chapter 6

CHAPTER 6: BASIC OPTION STRATEGIES

END-OF-CHAPTER QUESTIONS AND PROBLEMS

1.(Calls and Stock: The Covered Call) The covered call cuts losses on the downside and gains on the upside. In a bull market, the call is likely to be exercised and the stock called away. This limits the gain in a bull market to the amount of the premium plus the difference between the exercise price and the original stock price. A protective put cuts losses in a bear market but does allow gains in a bull market, which are higher the higher the stock price. Since a protective put is a synthetic call, the comparison is more appropriately seen as that of writing a covered call versus buying a call. Should we own stock and protect it with a short call or own simply the call? Both strategies are bullish, but the call (or protective put) would appeal more in situations where the investor is more concerned about taking advantage of a bull market than providing protection in a bear market.

2.(Calls and Stock: The Covered Call) You could concentrate on writing out-of-the-money calls; however, this would not guarantee that the stock would not be called away. The position must be carefully monitored. If the call moves in-the-money, it could be repurchased. Then you should write another out-of-the-money call. If the stock price continues to move upward, you continue to roll out of one exercise price into a higher exercise price. Disadvantages of this strategy are the potential for generating high transaction costs and the trouble of monitoring the portfolio, as well as the fact that writing out-of-the money calls produces relatively small premiums.

3.(Synthetic Puts and Calls) If P + S0 C < X, then the synthetic call (put plus stock) is underpriced or the actual call is overpriced. So buy the synthetic call and sell the actual call. The payoffs from this portfolio at expiration are

ST ( X

ST > X

Short call

0

(ST X)

Long stock

ST

ST

Long put

X S T

0

X

X

The cost of this portfolio is P + S0 C, which we said was less than the present value of the exercise price. So we are essentially buying a portfolio that pays off X dollars for certain and paying a price that is less than the present value of X.

4.(Calls and Stock: The Covered Call) Ns = 1

Nc = 1

( = Max(0, ST X) C (ST S0)

If ST ( X, ( = C ST + S0

If ST > X, ( = ST X C ST + S0 = S0 X C

The breakeven is at a value of ST ( X. Setting this profit equation to zero

C ST* + S0 = 0

and solving for ST*

ST* = S0 C

Maximum profit occurs if ST = 0

( = C + S0

Minimum profit occurs if ST > X

( = S0 X C

which is negative since C must be greater than S0 X.

5.(Puts and Stock: The Protective Put) Ns = 1

Np = 1

( = Max(0, X ST) + P (ST S0)

If ST < X, ( = X + ST + P ST + S0 = P + S0 X

If ST ( X, ( = P ST + S0

The breakeven is at a value of ST > X. Setting the profit equation to zero

P ST* + S0 = 0

and solving for ST*

ST* = P + S0

Maximum profit occurs at ST < X

( = P + S0 X

Minimum profit occurs if ST = (

( = (6.(Calls and Stock: The Covered Call) Time value is directly related to time to expiration. If the covered writer closes out the position prior to expiration, the call will be more expensive to repurchase. This reduces the profit for a given stock price. The shorter the holding period, however, the less time the stock price has to move downward. Writers of options benefit from short holding periods by giving the stock less time to move, but they have to buy back more of the original time value they received.

7.(Calls and Stock: The Covered Call) The lower the exercise price, the more protection the option is providing. This is because the call with the lower exercise price commands a higher premium and the premium cushions the loss on the downside. The disadvantage of a lower exercise price is that the option is more likely to be exercised and the upside gain is lower.

8.(Buy a Call) C = 5.25

( = 100(Max(0,ST 165) 5.25)

Option Value at

STExpirationProfit

1500525

1550525

1600525

1650525

1705 25

17510475

18015975

Note: If you use Stratlyz8e.xls to generate the graphs in this chapter, the horizontal axis will be labeled as the Stock Price at Expiration. Here it has been changed to Stock Price at End of Holding Period.

Breakeven: X + C = 165 + 5.25 = 170.25

Maximum loss: 525 (the premium)

9.(Buy a Call) X = 165

rc = 0.0535( = 0.21

Tt = 20/365 = 0.0548 (based on 20 days between 8/1 and 8/21)

Plugging into the Black-Scholes-Merton model, we obtain the option values on August 1 for stock prices of 150, 155, , 180.

( = 100(Option Value on 8/1 5.25)

The results using a spreadsheet are

StOption Value on 8/21Profit

1500.0901515.99

1550.4295482.05

1601.4203382.97

1653.4782177.18

1706.7247 147.47

17510.8992 564.92

18015.59511,034.51


Approximate breakeven stock price: 168

Maximum loss: 525 (the premium)

10.(Buy a Put) P = 6.75

( = 100(Max(0,165 ST) 6.75)

Option Value at

STExpirationProfit

15015825

15510325

1605175

1650675

1700675

1750675

1800675


Breakeven: X P = 165 6.75 = 158.25

Maximum loss: 675

Maximum gain (if ST = 0): 15,825

11.(Calls and Stock: The Covered Call) C = 6

( = 100(ST 165.13 Max(0, ST 170) + 6)

Option ValueProfitProfit Overall

STat Expirationfrom Optionfrom Stock Profit

15006001,513 913

15506001,013 413

1600600513 87

165060013 587

17006004871,087

17551009871,087

180104001,4871,087


Breakeven: S0 C = 165.13 6 = 159.13

Maximum profit: 1,087

Maximum loss (if ST = 0):

Profit from call = 600

Profit from stock = 16,513

Overall profit = 15,913 (maximum loss = $15,913)

12.(Calls and Stock: The Covered Call) X = 170

( = 0.21

rc = 0.0571

Tt = 45/365 = 0.1233 (based on 45 days from 9/1 to 10/16)

Plugging into the Black-Scholes-Merton model, we obtain the option values on August 1 for stock prices of 150, 155, , 180.

