23: Chapter Options and Corporate Finance: Basic Concepts
23.1 a. An option is a contract giving its owner the right to buy or sell an 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 in the option contract at which the 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 and the only date on which a European option can be exercised. e. A call option gives the owner the right to buy an asset at a fixed price during a particular time period. f. A put option gives the owner the right to sell an asset at a fixed price during a particular time period. 23.2 An American option can be exercised on any date up to and including the expiration date. A European option can only be exercised on the expiration date. Since an American option gives its owner the right to exercise on any date up to and including the expiration date, it must be worth at least as much as a European option, if not more. The put is not correctly priced. An American put option must always be worth more than the value of immediate exercise. The value of immediate exercise for a put option equals the strike price minus the current stock price. In this problem, the value of immediate exercise is $5 (= $40 - $35). Since the option is currently selling for $4.50, less than the value of immediate exercise, the option is underpriced. Consider the following investment strategy designed to take advantage of the mispricing: Strategy 1. Buy put option 2. Buy stock 3. Exercise put option Arbitrage Profit Cash Flow -$4.50 -$35.00 +$40.00 +$0.50
23.3
Therefore, Mr. Nash should buy the option for $4.50, buy the stock for $35, and immediately exercise the put option to receive its strike price of $40. This strategy yields a riskless, arbitrage profit of $0.50 (=$5 - $4.50). 23.4 a. If the option is American, it can be exercised on any date up to and including its expiration on February 25. b. If the option is European, it can only be exercised on its expiration date, February 25. c. The option is not worthless. There is a chance that the stock price of Futura Corporation will rise above $45 sometime before the options expiration on February 25. In this case, a call option with a strike price of $45 would be valuable at expiration. The probability of such an event happening is built into the current price of the option.
Answers to End-of-Chapter Problems
B-336
23.5
a. The payoff to the owner of a call option at expiration is the maximum of zero and the current stock price minus the strike price. The payoff to the owner of a call option on Stock A on December 21 is: max[0, ST - K] = max[0, 55-50] = $5 where ST = the price of the underlying asset at expiration K = the strike price
b. The payoff to the seller of a call option at expiration is the minimum of zero and the strike price minus the current stock price. The payoff to the seller of a call option on Stock A on December 21 is: min[0, K- ST] = min[0, 50-55] = -$5 In other words, the seller must pay $5. c. The payoff to the owner of a call option at expiration is the maximum of zero and the current stock price minus the strike price. The payoff to the owner of a call option on Stock A on December 21 is: max[0, ST - K] = max[0, 45-50] = $0 d. The payoff to the seller of a call option at expiration is the minimum of zero and the strike price minus the current stock price. The payoff to the seller of a call option on Stock A on December 21 is: min[0, K- ST] = min[0, 50-45] = $0 e.
25 20
Payoff to Owner
15 10 5 0 30 35 40 45 50 55 60 65 70 Stock Price at Expiration
Answers to End-of-Chapter Problems
B-337
f.
0 30 -5 35 40 45 50 55 60 65 70
Payoff to Seller
-10 -15 -20 -25 Stock Price at Expiration
g. The seller of a call option receives a premium, the price of the option, at the time of sale. At expiration, if the buyer chooses not to exercise, the premium becomes pure profit for the seller. Therefore, an individual will write (sell) a call option if he does not believe the stock price will rise above the strike price before expiration. 23.6 a. The payoff to the owner of a put option at expiration is the maximum of zero and the strike price minus the current stock price. The payoff to the owner of a put option on Stock A on December 21 is: max[0, K- ST] = max[0, 50-55] = $0 where ST = the price of the underlying asset at expiration K = the strike price
b. The payoff to the seller of a put option at expiration is the minimum of zero and the current stock price minus the strike price. The payoff to the seller of a call option on Stock A on December 21 is: min[0, ST- K] = min[0, 55-50] = $0 c. The payoff to the owner of a put option at expiration is the maximum of zero and the strike price minus the current stock price. The payoff to the owner of a put option on Stock A on December 21 is: max[0, K- ST ] = max[0, 50-45] = $5 d. The payoff to the seller of a put option at expiration is the minimum of zero and the current stock price minus the strike price. The payoff to the seller of a call option on Stock A on December 21 is:
Answers to End-of-Chapter Problems B-338
min[0, ST- K] = min[0, 45-50] = -$5 In other words, the seller must pay $5. e.
25 20
Payoff to Owner
15 10 5 0 30 35 40 45 50 55 60 65 70 Stock Price at Expiration
f.
0 30 -5 35 40 45 50 55 60 65 70
Payoff to Seller
-10 -15 -20 -25 Stock Price at Expiration
Answers to End-of-Chapter Problems
B-339
23.7
Let ST = the stock price at expiration K = the strike price Payoffs to Ms. Eisner's Portfolio Expiration If ST > $80 0 5(ST - $80) 5ST - $400
Sell 10 Puts (K=$80) Buy 5 Calls (K=$80) Total
If ST < $80 10(ST -$80) 0 10ST - $800
200 150 100 50 0 -50 -100 -150 -200 -250 -300 -350
Payoff
50
55
60
65
70
75
80
85
90
95
100 105 110
Stock Price at Expiration
23.8
a. If the stock price is $65 at expiration: The payoff to the owner of a call option at expiration is the maximum of zero and the current stock price minus the strike price. The payoff to each of Mr. Changs call options is: max[0, ST - K] = max[0, 65-70] = $0 where ST = the price of the underlying asset at expiration K = the strike price Since Mr. Chang bought 2 call contracts and each contract is for 100 options, the payoff to Mr. Changs position in call options is: # call contracts * # options per call contract * payoff per call option = 2 * 100 * $0 = $0 The payoff to the owner of a put option at expiration is the maximum of zero and the strike price minus the current stock price. The payoff to each of Mr. Changs put options is:
Answers to End-of-Chapter Problems
B-340
max[0, K- ST] = max[0, 75-65] = $10 Since Mr. Chang bought 1 put contract and each contract is for 100 options, the payoff to Mr. Changs position in put options is: # put contracts * # options per put contract * payoff per put option = 1 * 100 * $10 = $1,000 The total payoff of Mr. Changs position is the sum of the payoffs of his positions in call and put options. Total Payoff if the stock price is $65 at expiration = $0 + $1000 = $1,000 If the stock price is $72 at expiration: The payoff to each call option is: The payoff to each put option is: max[0, ST - K] = max[0, 72-70] = $2 max[0, K- ST] = max[0, 75-72] = $3
The payoff to the position in calls is:2 * 100 * $2 = $400 The payoff to the position in puts is: 1 * 100 * $3 = $300 Total Payoff if the stock price is $72 at expiration = $400 + $300 = $700 If the stock price is $80 at expiration: The payoff to each call option is: The payoff to each put option is: max[0, ST - K] = max[0, 80-70] = $10 max[0, K- ST] = max[0, 75-80] = $0
The payoff to the position in calls is:2 * 100 * $10 = $2,000 The payoff to the position in puts is: 1 * 100 * $0 = $0 Total Payoff if the stock price is $80 at expiration = $2,000 + $0 = $2,000
Answers to End-of-Chapter Problems
B-341
b.
