00

Lending position –––––––––––––––––– TOTAL C – P + = 50.00

CF in 3 months ST < X –––––– 0

ST > X –––––– ST – 50

–(50 – ST)

0

50/(1.10)1/4 = 48.82 ––––––– ––––––– ST ST

50

50

1

By the put-call parity theorem, the initial outlay equals the stock price, S0, or 50. In either scenario, you end up with the same payoff as you would if you bought the stock itself. 3. a. A butterfly spread is the purchase of one call at exercise price X 1, sale of two calls at X2 and the purchase of one call at X 3. X1 < X2 and X2< X3 by equal amounts and all calls have the same expiration date. Graph the payoff diagram in this strategy. b. A vertical combination is the purchase of a call with exercise price X 2 and a put with exercise price X1 with X2> X1. Graph the payoff to this strategy a. Butterfly Spread ST < X 1

Position

X1 ≤ ST ≤ X2

X2 < S T ≤ X 3

X3 < ST

Long call (X1)

0

ST – X1

ST – X 1

ST – X1

Short 2 calls

0

0

–2(ST – X2)

–2(ST – X2)

Long call (X3)

0

0

0

ST – X3

Total

0

ST – X1

2X2 – X1 – ST

(X2–X1 ) – (X3–X2) = 0

(X2)

Payoff

X2 – X 1

ST X1

X2

X3

4. A bearish spread is the purchase of a call with exercise price X 2 and the sale of a call with exercise price X1 with X2 > X1. Graph the payoff of this strategy Bearish spread Position ––––––––––– Buy Call (X2)

ST < X 1 –––––– 0

X1 ≤ ST ≤ X2 –––––––––– 0

X2 < ST ––––––––– ST – X 2

Sell Call (X1) –––––––––––

0 ––––––

–( ST – X1) ––––––––––

–( ST – X1) –––––––––

0

X 1 – ST

Total

X 1 – X2

P a y o f f 0 X 1

X 2

S T

P a y o f f ) – ( X – X 2 1

2

5. JJ received 10000 shares of company stock as his compensation package. The stock currently sells at 40 a share. He would like to defer selling the stock until the next tax year in April. In April he would have to sell all his holdings for a down payment on his house. JJ is worried about the price risk involved in keeping his stock. At current prices he would receive 400000 for the stock. If the value of the stock falls below 350000 his ability to come up with the necessary down payment would be jeopardized. On the other hand if the stock value rises to 450000 he would be able to maintain a small cash reserve even after making the down payment. JJ considers three strategies a. Write call options on the stock with strike price 45 selling at 3 each b. Buy April put options on the stock with strike price 35 also selling at 3 each c. Establish a zero cost collar by writing April calls and buying April puts. Evaluate each of these strategies with respect to JJ’s goals. What are the advantages and disadvantages of each strategy and which one would you recommend? a. By writing covered call options, JJ takes in premium income of 30,000. If the price of the stock in January is less than or equal to 45, he will have his stock plus the premium income. But the most he can have is 450,000 + 30,000 because the stock will be called away from him if its price exceeds 45. (We are ignoring interest earned on the premium income from writing the option over this short time period.) The payoff structure is: Stock price Portfolio value less than 45 10,000 times stock price + 30,000 more than 45 450,000 + 30,000 = 480,000 This strategy offers some extra premium income but leaves substantial downside risk. At an extreme, if the stock price fell to zero, JJ would be left with only 30,000. The strategy also puts a cap on the final value at 480,000, but this is more than sufficient to purchase the house. b. By buying put options with a 35 exercise price, JJ will be paying 30,000 in premiums to insure a minimum level for the final value of his position. That minimum value is 35  10,000 – 30,000 = 320,000. This strategy allows for upside gain, but exposes JJ to the possibility of a moderate loss equal to the cost of the puts. The payoff structure is: Stock price Portfolio value less than 35 350,000 – 30,000 = 320,000 more than 35 10,000 times stock price – 30,000 c. The net cost of the collar is zero. The value of the portfolio will be as follows: Stock price Portfolio value less than 35 350,000 between 35 and 45 10,000 times stock price more than 45 450,000 If the stock price is less than or equal to 35, the collar preserves the 350,000 in principal. If the price exceeds 45 JJ gains up to a cap of 450,000. In between, his proceeds equal 10,000 times the stock price. The best strategy in this case would be (c) since it satisfies the two requirements of preserving the 350,000 in principal while offering a chance of getting 450,000. Strategy (a) seems ruled out since it leaves JJ exposed to the risk of substantial loss of principal. Our ranking would be:

