Unformatted text preview: Chapter 22: Options and Corporate Finance 22.1 a. An option is a contract which gives its owner the right to buy or sell an underlying asset at a fixed price on or before a given date. b. Exercise is the act of buying or selling the underlying asset under the terms of the option contract. c. The strike price is the fixed price at which the option holder can buy or sell the underlying asset. The strike price is also called the exercise price. d. The expiration date is the maturity date of the option. It is the last date on which an American option can be exercised. It is the only date on which a European option can be exercised. e. A call is an option contract, which gives its owner the right to buy an underlying asset at a fixed price on or before a given date. f. A put is an option contract, which gives its owner the right to sell an underlying asset at a fixed price on or before a given date. American options can be exercised on any date up to and including the exercise date. A European option can be exercised only on the expiration date. The lower bound of the value of the put is Max {E  S, 0} = $40  $35 = $5. Since the option is selling for $4.5, the best strategy is (1) buy more put options at $4.5 and (2) exercise them to get arbitrage. a. b. c. Payoff Short 10 puts Long 5 calls Total payoff S < $80 10 ($80  S) $0 10 ($80  S) S = $80 $0 $0 $0 S > $80 0 5 (S  $80) 5 (S  $80) It can be exercised on any date up to and including the expiration date. It can be exercised only on February 18 of next year. The option is not worthless. 22.2 22.3 22.4 22.5 5(S$80) 0 80 $400 S 10($80S)
$800 Answers to EndofChapter Problems B193 22.6 a. b. c. Payoff = $55  $50 = $5 Payoff = max {$45  $50, $0} = $0 Payoff $0 $50 Value of stock at expiration 22.7 a. b. c Payoff = $0 Payoff = $50  $45 = $5 $50 $0 $50 Value of stock at expiration 22.8 a. Long 2 calls Long 1 put Total payoff b. S = $65 S = $72 S = $80 $0 200($72$70)=$400 200($80$70)=$2,000 100($75$65)=$1,000 100($75$72)=$300 $0 $1,000 $700 $2,000 B194 Answers to EndofChapter Problems Long 2 calls Long 1 put Total payoff S < $70 $0 100($75S) 100($75S) $70 < S < $75 200(S$70) 100($75S) 100(S$65) S $75 200(S$70) $0 200(S$70) Payoff $7,500 $1,000 $500 $0 $70 $75 S 22.9 a. b. You would exercise the call option contract by paying the strike price of $100/share and receiving stock worth $130 / share. The total profit from the contract is ($130  100) x 100 = $3,000. If the stock price is lower than the strike price, the option will be expired unexercised. 22.10 Assume the options will expire in one year, apply the putcall parity formula. S + P  C = PV(E) S = $2 + $8 + $40 / 1.1 = $42.36 22.11 a. To eliminate the risk of the put, sell a call and buy stock. Buying stock makes stock available to you in the event that its price falls. If the price falls below the exercise price, the put will be exercised against you. The call provides additional income if the price of the stock rises above the exercise price. The combination, buy stock, buy a put and sell a call, ensures that the net payoff is the same whether the stock price rises to $172 or falls to $138. Answers to EndofChapter Problems B195 b. Payoffs at expiration Buy stock Buy put Sell call Net payoff S = $172 $172 expires give up stock worth $172, receive $160 $160 S = $138 $138 give up stock worth $138, receive $160 $160 22.12 a. b. c. d. You should buy the call and exercise it immediately. Profit = ($60  $50)  $8 = $2 The lower bound on the price of American calls is [Stock price  Exercise price]. Upper bound = stock price because no one would be willing to pay more than the stock price for the right to receive the stock. 21.13 Factors determining the value of an American call: 1. Strike price: The value of an American call must be at least the difference between the stock price and the exercise price [S  E]. For a given stock price, a higher exercise price will reduce the value of the call. Expiration date: The time to expiration of a call affects the price of the option. Compare two calls, which are identical except for the time to expiration. The longerterm option has all the rights and benefits of the shorterterm option, plus more. It has all of those benefits and rights for a longer period of time. Thus, as the time to expiration increases, the value of the call increases. Stock price: The value of an American call must be at least the difference between the stock price and the exercise price [S  E]. For a given exercise price, a higher stock price will increase the value of the call. 2. 3. 4. Variability of the price of the underlying asset: The higher the variability of the price of the underlying asset is, the higher is the probability that the call will be in the money at the expiration date. Thus, higher variability of the asset's price will enhance the option's value. 5. Interest rate: If you buy a call, you do not have to pay the strike price until the expiration date. The delay in the payment has value. As interest rates rise, the delayed payment has more value. To convince yourself, consider what else you can do with the strike price until the expiration date (your opportunity cost). You can put that money in an account and earn interest on the amount until the expiration date. If the interest rate increases, you will earn more interest. 21.14 Factors determining the value of an American put: 1. Strike price: The value of an American put must be at least the difference between the exercise price and the stock price [E  S]. For a given stock price, a higher exercise price will increase the value of the put.