7000 6000 5000
Payoff
4000 3000 2000 1000 0 50 55 60 65 70 75 80 85 90 95 100 Stock Price at Expiration
23.9 a. If the stock is selling for $130, Louise will immediately exercise since the stock price is greater than the $100 strike price of the call options. If she then sells the stock at the market price of $130, her immediate cash flow is $30 (= $130 - $100) per option. Since Louise owns 1 contract and each contract is for 100 options, her cash flow at expiration is $3,000 (= $30 * 100). b. If the stock is selling for $90, Louise will choose not to exercise her options since exercising would require her to pay $100 (the strike price) for a stock that is only worth $90. 23.10 a. Yes, there is an arbitrage opportunity. You should buy the American call option for $8, exercise the option (buy the underlying stock for the options strike price of $50), and sell the stock at the market price of $60. This strategy yields a riskless arbitrage profit of $2 (= $60 - $50 - $8). b. Arbitrage opportunities such as this imply that the lower bound on the price of an American call option is the value of immediate exercise, which is equal to the current stock price minus the strike price of the option (S K). c. An upper bound on the price of an American call option is the current price of the underlying asset. A call option, which gives its owner the right to buy an underlying asset, cannot cost more than the underlying asset. If it did, selling the option, using the proceeds to purchase the underlying asset, and pocketing the difference would yield an arbitrage profit. Suppose Stock A is trading for $8 and a call option on Stock A with a strike price of $6 is selling for $10. Consider the following investment strategy designed to take advantage of the mispricing: Strategy 1. Sell call option 2. Buy stock Arbitrage Profit Cash Flow +$10.00 -$8.00 +$2.00
B-342
Answers to End-of-Chapter Problems
If the buyer of the call option decides to exercise, the seller will exchange the stock for the options strike price of $6. Since the seller already owns the stock (and therefore does not need to purchase it), this results in an additional cash inflow of $6 for the seller, regardless of the price of the stock at the time of exercise. If the buyer decides not to exercise, the seller keeps both the stock and the $2 arbitrage profit. In either case, it is impossible for the seller of the call option to lose money. 23.11 1) The price of the underlying asset. Holding all else equal, a rise in the price of the underlying asset will increase the value of an American call option since, for a fixed strike price, the owner will receive a more valuable asset. 2) The strike price of the option. Holding all else equal, an increase in the strike price will decrease the value of an American call option since the owner must pay more to receive the underlying asset. 3) The time to expiration of the option. Holding all else equal, an increase in the time to expiration will increase the value of an American call option. A longer period until expiration gives the owner more time to decide whether or not he should exercise the option. Also, upon exercise, the holder of the option must pay the strike price. When the time to expiration of an option is more distant, the present value of this payment falls. 4) The volatility of the underlying asset. Holding all else equal, an increase in the volatility of the underlying asset will increase the value of an American call option. An increase in the volatility of the underlying asset increases the value of exercise should the option end up in-the-money, allowing the options owner to receive a larger positive payoff. 5) The interest rate. Holding all else equal, an increase in the interest rate will increase the value of an American call option. Upon exercise, the holder of the option must pay the strike price. When the interest rate rises, the present value of this future payment falls. 23.12 1) The price of the underlying asset. Holding all else equal, a rise in the price of the underlying asset will decrease the value of an American put option since the owner is giving away a more valuable asset for a fixed strike price at expiration. 2) The strike price of the option. Holding all else equal, an increase in the strike price will increase the value of an American put option since the owner receives more in exchange for the underlying asset at expiration. 3) The time to expiration of the option. Holding all else equal, an increase in the time to expiration will increase the value of an American put option. A longer period until expiration gives the owner more time to decide whether or not he should exercise the option. 4) The volatility of the underlying asset. Holding all else equal, an increase in the volatility of the underlying asset will increase the value of an American put option. An increase in the volatility of the underlying asset increases the value of exercise should the option end up in-the-money, allowing the options owner to exercise and receive a larger positive payoff.
Answers to End-of-Chapter Problems
B-343
5) The interest rate. Holding all else equal, an increase in the interest rate will decrease the value of an American put option. The owner of a put option will receive the strike price at expiration if he chooses to exercise. A higher interest rate decreases the present value of the strike price the owner hopes to receive. 23.13 According to Put-Call Parity, for two options with the same strike price and time to expiration, the cost of a call must equal the cost of a put plus the cost of the stock minus the present value of the strike price: According to Put-Call Parity: C = P + S PV(K) where C P S PV(K) = the cost of a call option = the cost of a put option = the current price of the underlying asset = the present value of the strike price
Solving for the stock price, this equation shows that the stock price must equal the cost of a call minus the cost of a put plus the present value of the strike price: Put-Call Parity: S = C P + PV(K) The cost of a call with a strike of $40 written on General Eclectic Stock is $8. The cost of a put with a strike of $40 written on General Eclectic Stock is $2. The present value of the strike price is $ 38.10 (= $40 / 1.05). S = C P + PV(K) = $8 - $2 + $38.10 = $44.10 The price of General Eclectic stock must be $44.10 per share in order to prevent arbitrage. 23.14 a. While the market value of the put ($2) is less than the value of immediate exercise of $5 (= $140 - $135), there is no arbitrage opportunity since the option is European and cannot be exercised immediately. If this were an American option, one could buy the put for $2 and immediately exercise it for $5, yielding a riskless profit of $3. However, since this is a European option, the buyer must wait until the expiration date to exercise the put. If the stock price rises above $140 in one year, a put option with a strike of $140 will have no value at expiration. b. Yes, there is a way for Kevin to create a synthetic call option. A payoff structure identical to a call option can be created by purchasing a put option, purchasing a share of stock, and borrowing the present value of the strike price. In order to create a synthetic call option with a strike price of $140 and one year until expiration, Kevin should: Buy a put option with a strike price of $140 and one year until expiration for $2. Buy one share of Gimpellian Softwares stock for $135.