(1) c

(2) b

(3) a

6. You are attempting to formulate an investment strategy. On the one hand you think there is a great upward potential in the stock market and would like to participate in the upward movement if it materializes. However you are not able to afford substantial losses in the market and so can not run the risk of a stock market collapse, which you also think is a possibility. Your investment advisor suggests a protective put position: Buy both shares in a market index stock fund and put options on those shares with three month maturity and exercise price 780. The stock index is currently selling for 900. However your uncle suggests you

3

instead buy a three month call option on the index fund with exercise price 840 and buy three month T-bills with face value of 840. a. On the same graph draw the payoffs to each of these strategies as a function of the stock fund value in three months. (Hint Think of the options as being on one share of the stock index fund with the current price of each share of the index fund equal to 900) b. Which portfolio must require a greater initial outlay to establish? (Hint; Does either portfolio provide a final payout that is always at least as great as the payoff of the other portfolio? c. Suppose the market price of the securities are as follows: Stock fund T Bill (FV 840) Call (Exercise Price)

90 0 81 0 12 0

Put (Exercise Price)

6 Make a table of the profits realised for each portfolio for the following values of the stock price in three months S T =700, 840, 900, 960. Graph the profits to each portfolio as a function of S T on a single graph. d. Which strategy is riskier? Which should have a higher beta? e. Explain why the data given for the securities in part c above does not violate the put call parity relationship? a.

Protective Put ST ≤ 780 ST > 780 –––––––––––––––––––––––––––––––––––––––––––– Stock ST ST Put 780 – ST 0 ––––– ––––––– ––– Total 780 ST Bills and Call ST ≤ 840 ST > 840 –––––––––––––––––––––––––––––––––––––––––––– Bills 840 840 Call 0 ST – 840 ––––– –––– ––––––– Total 840 ST

4

Payoff Bills plus calls

840

780

Protective put strategy

ST 780

840

b. The bills plus call strategy has a greater payoff for some values of ST and never a lower payoff. Since its payoffs are always at least as attractive and sometimes greater, it must be more costly to purchase. c. The initial cost of the stock plus put position is 906; that of the bills plus call position is 930.

Stock +Put Payoff Profit

ST = 700 ––––––– 700 80 780 –126

ST = 840 ––––––– 840 0 840 –66

ST = 900 ––––––– 900 0 900 –6

ST = 960 ––––––– 960 0 960 54

Bill +Call Payoff Profit

840 0 840 –90

840 0 840 –90

840 60 900 –30

840 120 960 +30

P r o f i t P r o t e c t i v e p u t B i l l s p l u s c a l l s 7 8 09 0 0

S T

3 0 1 2 6

d. The stock and put strategy is riskier. It does worse when the market is down and better when the market is up. Therefore, its beta is higher.

5

e. Parity is not violated because these options have different exercise prices. Parity applies only to puts and calls with the same exercise price and expiration date. 7.