B196 Answers to EndofChapter Problems 2. Expiration date: The time to expiration of a put affects the price of the option. Compare two puts which are identical except for the time to expiration. The longerterm option has all the rights and benefits of the shorterterm option, plus more. It has all of those benefits and rights for a longer period of time. Thus, as the time to expiration increases, the value of the put increases. Stock price: The value of an American put must be at least the difference between the exercise price and the stock price [E  S]. For a given exercise price, a higher stock price will reduce the value of the put. Variability of the price of the underlying asset: The higher the variability of the price of the underlying asset is, the higher is the probability that the put will be in the money at the expiration date. Thus, higher variability of the asset's price will enhance the option's value. Interest rate: If you buy a put, you have the right to sell the stock for a fixed price in the future. The present value of the delayed receipt decreases as the interest rises. Thus, if the interest rate rises, the value of the put will fall. An increase in the risk of the stock implies an increase in the volatility of the stock price. As the volatility of the stock price rises, the value of a call increases. Call holders gain only if the stock price is greater than the exercise price. Call holders do not lose if the stock price is less than the exercise price. The volatility increases the probability that the call will be in the money. An increase in the risk of the stock implies an increase in the volatility of the stock price. As the volatility of the stock price rises, the value of a put increases. Put holders gain only if the stock price is less than the exercise price. Put holders do not lose if the stock price is greater than the exercise price. The volatility increases the probability that the put will be in the money. 3. 4. 5. rate 22.15 a. b. 22.16 Value the call by examining the value of the investment combination, which duplicates the payoffs of the call. The investment strategy, which duplicates the payoffs of the call, is buy stock and borrow money. Payoffs of buying a call: Call (A contract covers 100 shares) Payoff Payoffs of the strategy: Buy 32 shares of the stock Borrow $3,012.26* Net payoff ST = $120 100 ($120  $112) = $800 32 $120 = $3,840 $3,040 $800 ST = $95 Expires = $0 32 $95 = $3,040 $3,040 $0 * The net payoffs of the duplicating strategy must be the same as the payoffs of the call. To have the payoff be $800 when the stock price is $120, the repayment of the loan and its interest must be $3,040. If the annual interest rate is 10%, then the interest rate applicable Answers to EndofChapter Problems B197 for the five week life of the call is (1.10)5/52  1 = 0.00921. Thus, the amount which you must borrow to be sure you repay $3,040 is $3,012.26 [= $3,040 / 1.00921]. To prevent arbitrage, the value of the call must be equal to the value of setting up this strategy. The cost today of purchasing 32 shares of stock is 32 $96 = $3,072. In addition you will borrow $3,012.26. The borrowing generates a cash inflow. The cost of establishing this strategy is $3,072  $3,012.26 = $59.74. $59.74 is the cost of setting up a strategy, which duplicates the contract. Since the contract covers 100 shares, each call is worth $0.5974 [= $59.74 / 100]. 22.17 Payoff at expiration 1. Call 2. Stock (N shares) Borrow ($25N/1.05) Net payoff S = $25 $0 25 N 25 N $0 S = $35 $3 100 = $300 $35N $25N $10N Duplicating amount = $25 N / 1.05 where $10 N = $300 N = 30 shares Borrow $25 30 shares / 1.05 = $714.29. Thus, buying one call contract = (1) buy 30 shares of stock $900 (2) borrow $714.29 $714.29 $185.71 Call option value = $185.71 Call price per share = $1.857 From putcall parity, P = C + PV(E)  S = $1.857 + $32 / 1.05  $30 = $2.333 22.18 Payoff at expiration 1. Call 2. Buy N shares Borrow Net payoff ST = $40 $0 $40N $40N $0 ST = $60 100 ($60  $50) = $1,000 $60N $40N $20N To equate, $20N = $1,000 N = 50 shares Thus, borrowing amount = $40 50 / 1.09 = $1,834.86 Call value = Value of 50 shares + Borrowing $1,834.86 = 50 $55  $1,834.86