Answers to End-of-Chapter Problems B-344
Borrow $ 132.08 (= $140 / 1.06), which equals the present value of the strike price. This synthetic call position costs $ 4.92 (= 2 + 135 132.08). In order to verify that this is in fact a synthetic call option, draw the payoff diagram and check that it looks exactly like the payoff diagram of a call option. This is done in part c. c. Expiration if ST < $140 if ST > $140 $140 - ST 0 +ST +ST -$140 -$140 0 ST - $140
Buy 1 Put(K=$140) Buy 1 Share Borrow PV($140) Total
25 20
Payoff
15 10 5 0 120 125 130 135 140 145 150 155 160 Stock Price at Expiration
23.15
a. In order to solve a problem using the two-state option model, first draw a stock price tree containing both the current stock price and the stocks possible values at the time of the options expiration. Next, draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible stock price movements. Eastjets stock price today is $100. It will either increase to $120 or decrease to $80 in one year. If the stock price rises to $120, Ken will exercise his call option for $110 and receive a payoff of $10 at expiration. If the stock price falls to $80, Ken will not exercise his call option, and he will receive no payoff at expiration.
Answers to End-of-Chapter Problems
B-345
Eastjet's Stock Price Today 1 Year 120 100 80
Ken's European Call Option with a Strike of 110 Today 1 Year 10 ? 0 = max(0, 80 -110) = max(0, 120 -110)
If Eastjets stock price rises, its return over the period is 20% [= (120/100) 1]. If Eastjets stock price falls, its return over the period is 20% [= (80/100) 1]. Use the following expression to determine the risk-neutral probability of a rise in the price of Eastjets stock: Risk-Free Rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) = (ProbabilityRise)(ReturnRise) + (1 - ProbabilityRise)(ReturnFall) 0.025 = (ProbabilityRise)(0.20) + (1 ProbabilityRise)(-0.20)
ProbabilityRise = 0.5625 ProbabilityFall = 1 - ProbabilityRise = 1 0.5625 = 0.4375 The risk-neutral probability of a rise in Eastjet stock is 50%, and the risk-neutral probability of a fall in Eastjet stock is 50%. Using these risk-neutral probabilities, determine the expected payoff to Kens call option at expiration. Expected Payoff at Expiration = (.5625)($10) + (.4375)($0) = $5.625 Since this payoff occurs 1 year from now, it must be discounted at the risk-free rate of 2.5% in order to find its present value: PV(Expected Payoff at Expiration) = ($5.625 / 1.025) = $5.49 Therefore, given the information Ken has about Eastjets stock price movements over the next year, a European call option with a strike price of $110 and one year until expiration is worth $5.49 today. b. Yes, there is a way for Ken to create a synthetic call option with identical payoffs to the call option described above. In order to do this, Ken will need to buy shares of Eastjets stock and borrow at the risk-free rate. The number of shares that Ken should buy is based on the delta of the option, where delta is defined as: Delta = (Swing of option) / (Swing of stock)
Answers to End-of-Chapter Problems
B-346
Since the call option will be worth $10 if Eastjets stock price rises and $0 if it falls, the swing of the call option is 10 (= 10 0). Since the stock price will either be $120 or $80 at the time of the options expiration, the swing of the stock is 40 (= 120 - 80). Given this information: Delta = (Swing of option) / (Swing of stock) = (10 / 40) = 1/4
Therefore, Kens first step in creating a synthetic call option is to buy 1/4 of a share of Eastjets stock. Since Eastjets stock is currently trading at $100 per share, this will cost him $25.00 [= (1/4)($100)]. In order to determine the amount that Ken should borrow, compare the payoff of the actual call option to the payoff of delta shares at expiration. Call Option If the stock price rises to $120: If the stock price falls to $80: Delta Shares If the stock price rises to $120: If the stock price falls to $80:
payoff = $10 payoff = $0
payoff = (1/4)($120) = $30.00 payoff = (1/4)($80) = $20.00
Ken would like the payoff of his synthetic call position to be identical to the payoff of an actual call option. However, owning 1/4 of a share leaves him exactly $20.00 above the payoff at expiration, regardless of whether the stock price rises or falls. In order to reduce his payoff at expiration by $20.00, Ken should borrow the present value of $20.00 now. In one year, his obligation to pay $20.00 will reduce his payoffs so that they exactly match those of an actual call option. Ken should purchase 1/4 of a share of Eastjets stock and borrow $19.51 (= $20.00 / 1.025) in order to create a synthetic call option with a strike price of $110 and 1 year until expiration. c. Since Ken pays $25.00 to purchase 1/4 of a share and borrows $19.51, the total cost of the synthetic call option is $5.49 (= $25.00 - $19.51). This is exactly the same price that Ken would pay for an actual call option. Since an actual call option and a synthetic call option provide Ken with identical payoff structures, he should not expect to pay more for one than the other. In order to solve a problem using the two-state option model, first draw a stock price tree containing both the current stock price and the stocks possible values at the time of the options expiration. Next, draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible stock price movements. BioLabs stock price today is $30. It will either decrease to $15 or increase to $60 in six months. If the stock price falls to $15, Dana will exercise her put option for $40 and receive
Answers to End-of-Chapter Problems B-347
23.16 a.
a payoff of $25 at expiration. If the stock price rises to $60, Dana will not exercise her put option, and she will receive no payoff at expiration.
BioLab's Stock Price Today 6 months 60 30 15
Dana's European Put Option with a Strike of 40 Today 6 months 0 ? 25 = max(0, 40-15) = max(0, 40-60)
If BioLabs stock price rises, its return over the period is 100% [= (60/30) 1]. If BioLabs stock price falls, its return over the period is 50% [= (15/30) 1]. Use the following expression to determine the risk-neutral probability of a rise in the price of BioLabs stock: Risk-Free Rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) = (ProbabilityRise)(ReturnRise) + (1 - ProbabilityRise)(ReturnFall) The risk-free rate over the next six months must be used in the order to match the timing of the expected stock price change. Since the risk-free rate per annum is 21%, the risk-free rate over the next six months is 10% [= (1.21)1/2 1]. 0.10 = (ProbabilityRise)(1) + (1 ProbabilityRise)(-0.50)
ProbabilityRise = 0.40 ProbabilityFall = 1 - ProbabilityRise = 1 0.40 = 0.60 The risk-neutral probability of a rise in BioLabs stock is 40%, and the risk-neutral probability of a fall in BioLabs stock is 60%. Using these risk-neutral probabilities, determine the expected payoff to Robs put option at expiration. Expected Payoff at Expiration = (.40)($0) + (.60)($25) = $15.00 Since this payoff occurs 6 months from now, it must be discounted at the risk-free rate of 21% per annum in order to find its present value: PV(Expected Payoff at Expiration) = [$15.00 / (1.21)1/2 ] = $13.64 Therefore, given the information Dana has about BioLabs stock price movements over the next six months, a European put option with a strike price of $40 and six months until expiration is worth $13.64 today.