In what ways is owning a corporate bond similar to writing a put option? A call option? The bondholders have in effect made a loan which requires repayment of B dollars, where B is the face value of bonds. If, however, the value of the firm, V, is less than B, the loan is satisfied by the bondholders taking over the firm. In this way, the bondholders are forced to “pay” B (in the sense that the loan is cancelled) in return for an asset worth only V. It is as though the bondholders wrote a put on an asset worth V with exercise price B. Alternatively, one may view the bondholders as giving the right to the equityholders to reclaim the firm by paying off the B dollar debt. They’ve issued a call to the equity holders. 8. A member of an investment committee interested in learning more about fixed income investment procedures, recalls that a fixed income manager recently stated that derivative instruments could be used to control portfolio duration, saying “ a futures –like position can be created in a portfolio by using put and call options on Treasury bonds” a. Identify the option market exposure or exposures that create a “futures – like position” similar to being long Treasury bond futures. Explain why the position you created is similar to being long Treasury bond futures b. Explain in which direction and why the exposure(s) you identified in part (a) would affect the portfolio duration. c. Assume that a pension plan’s investment policy requires the fixed income manager to hold portfolio duration within a narrow range. Identify and briefly explain circumstances or transactions in which the treasury bond futures would be helpful in managing a fixed income portfolio when duration is constrained. a. If one buys a call option and writes a put option on a T-bond, the total payoff to the position at maturity is ST – X, where ST is the price of the T-bond at the maturity date, time T. This is equivalent to the profits on a forward or futures position with futures price X. If you choose an exercise price, X, equal to the current T-bond futures price, the profit on the portfolio will replicate that of market-traded futures. b. Such a position will increase the portfolio duration, just as adding a T-bond futures contract would increase duration. As interest rates fall, the portfolio gains additional value, so duration is longer than it was before the synthetic futures position was established. d. Futures can be bought and sold very cheaply and quickly. They give the manager flexibility to pursue strategies or particular bonds that seem attractively priced without worrying about the impact on portfolio duration. The futures can be used to make any adjustments to duration necessitated by other portfolio actions 9. Consider the following portfolio. You write a put option with exercise price 90 and buy a put option on the same stock with exercise price 95 with the same maturity date. a. Plot the value of the portfolio at the maturity date of the options b. On the same graph, plot the profit of the portfolio. Which option must cost more? a. Written Call Written Put Total

105 ≤ ST ≤ 110  0 0 – (105 – ST) 0   ST – 105 0 ST < 105 

ST > 110  – (ST – 110) 0  110 – ST

6

P a y o f f 1 0 5

1 1 0

W r i t e p u t

b.

S T W r i t e c a l l

Proceeds from writing options: Call Put Total

=

2.85 = =

4.40 7.25

If stock sells at 107, both options expire out of the money, and profit = 7.25. If it sells at 120 the call written results in a cash outflow of 10 at maturity, and an overall profit of 7.25 – 10 = −2.75. c.

You break even when either the put or the call written results in a cash outflow of 10.25. For the put, this would require that 7.25 = 105 – S, or S = 97.75. For the call this would require that 7.25 = S – 110, or S = 117.25.

d. The investor is betting that stock price will have low volatility. This position is similar to a straddle.

10. You buy a share of stock, write a one year call option with X = 10 and buy a one year put option with X=10. Your net outlay to establish the entire portfolio is 9.50. What is the risk free rate? The stock pays no dividends. The following payoff table shows that the portfolio is riskless with time-T value equal to 10. Therefore, the risk-free rate must be 10/9.50 – 1 = . 0526 or 5.26%. ST ≤ 10 ST > 10 –––––––––––––––––––––––––––––––––––––––––––– Buy stock ST ST Write call 0 – (ST – 10) Buy put 10 – ST 0 ––––––– –––––––– Total 10 10 11. Demonstrate that an at- the - money call option on a given stock must cost more than an at – the – money. From put-call parity, C – P = S0 – X/(l + rf)T If the options are at the money, then S0 = X and therefore, C – P = X – X/(l + rf)T which must be positive. Therefore, the righthand side of the equation is positive, and we conclude that C > P.

7

12. John has purchased a stock index fund currently trading at 400 per share. To protect against losses he purchased an at – the – money European put option on the fund for 20 with exercise price 400 and three months to expiration. His financial advisor points out that Joe is spending a lot of money on the put and observes that three months puts with strike price of 390 cost only 15 and suggests that he use the cheaper put. a. Analyse the strategies by drawing the profit diagrams for various value of the funds in three months b. When does the financial advisor’s strategy do better? When does it do worse? c. Which strategy entails greater systematic risk? a. Joe’s strategy Payoff Cost ST < 400 ST > 400 Stock index 400 ST S Put option (X=400)20 400 – ST 0 Total 420 400 S Profit = payoff – 420 –20 ST – 420 Sally’s Strategy Payoff Cost ST < 390 Stock index 400 ST Put option (X=390)15 390 – ST Total 415 390 Profit = payoff – 415 –25

ST > 390 ST 0 S ST – 415

b. Sally does better when the stock price is high, but worse when the stock price is low. (The break-even point occurs at S = 395, when both positions provide losses of 20.) c. Sally’s strategy has greater systematic risk. Profits are more sensitive to the value of the stock index.