B198 Answers to EndofChapter Problems = $915.14 Each call is worth $9.1514. 22.19 d1 d2 N(d1) N(d2) C = [ ln ($62 / $70) + (0.05 + 0.35 / 2) (4 / 52)] / [0.35 (4 / 52)]0.5 = 0.6342 = 0.6342  [0.35 (4 / 52)]0.5 = 0.7982 = 0.2643 = 0.2119 = [$62 N(d1)]  [$70 e(0.05)(4/52) N(d2)] = [$62 (0.2643)]  [$70 e(0.05)(4/52) (0.2119)] = $1.61 = [ ln ($52 / $48) + (0.05 + 0.02) (1 / 3)] / [0.04 (1 / 3)]0.5 = 0.8953 = 0.8953  [0.04 (1 / 3)]0.5 = 0.7798 = 0.8147 = 0.7822 = [$52 0.8147]  [$48 e(0.05)(1/3) 0.7822] = $5.4394 = [ ln ($45 / $52) + (0.065 + 0.4 / 2) 0.5] / [0.4 0.5]0.5 = 0.0270 d2 = 0.0270  [0.4 0.5]0.5 = 0.4742 N(d1) = 0.4880 N(d2) = 0.3192 C = $45 0.4880 $52 e(0.065)(0.5) 0.3192 = $5.89 Since the time period is only six months, the exponent used in the calculation of the PV of the exercise price is 0.5. P = C + PV(E)  S = $5.89 + $52 / (1 + 0.065)0.5  $45 = $11.28 d1 = [ ln ($70 / $90) + (0.06 + 0.25 / 2) 0.5] / [0.25 0.5]0.5 = 0.4492 d2 = 0.4492  [0.25 0.5]0.5 = 0.8028 N(d1) = 0.3267 N(d2) = 0.2111 C = $70 0.3267  $90 e(0.06)(0.5) 0.2111 = $4.4315 P = C + PV(E)  S = $4.4315 + $90 / (1 + 0.06)0.5  $70 = $21.8472 d1 22.20 d1 d2 N(d1) N(d2) C 22.21 a. b. 22.22 a. b. Answers to EndofChapter Problems B199 22.23 a. d1 d2 N(d1) N(d2) C b. d1 d2 N(d1) N(d2) C c. d1 d2 N(d1) N(d2) C = [ ln ($37 / $35) + (0.07 + 0.004 / 2)] / 0.0632 = 2.0185 = 2.0185  0.0632 = 1.9553 = 0.9782 = 0.9747 = $37 0.9782  $35 e0.07 0.9747 = $4.3853 = [ ln ($37 / $35) + (0.07 + 0.0064 / 2)] / 0.08 = 1.6096 = 1.6096  0.08 = 1.5296 = 0.9463 = 0.9369 = $37 0.9463  $35 e0.07 0.9369 = $4.4385 = [$0 + (0.07 + 0.0064 / 2)] / 0.08 = 0.915 = 0.915  0.08 = 0.835 = 0.8199 = 0.7981 = $35 0.8199  $35 e0.07 0.7981 = $2.6515 22.24 = $27 = $25 = 120 / 365 = 0.0576 = 0.07 = [ ln ($27 / $25) + (0.07 + 0.0576 / 2) (120 / 365)] / [0.0576 (120 / 365)]0.5 = 0.7953 d2 = 0.7953  [0.0576 (120 / 365)]]0.5 = 0.6577 N(d1) = 0.7867 N(d2) = 0.7447 C = $27 0.7867  $25 e(0.07)(120/365) 0.7447 = $3.0470 22.25 For a firm with debt, shares of stock can be thought of as call options on the assets of the firm due to the limited liability of stock. If the value of the assets exceeds the value of the debt, the stockholders will pay the debtholders and enjoy the benefits of ownership of the remaining assets. The stockholders have the property rights over the assets in excess of the debt. If the value of the assets of the firm is less than the value of the debt, the stockholders will simply give the bondholders the assets of the firm. The shareholders will walk away owning nothing. Thus, the return to the shareholders at the end of a period is the maximum between zero and the excess of assets over debt, [V  B] where V is the value of the assets of the firm S E t 2 rf d1 B200 Answers to EndofChapter Problems and B is the value of the debt. This payoff is exactly the payoffs of a call where V is analogous to the stock price and B is analogous to the exercise price. 22.26 The equity of the firm is regarded as a call option. Payoff at expiration (million) 1. Call 2. Buy N shares Borrow Net payoff $250 $0 $250N $250N $0 $650 $350 $650N $250N $400N To equate, $400N = $350 N = 0.875 shares of assets. Thus, borrowing amount = $250 0.875 / 1.07 = $204.44 million Call value = Value of 0.875 shares of the asset + Borrowing $204.44 million = $350 million  $204.44 million = $145.56 million The value of the equity = $145.56 million The value of the debt = $400  $145.56 = $254.44 million 22.27 Payoff at expiration (million) 1. Call 2. Buy N shares Borrow Net payoff $100 $0 $100N $100N $0 $800 $500 $800N $100N $700N To equate, $700N = $550 N = (5 / 7) shares of assets. Thus, borrowing amount = $100 (5 / 7) / 1.07 = $66.76 million Call value = Value of (5 / 7) shares of the asset + Borrowing $66.76 million = $285.71 million  $66.76 million = $218.95 million The value of the equity = $218.95 million The value of the debt = $400  $218.95 = $181.05 million Thus, bondholders prefer the less risky project. Answers to EndofChapter Problems B201 ...
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This note was uploaded on 05/07/2010 for the course FIN 302 taught by Professor Corporationfinance during the Spring '10 term at Uni Potsdam.
 Spring '10
 corporationfinance
 Finance, Corporate Finance, Options

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