Answers to End-of-Chapter Problems
B-348
b. Yes, there is a way for Dana to create a synthetic put option with identical payoffs to the put option described above. In order to do this, Dana will need to short shares of BioLabs stock and lend at the risk-free rate. The number of shares that Dana should sell is based on the delta of the option, where delta is defined as: Delta = (Swing of option) / (Swing of stock) Since the put option will be worth $0 if BioLabs stock price rises and $25 if it falls, the swing of the call option is -25 (= 0 25). Since the stock price will either be $60 or $15 at the time of the options expiration, the swing of the stock is 45 (= 60 - 15). Given this information: Delta = (Swing of option) / (Swing of stock) = (-25 / 45) = -5/9
Therefore, Danas first step in creating a synthetic put option is to short 5/9 of a share of BioLabs stock. Since BioLabs stock is currently trading at $30 per share, Dana receives $16.67 (= 5/9 * $30) as a result of his short sale. In order to determine the amount that Dana should lend, compare the payoff of the actual put option to the payoff of delta shares at expiration. Put Option If the stock price rises to $60: If the stock price falls to $15: Delta Shares If the stock price rises to $60: If the stock price falls to $80:
payoff = $0 payoff = $25
payoff = (-5/9)($60) = -$33.33 payoff = (-5/9)($15) = -$8.33
Dana would like the payoff of his synthetic put position to be identical to the payoff of an actual put option. However, shorting 5/9 of a share leaves her exactly $33.33 below the payoff at expiration, regardless of whether the stock price rises or falls. In order to increase his payoff at expiration by $33.33, Dana should lend the present value of $33.33 now. In six months, she will receive $33.33, which will increase her payoffs so that they exactly match those of an actual put option. Dana should short 5/9 of a share of BioLabs stock and lend $30.30 [= $33.33 / (1.21)1/2] in order to create a synthetic put option with a strike price of $40 and 6 months until expiration. c. Since Dana receives $16.67 as a result of the short sale and lends $30.30, the total cost of the synthetic put option is $13.63 (= $30.30 16.66). This is the same price that Dana would pay for an actual put option. Since an actual put option and a synthetic put option
Answers to End-of-Chapter Problems
B-349
provide Dana with identical payoff structures, she should not expect to pay more for one than the other. 23.17 a. Maverick would be interested in purchasing a call option on the price of gold with a strike price of $375 per ounce and 3 months until expiration. This option will compensate Maverick for any increases in the price of gold above the strike price and places a cap on the amount the firm must pay for gold at $375 per ounce. In order to solve a problem using the two-state option model, first draw a price tree containing both the current price of the underlying asset and the underlying assets possible values at the time of the options expiration. Next, draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible stock price movements. The price of gold is $350 per ounce today. If the price rises to $400, Maverick will exercise its call option for $375 and receive a payoff of $25 at expiration. If the price of gold falls to $325, Maverick will not exercise its call option, and the firm will receive no payoff at expiration.
Price of Gold (per ounce) Today 3 months 400 350 325 ? 0 = max(0, 325-375) Maverick's Call Option with a Strike of 375 Today 3 months 25 = max(0, 400-375)
b.
If the price of gold rises, its return over the period is 14.29% [= (400/350) 1]. If the price of gold falls, its return over the period is -7.14% [= (325/350) 1]. Use the following expression to determine the risk-neutral probability of a rise in the price of gold: Risk-Free Rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) = (ProbabilityRise)(ReturnRise) + (1 - ProbabilityRise)(ReturnFall) The risk-free rate over the next three months must be used in the order to match the timing of the expected price change. Since the risk-free rate per annum is 16.99%, the risk-free rate over the next three months is 4% [= (1.1699)1/4 1]. 0.04 = (ProbabilityRise)(0.1429) + (1 ProbabilityRise)(-0.0714)
ProbabilityRise = 0.5198 ProbabilityFall = 1 - ProbabilityRise = 1 0.5198 = 0.4802
Answers to End-of-Chapter Problems
B-350
The risk-neutral probability of a rise in the price of gold is 51.98%, and the risk-neutral probability of a fall in the price of gold is 48.02%. Using these risk-neutral probabilities, determine the expected payoff to Mavericks call option at expiration. Expected Payoff at Expiration = (.5198)($25) + (.4802)($0) = $13.00 Since this payoff occurs 3 months from now, it must be discounted at the risk-free rate of 16.99% per annum in order to find its present value: PV(Expected Payoff at Expiration) = [$13.00 / (1.1699)1/4 ] = $12.50 Therefore, given the information Maverick has about golds price movements over the next three months, a European call option with a strike price of $375 and three months until expiration is worth $12.50 today. c. Yes, there is a way for Maverick to create a synthetic call option with identical payoffs to the call option described above. In order to do this, Maverick will need to buy gold and borrow at the risk-free rate. The amount of gold that Maverick should buy is based on the delta of the option, where delta is defined Delta as: = (Swing of option) / (Swing of price of gold) Since the call option will be worth $25 if the price of gold rises and $0 if it falls, the swing of the call option is 25 (= 25 0). Since the price of gold will either be $400 or $325 at the time of the options expiration, the swing of the price of gold is 75 (= 400 - 325). Given this information: Delta = (Swing of option) / (Swing of price of gold) = (25 / 75) = 1/3
Therefore, Mavericks first step in creating a synthetic call option is to buy 1/3 of an ounce of gold. Since gold currently sells for $350 per ounce, Maverick must pay $116.67 (= 1/3 * $350) to purchase 1/3 of an ounce of gold. In order to determine the amount that Maverick should borrow, compare the payoff of the actual call option to the payoff of delta shares at expiration. Call Option If the price of gold rises to $400: If the price of gold falls to $325: Delta Shares If the price of gold rises to $400:
Answers to End-of-Chapter Problems
payoff = $25 payoff = $0
payoff = (1/3)($400) = $133.33
B-351
If the price of gold falls to $325:
payoff = (1/3)($325) = $108.33
Maverick would like the payoff of his synthetic call position to be identical to the payoff of an actual call option. However, buying 1/3 of a share leaves him exactly $108.33 above the payoff at expiration, regardless of whether the price of gold rises or falls. In order to decrease the firms payoff at expiration by $108.33, Maverick should borrow the present value of $108.33 now. In three months, the firm must pay $108.33, which will decrease its payoffs so that they exactly match those of an actual call option. Maverick should buy 1/3 of an ounce of gold and borrow $104.17 [= $108.33 / (1.1699)1/4] in order to create a synthetic call option with a strike price of $375 and 3 months until expiration. d. Since Maverick pays $116.67 in order to purchase gold and borrows $104.17, the total cost of the synthetic call option is $12.50 (= $116.67 $104.17). This is exactly the same price that Maverick would pay for an actual call option. Since an actual call option and a synthetic call option provide Maverick with identical payoff structures, the firm should not expect to pay more for one than the other. Textbook p. 686: The index fund is trading at $1,400 today should be added to the question. Liz would be interested in purchasing a put option on the index fund with a strike price of $1,300 and 1 year until expiration. This option will compensate Liz for any decreases in value of the index fund below the strike price and places a floor of $1,300 on the net worth of her position. In order to solve a problem using the two-state option model, first draw a stock price tree containing both the current stock price and the stocks possible values at the time of the options expiration. Next, draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible stock price movements. The index fund is trading today at $1,400 per share. It will either increase by 25% or decrease by 20% in one year. If the fund increases by 25%, its value will be $1,750 (= $1,400 * 1.25) per share. If it decreases by 20%, its value will be $1,120 (= $1,400 * 0.80) per share. If the fund falls to $1,120, Liz will exercise her put option for $1,300 and receive a payoff of $180 at expiration. If the fund rises to $1,750, Liz will not exercise her put option, and she will receive no payoff at expiration.