Profit

Sally 390

400

Joe ST

-20 -25

13. You write a call option with X=50 and buy a call with X=60. the options are on the same stock and have the same maturity dates. One of the calls sells for 3 and the other sells for 9. a. Draw the payoff graph for this strategy at the option maturity date. b. Draw the profit graph for this strategy c. What is the break even point for this strategy? Is the investor bullish or bearish in the stock? This strategy is a bear spread. The initial proceeds are 9 – 3 = 6. The ultimate payoff is either negative or zero: ST < 50 50 ≤ ST ≤ 60 ST > 60

8

––––––––––––––––––––––––––––––––––––––––– Buy call (X = 60) 0 0 ST – 60 Write call (X = 50) 0 – (ST – 50) – (ST – 50) –––––– –––––––––– –––––––– TOTAL 0 – (ST – 50) –10 c. Breakeven occurs when the payoff offsets the initial proceeds of 6, which occurs at a stock price of ST = 56. The investor must be bearish: the position does worse when the stock price increases.

6

0

50

60

ST

-4

Profit

-10

Payoff

14. Which one of the following comparative statements about common stock call options and warrants is correct Call Warra Option nt 1 Issued by the company No Yes 2 Sometimes attached to Yes Yes bonds 3 Maturity greater than one Yes No year 4 Convertible into stock Yes No a. Consider a bullish spread option strategy using a call option with 25 strike price priced at 4 and a call option with strike price 40 priced at 2.50. If the price of the stock increases to 50 at expiration and the option is exercised on the expiration date , the net profit per share at expiration is i. 8.50 ii. 13.50 iii. 16.50 iv. 23.50 b. A convertible bond sells at 100 par with a conversion ratio of 40 and an accompanying stock price of 20 per share. The conversion premium and percentage conversion premium are i. 200 and 20% ii. 200 and 25% iii. 250 and 20% iv. 250 and 25% c. A put on XYZ stock with a strike price of 40 is priced at 2, while a call with a strike price of 40 is priced at 3.50. What is the maximum per share loss to the writer of the uncovered put and the maximum per share gain to the writer of the uncalled call? Maximum loss to put Maximum gain to call writer writer 1 38 3.5 2 38 36.50 3 40 3.50 4 40 40

9

d. You create a strap by buying two calls and one put on ABC stock with a strike price of 45. The calls cost 5 each and the put costs 4. If you close your position when ABC is priced at 55. your per share gain or loss is: i. 4 loss ii. 6 gain iii. 10 gain iv. 20 gain a.(i) b.(ii) [Profit = 40 – 25 + 2.50 – 4.00] c.(ii)[Conversion premium is 200, which is 25% of 800] d.(i) 15. The value of a call option increases with the volatility of the stock. Is this also true for put option values. Use the put call parity theorem as well as a numerical example to prove your answer. Put values also must increase as the volatility of the underlying stock increases. We see this from the parity relation as follows: P = C + PV(X) – S0 + PV(Dividends). Given a value of S and a risk-free interest rate, if C increases because of an increase in volatility, so must P to keep the parity equation in balance. 16. In each of the following questions you are asked to compare two options with parameters as given. The risk free rate for all cases is 6%. Assume that the stocks on which the options are written pay no dividends. a. PU T X σ Price of the T option A . 5 0.2 10 5 0 B . 5 0.2 10 5 0 5 Which put option is written on the stock with the lower price? b. PU T A

Which c. Which

d. Which

e. Which

T

X

σ

Price of option 10

the

. 5 0. 5 0 2 B . 5 0. 12 5 0 2 put option must be written on the stock with the lower price? call option must have the lower time to maturity S X σ Price of the Call option A 5 5 0. 12 0 0 2 B 5 5 0. 10 5 0 2 call option is written on the stock with higher volatility Ca T X S Price of the ll option A . 5 5 10 5 0 5 B . 5 5 12 5 0 5 call option is written on the stock with higher volatility Ca T X S Price of the ll option A . 5 5 10 5 0 5 B . 5 5 7 5 0 5