Profit = 100(St 165.13 (Option Value on 9/1 6))

The results using a spreadsheet are

Option ValueProfitProfitOverall

Ston 9/1from Option from Stock Profit

1500.2714572.861,513 940.14

1550.7297527.031,013 485.97

1601.6540434.60513 78.40

1653.2381276.1913 263.19

1705.601439.86487 526.86

1758.7436274.36987 712.64

18012.5540655.401,487 831.60


Approximate breakeven stock price: 162

13.(Puts and Stock: The Protective Put) P = 4.75

( = 100(ST 165.13 + Max(0,165 ST) 4.75)

Option Value atProfitProfit Overall

STExpirationfrom Putfrom Stock Profit

150151,0251,513488

155105251,013488

160525513488

165047513488

1700475487 12

1750475987 512

18004751,4871,012


Breakeven: P + S0 = 4.75 + 165.13 = 169.88

Maximum profit:

Maximum loss: 487.50

14.(Buy a Call) C = $0.0385X = $1.00

Contract size is 100,000

Premium is 100,000($0.0385) = $3,850

Option Value

STat ExpirationProfit

$0.900.00$3,850

$0.950.00 3,850

$1.000.00 3,850

$1.050.051,150

$1.100.106,150

Note: If you use Stratlyz8e.xls to generate the graphs in this chapter, the horizontal axis will be labeled as the Stock Price at Expiration. Here it has been changed to Spot Exchange Rate at End of Holding Period.

Breakeven exchange rate: X + C = $1.00 + $0.0385 = $1.0385

15.(Buy a Put) P = $0.0435X = $1.00

Premium = 100,000($0.0435) = $4,350

Option Value


$0.900.10$5,650

$0.950.05 650

$1.000.00

4,350

$1.050.00 4,350

$1.100.00 4,350


Breakeven exchange rate: X P = $1.00 $0.0435 = $0.9565

16.(Calls and Stock: The Covered Call) C = $0.0385X = $1.00S0 = $0.9825

Contract size is 100,000

Premium is 100,000($0.0385) = $3,850

The currency costs 100,000($0.9825) = $98,250

Net cost = $98,250 $3,850 = $94,400

Option Value


$0.900.00$4,400

$0.950.00 600

$1.000.00

5,600

$1.050.055,600

$1.100.105,600

Note: If you use Stratyz8e.xls to generate the graphs in this chapter, the horizontal axis will be labeled as the Stock Price at Expiration. Here it has been changed to Stock Price at End of Holding Period.

Breakeven exchange rate: S0 C = $0.9825 $0.0385 = $0.9440

17.(Calls and Stock: The Covered Call) Covered call writing refers to owning the portfolio and writing call options. The key word to describe this strategy is ceiling, writing calls places a ceiling on the value of your portfolio. Covered call writing reduces the expected return and standard deviation of a portfolio as well as negative skewness. The following probability density function illustrates this strategy:

18.(Puts and Stock: The Protective Put) Portfolio insurance with options involves owning the portfolio and buying put options. The key word to describe this strategy is floor, buying puts places a floor on the value of your portfolio. Protective put buying reduces the expected return and standard deviation of a portfolio as well as positive skewness. The following probability density function illustrates this strategy:

19.(Synthetic Puts and Calls) See excerpt from BSMbin8e.xls:

Synthetic call can be created by purchasing the stock and purchasing the put option. The cost would be $100 for the stock and $9.35 for the put. Thus, the most that would be lost is $9.35 that occurs if the stock price remains at or below $100. If the present value of the exercise price is borrowed, then the payoff will be identical to that of purchasing a call option.20.(Synthetic Puts and Calls) Recall from put-call parity that

Solving for the stock price, we have

Synthetic stock can be created by purchasing the call and selling the put. The cost would be $14.20 less $9.30 or $4.90. The present value of the exercise price is $95.12. Thus, the cost of exactly replicating a stock is $100.02 (=$95.12 + $14.20 - $9.30). Assuming no transaction costs, if the stock price is $100, then you should buy the stock, sell the call, purchase the put, and lend the present value of the exercise price. This transaction would generate $0.02 today with no future liability.

21.(Synthetic Puts and Calls) This is a reverse conversion

Sell put

+4.75

Buy synthetic put

Buy call5.25

Sell stock+165.125

Net cash in164.625

Invest at 5.35% to grow to 164.625(1.0535)0.1260 = 165.71

Amount owed at expiration = 165

Net gain = 0.71

22.(Buy a Call) a. ( = Max(0, ST X) C TC

where TC = transaction costs

ST ( X

( = ST X 0.005X 0.005ST C 0.01C

ST < X

( = C 0.01C

(Buy a Put) b. ( = Max(0, X ST) P TC

ST ( X

( = P 0.01P

ST < X

( = X ST 0.005X 0.005ST P 0.01P

(Calls and Stock) c. ( = ST S0 Max(0, ST X) + C TC

ST ( X

( = X S0 + C 0.005S0 0.005X 0.01C

ST < X

( = ST S0 + C 0.005ST 0.005 S0 0.01C

(Puts and Stock) d. ( = ST S0 + Max(0, X ST) P TC

ST ( X

( = ST S0 P 0.005ST 0.005S0 0.01P

ST < X

( = X S0 P 0.005X 0.005S0 0.01P

Probability

Change in Portfolio Value

Probability

Terminal portfolio value

Chapter 635End-of-Chapter Solutions

2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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chapter 6

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