Index Fund's Value Today Today 1 Year 1750 1400 1120 ? 180 = max(0, 1300-1120) Liz's Put Option with a Strike of 1300 Today 1 Year 0 = max(0, 1300-1750)
23.18 a.
b.
Answers to End-of-Chapter Problems
B-352
If the price of the index fund rises, its return over the period is 25% [= (1750/1400) 1]. If the price falls, its return over the period is 20% [= (1120/1400) 1]. Use the following expression to determine the risk-neutral probability of a rise in the index fund:
Risk-Free Rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) = (ProbabilityRise)(ReturnRise) + (1 - ProbabilityRise)(ReturnFall) 0.07 = (ProbabilityRise)(0.25) + (1 ProbabilityRise)(-0.20)
ProbabilityRise = 0.60 ProbabilityFall = 1 - ProbabilityRise = 1 0.60 = 0.40 The risk-neutral probability of a rise in the index fund is 60%, and the risk-neutral probability of a fall in the index fund is 40%. Using these risk-neutral probabilities, determine the expected payoff to Lizs put option at expiration. Expected Payoff at Expiration = (.60)($0) + (.40)($180) = $72.00 Since this payoff occurs 1 year from now, it must be discounted at the risk-free rate of 7% per annum in order to find its present value: PV(Expected Payoff at Expiration) = ($72.00 / 1.07 ) = $67.29 Therefore, given the information Liz has about the index funds price movements over the next year, a put option with a strike price of $1,300 and 1 year until expiration is worth $67.29 today. c. Yes, there is a way for Liz to create a synthetic put option with identical payoffs to the put option described above. In order to do this, Liz will need to short shares of the index fund and lend at the risk-free rate. The number of shares that Liz should sell is based on the delta of the option, where delta is defined as: Delta = (Swing of option) / (Swing of stock) Since the put option will be worth $0 if the index fund rises and $180 if it falls, the swing of the put option is -180 (= 0 180). Since the index fund will either be worth $1,750 or $1,120 at the time of the options expiration, the swing of the stock is 630 (= 1,750 1,120). Given this information: Delta = (Swing of option) / (Swing of stock)
B-353
Answers to End-of-Chapter Problems
= (-180/630) = -2/7 Therefore, Lizs first step in creating a synthetic put option is to short 2/7 of a share of the index fund. Since the fund is currently trading at $1,400 per share, Liz receives $400 (= 2/7 * $1,400) as a result of her short sale. In order to determine the amount that Liz should lend, compare the payoff of the actual put option to the payoff of delta shares at expiration. Put Option If the index fund rises to $1,750: If the index fund falls to $1,120: Delta Shares If the index fund rises to $1,750: If the index fund falls to $1,120:
payoff = $0 payoff = $180
payoff = (-2/7)($1,750) = -$500 payoff = (-2/7)($1,120) = -$320
Liz would like the payoff of her synthetic put position to be identical to the payoff of an actual put option. However, shorting 2/7 of a share of the index fund leaves her exactly $500 below the payoff at expiration, regardless of whether the fund rises or falls. In order to increase his payoff at expiration by $500, Liz should lend the present value of $500 now. In one year, she will receive $500, which will increase her payoffs so that they exactly match those of an actual put option. Liz should short 2/7 of a share of the index fund and lend $467.29 (= $500 / 1.07) in order to create a synthetic put option with a strike price of $1,300 and 1 year until expiration. d. Since Liz receives $400 as a result of the short sale and lends $467.29, the total cost of the synthetic put option is $67.29 (= $467.29 - $400). This is exactly the same price that Liz would pay for an actual put option. Since an actual put option and a synthetic put option provide Liz with identical payoff structures, she should not expect to pay more for one than the other. The inputs to the Black-Scholes model are the current price of the underlying asset (S), the strike price of the option (K), the time to expiration of the option in fractions of a year (t), the variance of the underlying asset (s2), and the continuously-compounded risk-free interest rate (r). In this problem, the inputs are: S = $55 K =$50 t=1 s2 = 0.0625 r = 0.10
23.19
After identifying the inputs, solve for d1 and d2: d1 = [ln(S/K) + (r + s2)(t) ] / (s2t)1/2 = [ln(55/50) + {0.10 + (0.0625)}(1) ] / (0.0625*1)1/2 = 0.9062
Answers to End-of-Chapter Problems
B-354
d2 = d1 - (s2t)1/2 = 0.9062 - (0.0625*1)1/2 = 0.6562 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. N(d1) = N(0.9062) = 0.8176 N(d2) = N(0.6562) = 0.7442 According to the Black-Scholes formula, the price of a European call option (C) on a nondividend paying common stock is: C = SN(d1) Ke-rtN(d2) = (55)(0.8176) (50)e-(.10)(1) (0.7442) = $11.30 The Black-Scholes Price of the call option is $11.30. 23.20 The inputs to the Black-Scholes model are the current price of the underlying asset (S), the strike price of the option (K), the time to expiration of the option in fractions of a year (t), the variance of the underlying asset (s2), and the continuously-compounded risk-free interest rate (r). In this problem, the inputs are: S = $15 K =$25 t = 0.50 s2 = 0.25 r = 0.08
After identifying the inputs, solve for d1 and d2: d1 = [ln(S/K) + (r + s2)(t) ] / (s2t)1/2 = [ln(15/25) + {0.08 + (0.25)}(0.50) ] / (0.25*.50)1/2 = -1.1549 d2 = d1 - (s2t)1/2 = -1.1549- (0.25*.50)1/2 = -1.5085 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. N(d1) = N(-1.1549) = 0.1241 N(d2) = N(-1.5085) = 0.0657 According to the Black-Scholes formula, the price of a European call option (C) on a nondividend paying common stock is: C = SN(d1) Ke-rtN(d2)