10

a. Put A must be written on the lower-priced stock. Otherwise, given the lower volatility of stock A, put A would sell for less than put B. b. Put B must be written on the stock with lower price. This would explain its higher value. c. Call B must be written on the stock with lower time to expiration. Despite the higher price of stock B, call B is cheaper than call A. This can be explained by a lower time to expiration. d. Call B must be written on the stock with higher volatility. This would explain its higher price. e. Call A must be written on the stock with higher volatility. This would explain the higher option premium. 17. We will derive a two state put option value in this problem. Data S o = 100, X= 110, 1+r=1.10. The two possibilities for ST are 130 and 80. a. Show that the range of S is 50 whereas that of P is 30 across the two states. What is the hedge ratio b. Form a portfolio of three stocks and five puts. What is the non random payoff to the portfolio? What is the present value of the portfolio? c. Given that the stock is currently trading at 100, solve for the value of the put. a.

b.

When ST is 130, P will be 0. When ST is 80, P will be 30. The hedge ratio is (P+ – P–)/(S+ – S–) = (0 – 30)/(130 – 80) = –3/5. Riskless Portfolio 3 shares 5 puts 150 Total 390

S = 80 240 390 0 390

S = 130

Present value = 390/1.10 = 354.545 c. The portfolio cost is 3S + 5P = 300 + 5P, and it is worth 354.545 . Therefore P must be 54.545/5 = 10.91. 18. Calculate the value of a call option on the stock in the previous problem with an exercise price of 110.. Verify that the put call parity is satisfied by your answers to problem 17 and 18. (Do not use continuous compounding to calculate the present value of X in this example because we are using a two state model here and not continuous time) The hedge ratio for the call is (C+ – C–)/(S+ – S–) = (20 – 0)/(130 – 80) = 2/5. Riskless Portfolio

S = 80

S = 130

2 shares 160 260 5 calls written 0 –100 Total 160 160 –5C + 200 = 160/1.10. Therefore, C = 10.91. Does P = C + PV(X) – S? 10.91 = 10.91 + 110/1.10 – 100 Hence 10.91 = 10.91 19. Use the Black Scholes formula to value a call on the following stock: Time to Maturity Standard Deviation

6 months 50% p.a.

11

Exercise Price Stock Price Interest rate d1 = .3182 d2 = – .0354 Xe- rT = 47.56 C = 8.13

50 50 10% N(d1) = .6248 N(d2) = .4859

20. Find the value of a put option on the above stock P = 5.69. This value is from our Black-Scholes spreadsheet, but note that we could have derived the value from put-call parity: P = C + PV(X) – S0 = 8.13 + 47.56 − 50 = 5.69 21. Recalculate the value of the call option in problem 19 by submitting one of the changes keeping the other parameters the same as in problem 19 Time to Maturity

3 months Standard 25% Deviation p.a. Exercise Price 55 Stock Price 55 Interest rate 15% a. C falls to 5.5541 b. C falls to 4.7911 c. C falls to 6.0778 d. C rises to 11.5066 e. C rises to 8.7187 22. A call option with X = 50 on a stock currently priced at S=55 is selling for 10. Using a volatility estimate of σ =0.3 you find that N(d1) = 0.6 and N(d2) = 0.5. the risk free rate is zero. Is the implied volatility based on the option price more or less than 0.3? Explain. According to the Black-Scholes model, the call option should be priced at 55 X N(d1) – 50 X N(d2) = 55 X .6 – 50 X .5 = 8 Because the option actually sells for more than 8, implied volatility is higher than .30 23. Would you expect a 1 increase in the call option’s exercise price to lead to a decrease in the option’s value of more or less than 1 Less. The change in the call price would be 1 only if (i) there were 100% probability that the call would be exercised, and (ii) the interest rate were zero 24. Is the put option on a high beta stock worth more than one on a low beta stock? the firms have identical firm specific risk Holding firm-specific risk constant, higher beta implies higher total stock volatililty. Therefore, the value of the put option will increase as beta increases 25. All else equal is a call option on a stock with a lot of firm specific risk worth more than one on a stock with little firm specific risk. The betas of the two stocks are equal Holding beta constant, the high firm-specific risk stock will have higher total volatility. The option on the stock with higher firm-specific risk will be worth more. 26. If the stock price falls and the call price rises, then what has happened to the call option’s implied volatility? Implied volatility has increased. If not, the call price would have fallen. 27. If the time to maturity falls and the put price rises, then what has happened to the put option’s implied volatility Implied volatility has increased. If not, the put price would have fallen

12

28. According to the Black Scholes formula what will be the value of the hedge ratio of a call option as the stock price become infinitely large. The hedge ratio approaches one. As S increases, the probability of exercise approaches 1.0. N(d1) approaches 1.0. 29. According to the Black Scholes formula what will be the value of the hedge ratio of a put option on a very small exercise price. The hedge ratio approaches –1.0. As S decreases, the probability of exercise approaches 1. [N(d1) –1] approaches –1 as N(d1) approaches 0. 30. The hedge ratio of an at the money call option is 0.4. the hedge ratio of an at the money option is -0.6. What is the hedge ratio of an at the money straddle position. The hedge ratio of the straddle is the sum of the hedge ratios of the individual options: .4 + (–0.6) = –0.2. 31. A collar is established by buying a share of stock for 50, buying a six month put option with exercise price 45 and writing a six month call option with exercise price 55. Based on the volatility of the stock you calculate that for a strike price of 45 and maturity of 6 months N(d1) = 0.6 whereas for the exercise price of 55, n(d1) = 0.35. a. What will be the gain or loss on the collar if the stock price increases by 1 b. What happens to the delta of the portfolio if the stock price becomes very large? Very small? a. The delta of the collar is calculated as follows: Stock Put purchased (X = 45) Call written (X = 55) ––––––––––––––––– Total

Delta 1.0 N(d1) – 1 = –.40 – N(d1) = –.35 –––––––––––––– .25

If the stock price increases by 1, the value of your position will increase by .25. The stock will be worth 1 more, the loss on the purchased put will be .40, and the call written will represent a liability that increases by .35. b. If S becomes very large, then the delta of the collar approaches zero. Both N(d1) terms approach 1. Intuitively, for very large stock prices, the value of the portfolio is simply the (present value of the) exercise price of the call, and is unaffected by small changes in the stock price. As S approaches zero, the delta also approaches zero: both N(d1) terms approach 0. For very small stock prices, the value of the portfolio is simply the (present value of the) exercise price of the put, and is unaffected by small changes in the stock price. 32. The board of directors of ABC is concerned about the downside risk of a 100 million equity portfolio in its pension plan. The board’s consultant has proposed temporarily (one month) hedging the portfolio with either futures or options. referring to the following table the consultant states: a. “The 100 million equity portfolio can be fully protected on the downside by selling 2000 futures contract” b. “The cost of this protection is that the portfolio’s expected arte of return will be zero percent.” Market, Portfolio and Contract data Equity index level 99.00 Equity futures price 100.0 0 Futures contract 500 multiplier Portfolio beta 1.20 Contract expiration 3 (months) Critique the accuracy of each of the consultant’s two statements. Statement a: The hedge ratio (determining the number of futures contracts to sell) ought to be adjusted by the beta of the equity portfolio, which is given as 1.20. The proper hedge ratio would be:

13

× β = 2,000 × β = 2,000 × 1.2 = 2,400 Statement b: The portfolio will be hedged, and therefore should earn the riskfree rate, not zero, as the consultant claims. Given a futures price of 100 and an equity price of 100, the rate of return over the 3-month period is (100 − 99)/99 = 1.01% or about 4.1% annualized 33. These three put options are all written on the same stock. One has a delta pf -0.9, one a delta of -0.5 and one a delta of -0.1. Assign deltas to the three puts by filling in this table: Pu t A