Answers to End-of-Chapter Problems B-355
= (15)(0.1241) (25)e-(0.08)(0.50) (0.0657) = $0.28 The Black-Scholes Price of the call option is $0.28. 23.21 a. The inputs to the Black-Scholes model are the current price of the underlying asset (S), the strike price of the option (K), the time to expiration of the option in fractions of a year (t), the variance of the underlying asset (s2), and the continuously-compounded risk-free interest rate (r). In this problem, the inputs are: S = $100 K =$100 t=2 s2 = 0.04 r = 0.05
After identifying the inputs, solve for d1 and d2: d1 = [ln(S/K) + (r + s2)(t) ] / (s2t)1/2 = [ln(100/100) + {0.05 + (0.04)}(2) ] / (0.04*2)1/2 = 0.4950 d2 = d1 - (s2t)1/2 = 0.4950 - (0.04*2)1/2 = 0.2122 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. N(d1) = N(0.4950) = 0.6897 N(d2) = N(0.2122) = 0.5841 According to the Black-Scholes formula, the price of a European call option (C) on a nondividend paying common stock is: C = SN(d1) Ke-rtN(d2) = (100)(0.6897) (100)e-(0.05)(2) (0.5841) = $16.12 The Black-Scholes Price of the call option is $16.12. b. Put-Call Parity implies that the cost of a European call option (C) must equal the cost of a European put option with the same strike price and time to expiration (P) plus the current stock price (S) minus the present value of the strike price [PV(K)]. In this problem: C = $16.12 S = $100 PV(K) = $100 / e(.05*2) = $90.48
Answers to End-of-Chapter Problems
B-356
Rearranging the Put-Call Parity formula: P = C S + PV(K) = $16.12 - $100 + $90.48 = $6.60
Therefore, Put-Call Parity implies that the Black-Scholes price of a European put option with a strike price of $100 and 2 years until expiration should be $6.61. 23.22 a. The inputs to the Black-Scholes model are the current price of the underlying asset (S), the strike price of the option (K), the time to expiration of the option in fractions of a year (t), the variance of the underlying asset (s2), and the continuously-compounded risk-free interest rate (r). In this problem, the inputs are: S = $60 K =$30 t = 0.25 s2 = 0.36 r = 0.03
After identifying the inputs, solve for d1 and d2: d1 = [ln(S/K) + (r + s2)(t) ] / (s2t)1/2 = [ln(60/30) + {0.03 + (0.36)}(0.25) ] / (0.36*0.25)1/2 = 2.4855 d2 = d1 - (s2t)1/2 = 2.4855- (0.36*.25)1/2 = 2.1855 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. N(d1) = N(2.4855) = 0.9935 N(d2) = N(2.1855) = 0.9856 According to the Black-Scholes formula, the price of a European call option (C) on a nondividend paying common stock is: C = SN(d1) Ke-rtN(d2) = (60)(0.9935) (30)e-(0.03)(0.25) (0.9856) = $30.26 The Black-Scholes Price of the call option is $30.26. b. Put-Call Parity implies that the cost of a European call option (C) must equal the cost of a European put option with the same strike price and time to expiration (P) plus the current stock price (S) minus the present value of the strike price [PV(K)]. In this problem: C = $30.26 S = $60
B-357
Answers to End-of-Chapter Problems
PV(K) = $30 / e(.03*0.25) = $29.78 Rearranging the Put-Call Parity formula: P = C S + PV(K) = $30.26 - $60 + $29.78 = $0.04
Therefore, Put-Call Parity implies that the Black-Scholes price of a European put option with a strike price of $30 and 3 months until expiration should be $0.04. 23.23 a. The inputs to the Black-Scholes model are the current price of the underlying asset (S), the strike price of the option (K), the time to expiration of the option in fractions of a year (t), the variance of the underlying asset (s2), and the continuously-compounded risk-free interest rate (r). In this problem, the inputs are: S = $37 K =$35 t=1 s2 = 0.0225 r = 0.07
After identifying the inputs, solve for d1 and d2: d1 = [ln(S/K) + (r + s2)(t) ] / (s2t)1/2 = [ln(37/35) + {0.07 + (0.0225)}(1) ] / (0.0225*1)1/2 = 0.9121 d2 = d1 - (s2t)1/2 = 0.9121- (0.0225*1)1/2 = 0.7621 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. N(d1) = N(0.9121) = 0.8191 N(d2) = N(0.7621) = 0.7770 According to the Black-Scholes formula, the price of a European call option (C) on a nondividend paying common stock is: C = SN(d1) Ke-rtN(d2) = (37)(0.8191) (35)e-(0.07)(1) (0.7770) = $4.95 The Black-Scholes Price of the call option is $4.95. b. The inputs to the Black-Scholes formula are: S= $37 K =$35 t=1 s2 = 0.09 r = 0.07
Answers to End-of-Chapter Problems
B-358
After identifying the inputs, solve for d1 and d2: d1 = [ln(S/K) + (r + s2)(t) ] / (s2t)1/2 = [ln(37/35) + {0.07 + (0.09)}(1) ] / (0.09*1)1/2 = 0.5686 d2 = d1 - (s2t)1/2 = 0.9121- (0.09*1)1/2 = 0.2686 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. N(d1) = N(0.5686) = 0.7152 N(d2) = N(0.2686) = 0.6059 According to the Black-Scholes formula, the price of a European call option (C) on a nondividend paying common stock is: C = SN(d1) Ke-rtN(d2) = (37)(0.7152) (35)e-(0.07)(1) (0.6059) = $6.69 The Black-Scholes Price of the call option is $6.69. c. An increase in the volatility (variance) of the underlying asset increases the Black-Scholes price of a call option. An increase in variance increases the value of exercise should the option end up in-the-money. In this example, an otherwise identical European call option increases in price from $4.95 to $6.69 when Marilyns estimate of the variance of Scubas stock returns changes from 0.0225 to 0.09. The inputs to the Black-Scholes formula are: S= $20 K =$35 t=1 s2 = 0.09 r = 0.07
d.