Delt a

1 0 B 2 0 C 3 0 XDelta

Put A B C

X

10 20 30

–.1 –.5 –.9

34. Frank is a portfolio manager responsible for derivatives. He observes an American style option and a European style option with the same strike price, expiration, and underlying stock. He believes that the European style option will have a higher premium than the American style option. a. Critique Frank’s belief that the European style option will have a higher premium Frank is asked to value a one year European style call option for ABC Ltd which last traded at 43. Frank has collected the following information. Closing stock price 43 Call and Put exercise 45 price One year put option 4 price One year T-bill rate 5.5 Time to expiration 1 year b. Calculate using put call parity The European style call option value c. State the effect, if any, of each of the following three variables on the value of a call option (No calculations Please) i. an increase in short term interest rate ii. an increase in stock price volatility iii. A decrease in time to option expiration. a. American options should cost more (have a higher premium). They give the investor greater flexibility than the European option since the investor can choose whether to exercise early. When the stock pays a dividend, the right to exercise a call early can be valuable. But regardless of dividends, a European option (put or call) will never sell for more than an otherwise-identical American option. b. C = S0 + P − PV(X) [no dividends seem to be paid] = 43 + 4 − 45/1.055 = 4.346 c. Short-term interest rate higher ⇒ PV(exercise price) is lower, and call is worth more. Higher volatility makes the call worth more. Lower time to maturity makes the call worth less 35. Ken manages a 100 million equity portfolio benchmarked to the S&P 500 index. Over the past two years the S&P 500 index has appreciated 60%. Ken believes that the market is overvalued when measured by several traditional indicators. He

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is concerned about maintaining the excellent gains the portfolio ha experienced in the past two years but believes that the S&P could still move above its current 668 level. Ken is considering the following option call strategy: • Protection for the portfolio can be attained by purchasing an S&P 500 Index put with a strike price of 665 (just out of the money) • The put can be financed by selling two 665 calls (further out of the money) for every put purchased • Because the combined delta of the two calls is less than 1(2*0.36 =0.72) the options will not lose more than the underlying portfolio will gain if the market advances. The information in the following table describes the two options used to create the collar. Options to create the collar Characteristics 675 665 call Put Option price 4.30 8.05 Options implied 11% 14% volatility Options delta 0.36 -0.44 Contracts needed for 602 301 collar Time to expiration 30 days S&P historical volatility 12% a. Describe the potential return of the combined portfolio(the underlying portfolio plus the option collar) if after 30 days the S&P index has i. risen approx. 5% to 701 ii. remained at 668 iii. declined by approx 5% to 635 b. Discuss the effect of the hedge ratio (delta) of each option as the S&P approaches the level of each of the potential outcomes listed above c. Evaluate the pricing of the put and the call in relation to the volatility data provided. a.

(i) Index rises to 701. The combined portfolio will suffer a loss. The written calls will expire in the money; the protective put purchased will expire worthless. Let’s analyze on a per-share basis. The payout on each call option is 26, for a total cash outflow of 52. The stock is worth 701. The portfolio thus ends up worth 701 − 52 = 649. The net cost of the portfolio when the option positions were established was: 668 + 8.05 (put) − 2 × 4.30 (calls written) = 667.45 (ii) Index remains at 668. Both options expire out of the money. The portfolio ends up worth 668 (per share), compared to an initial cost 30 days earlier of 667.45. The portfolio experiences a very small gain of 0.55. (iii) Index declines to 635. The calls expire worthless. The portfolio will be worth 665, the exercise price of the protective put. This represents a very small loss of 2.45 compared to the initial cost 30 days earlier of 667.45. b. (i) Index rises to 701. The delta of the call will approach 1.0 as the stock goes deep into the money, while expiration of the call approaches and exercise becomes essentially certain. The put delta will approach zero. (ii) Index remains at 668. Both options expire out of the money. Delta of each will approach zero as expiration approaches and it becomes certain that the options will not be exercised. (iii) Index declines to 635. The call is out of the money as expiration approaches. Delta approaches zero. Conversely, the delta of the put approaches −1.0 as exercise becomes certain.

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c. The call sells at an implied volatility less than recent historical volatility; the put at a higher implied volatility. The call seems relatively cheap; the put seems expensive. ****************

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