After identifying the inputs, solve for d1 and d2: d1 = [ln(S/K) + (r + s2)(t) ] / (s2t)1/2 = [ln(20/35) + {0.07 + (0.09)}(1) ] / (0.09*1)1/2 = -1.4821 d2 = d1 - (s2t)1/2 = -1.4821- (0.09*1)1/2 = -1.7821 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively.
Answers to End-of-Chapter Problems
B-359
N(d1) = N(-1.4821) = 0.0691 N(d2) = N(-1.7821) = 0.0373 According to the Black-Scholes formula, the price of a European call option (C) on a nondividend paying common stock is: C = SN(d1) Ke-rtN(d2) = (20)(0.0691) (35)e-(0.07)(1) (0.0373) = $0.16 The Black-Scholes Price of the call option is $0.16. 23.24 To calculate the total cost of the position, consider what Marie must do in order to obtain a collar: a. Purchase one share of Hollywoods stock b. Sell a call option on Hollywoods stock with a strike price of $120 and 6 months until expiration c. Purchase a put option on Hollywoods stock with a strike price of $50 and 6 months until expiration a. To purchase one share of Hollywoods stock, Marie must pay the current price of $80 per share. b. Use the Black-Scholes model to calculate the proceeds that Marie will receive from the sale of a call option with a strike price of $120 and 6 months until expiration. The inputs to the Black-Scholes formula are: S= $80 K =$120 t = .50 s2 = 0.25 r = 0.10
After identifying the inputs, solve for d1 and d2: d1 = [ln(S/K) + (r + s2)(t) ] / (s2t)1/2 = [ln(80/120) + {0.10 + (0.25)}(0.50) ] / (0.25*0.50)1/2 = -0.8286 d2 = d1 - (s2t)1/2 = -0.8286- (0.25*0.50)1/2 = -1.1822 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. N(d1) = N(-0.8286) = 0.2037 N(d2) = N(-1.1822) = 0.1186 According to the Black-Scholes formula, the price of a European call option (C) on a nondividend paying common stock is:
Answers to End-of-Chapter Problems B-360
C = SN(d1) Ke-rtN(d2) = (80)(0.2037) (120)e-(0.10)(0.50) (0.1186) = $2.76 The Black-Scholes Price of the call option is $2.76. Marie will receive $2.76 as a result of selling a call option on Hollywoods stock with a strike price of $120 and 6 months until expiration. c. Use the Black-Scholes model to calculate the price that Marie will pay to purchase a put option with a strike price of $50 and 6 months until expiration. In order to this, calculate the cost of a call option with a strike price of $50 and 6 months until expiration. Then use PutCall Parity to find the cost of an otherwise identical put. The inputs to the Black-Scholes formula are: S= $80 K =$50 t = 0.50 s2 = 0.25 r = 0.10
After identifying the inputs, solve for d1 and d2: d1 = [ln(S/K) + (r + s2)(t) ] / (s2t)1/2 = [ln(80/50) + {0.10 + (0.25)}(0.50) ] / (0.25*0.50)1/2 = 1.6476 d2 = d1 - (s2t)1/2 = 1.6476 - (0.25*0.50)1/2 = 1.2940 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. N(d1) = N(1.6476) = 0.9503 N(d2) = N(1.2940) = 0.9022 According to the Black-Scholes formula, the price of a European call option (C) on a nondividend paying common stock is: C = SN(d1) Ke-rtN(d2) = (80)(0.9503) (50)e-(0.10)(0.50) (0.9022) = $33.11 The Black-Scholes Price of the call option is $33.11. According to Put-Call Parity: In this problem: C = P + S PV(K)
C = $33.11 S = $80 PV(K) = [$50 / e(0.10*0.50) ] = $47.56
B-361
Answers to End-of-Chapter Problems
Rearranging the Put-Call Parity equation: P = C S + PV(K) = $33.11 - $80 + $47.56 = $0.67 Marie must pay $0.67 in order to purchase a put option on Hollywoods stock with a strike price of $50 and 6 months until expiration. Total cost of collar = Price of Stock + Price of Put Price of Call = $80 + $0.67 - $2.76 = $77.91 Therefore, Marie must pay $77.91 in order to purchase a collar with the characteristics described above on Hollywoods stock. 23.25 The equityholders of a firm financed partially with debt can be thought of as holding a call option on the assets of the firm with a strike price equal to the debts face value and a time to expiration equal to the debts time to maturity. If the value of the firm exceeds the face value of the debt when it matures, the firm will pay off the debtholders in full, leaving the equityholders with the firms remaining assets. However, if the value of the firm is less than the face value of debt when it matures, the firm must liquidate all of its assets in order to pay off the debtholders, and the equityholders receive nothing. Let VL = the value of a firm financed with both debt and equity
Payoff to Debtholders Payoff to Equityholders Total
If VL < FV(Debt) VL 0 VL
If VL > FV(Debt) FV(Debt) VL - FV(Debt) VL
FV(Debt) = the face value of the firms outstanding debt at maturity
Notice that the payoff to equityholders is identical to a call option of the form max(0, ST K), where the stock price at expiration (ST) is equal to the value of the firm at the time of the debts maturity and the strike price (K) is equal to the face value of outstanding debt. 23.26 a. Since the equityholders of a firm financed partially with debt can be thought of as holding a call option on the assets of the firm with a strike price equal to the debts face value and a time to expiration equal to the debts time to maturity, the value of Strudlers equity equals a call option with a strike price of $380 million and 1 year until expiration. In order to value this option using the two-state option model, first draw a tree containing both the current value of the firm and the firms possible values at the time of the options expiration. Next, draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible changes in the firms value.
Answers to End-of-Chapter Problems
B-362
The value of Strudler today is $400 million. It will either increase to $500 million or decrease to $320 million in one year as a result of its new project. If the firms value increases to $500 million, the equityholders will exercise their call option, and they will receive a payoff of $120 million at expiration. However, if the firms value decreases to $320 million, the equityholders will not exercise their call option, and they will receive no
Value of Strudler, Inc. (in millions) Today 1 Year 500 400 320 ? 0 = max(0, 320-380) The Equityholders' Call Option with a Strike of $380 (in millions) Today 3 months 120 = max(0, 500-380)
payoff at expiration. If the project is successful and Strudlers value rises, the return on Strudler over the period is 25% [= (500/400) 1]. If the project is unsuccessful and Strudlers value falls, the return on Strudler over the period is 20% [= (320/400) 1]. Use the following expression to determine the risk-neutral probability of a rise in the value of Strudler: Risk-Free Rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) = (ProbabilityRise)(ReturnRise) + (1 - ProbabilityRise)(ReturnFall) 0.07 = (ProbabilityRise)(0.25) + (1 ProbabilityRise)(-0.20)
ProbabilityRise = 0.60 ProbabilityFall = 1 - ProbabilityRise = 1 0.60 = 0.40 The risk-neutral probability of a rise in the value of Strudler is 60%, and the risk-neutral probability of a fall in the value of Strudler is 40%. Using these risk-neutral probabilities, determine the expected payoff to the equityholders call option at expiration. Expected Payoff at Expiration = (.60)($120,000,000) + (.40)($0) = $72,000,000 Since this payoff occurs 1 year from now, it must be discounted at the risk-free rate of 7% in order to find its present value: PV(Expected Payoff at Expiration) = ($72,000,000 / 1.07) = $67,289,720 A call option with a strike price of $380 million and one year until expiration is worth $67,289,720 today. Therefore, the current value of the firms equity is $67,289,720.
Answers to End-of-Chapter Problems
B-363
The current value of the firm ($400 million) is equal to the value of its equity plus the value of its debt. In order to find the value of Strudlers debt, subtract the value of the firms equity from the total value of the firm: VL = Debt + Equity $400,000,000 = Debt + $67,289,720 Debt = $332,710, 280 Therefore, the current value of the firms debt is $332,710,280. b. Since the firms equity is worth $67,289,720 and there are 500,000 shares outstanding, each share is worth: Price Per Share = Equity Value / # shares outstanding = $67,289, 720 / 500,000 = $134.58 Therefore, the price of Strudlers equity is $134.58 per share. c. The market value of the firms debt is $332,710,280. The present value of the same face amount of riskless debt is $355,140,187 (= $380,000,000 / 1.07). The firms debt is worth less than the present value of riskless debt since there is a risk that it will not be repaid in full. In other words, the market value of the debt takes into account the risk of default. The value of riskless debt is $355,140,187. Since there is a chance that Strudler might not repay its debtholders in full, the debt is worth less than $355,140,187. The value of Strudler today is $400 million. It will either increase to $800 million or decrease to $200 million in one year as a result of the new project. If the firms value increases to $800 million, the equityholders will exercise their call option, and they will receive a payoff of $420 million at expiration. However, if the firms value decreases to $200 million, the equityholders will not exercise their call option, and they will receive no payoff at expiration.
Value of Strudler, Inc. (in millions) Today 1 Year 800 400 200 ? 0 = max(0, 200-380) The Equityholders' call Option with a Strike of $380 (in millions) Today 1 Year 420 = max(0, 800-380)
d.
If the project is successful and Strudlers value rises, the return on Strudler over the period is 100% [= (800/400) 1]. If the project is unsuccessful and Strudlers value falls, the return on Strudler over the period is 50% [= (200/400) 1]. Use the following expression to determine the risk-neutral probability of a rise in the value of Strudler:
Answers to End-of-Chapter Problems B-364
Risk-Free Rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) = (ProbabilityRise)(ReturnRise) + (1 - ProbabilityRise)(ReturnFall) 0.07 = (ProbabilityRise)(1) + (1 ProbabilityRise)(-0.50)
ProbabilityRise = 0.38 ProbabilityFall = 1 - ProbabilityRise = 1 0.38 = 0.62 The risk-neutral probability of a rise in the value of Strudler is 38%, and the risk-neutral probability of a fall in the value of Strudler is 62%. Using these risk-neutral probabilities, determine the expected payoff to the equityholders call option at expiration. Expected Payoff at Expiration = (.38)($420,000,000) + (.62)($0) = $159,600,000 Since this payoff occurs 1 year from now, it must be discounted at the risk-free rate of 7% in order to find its present value: PV(Expected Payoff at Expiration) = ($159,600,000 / 1.07) = $149,158,879 A call option with a strike price of $380 million and one year until expiration is worth $149,158,879 today. Therefore, the current value of the firms equity is $149,158,879. The current value of the firm ($400 million) is equal to the value of its equity plus the value of its debt. In order to find the value of Strudlers debt, subtract the value of the firms equity from the total value of the firm: VL = Debt + Equity $400,000,000 = Debt + $149,158,879 Debt = $250,841,121 Therefore, the current value of the firms debt is $250,841,121. The riskier project increases the value of the firms equity and decreases the value of the firms debt. If Strudler takes on the riskier project, the firm is less likely to be able to pay off its bondholders. Since the risk of default increases if the new project is undertaken, the value of the firms debt decreases. Bondholders would prefer Strudler to undertake the more conservative project. 23.27 Since the firm has 700 bonds outstanding, each with a face value of $1,000, the total face value of the firms outstanding debt is $700,000 (= 700 * $1,000). Given that the equityholders of a levered firm can be thought as holding a call option on the assets of the firm with a strike price equal to the debts face value and a time to expiration equal to the debts time to maturity, the value of this firms equity equals a call option with a strike
B-365
Answers to End-of-Chapter Problems
price of $700,000 and six months until expiration. Use the Black-Scholes formula to calculate the price of this call option. The inputs to the Black-Scholes formula are: S = $1,000,000 s2 = 0.16 K = $700,000 r = 0.08 t = 0.50
After identifying the inputs, solve for d1 and d2: d1 = [ln(S/K) + (r + s2)(t) ] / (s2t)1/2 = [ln(1,000,000 / 700,000) + {0.08 + (0.16)}(0.50) ] / (0.16*0.50)1/2 = 1.5439 d2 = d1 - (s2t)1/2 = 1.5439 - (0.16*0.50)1/2 = 1.2611 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. N(d1) = N(1.5439) = 0.9387 N(d2) = N(1.2611) = 0.8964 According to the Black-Scholes formula, the price of a European call option (C) on a nondividend paying common stock is: C = SN(d1) Ke-rtN(d2) = (1,000,000)(0.9387) (700,000)e-(0.08)(0.50) (0.8964) = $335,824 The Black-Scholes Price of the call option is $335,824. Therefore, the current value of the firms equity is $335,824. The current value of the firm ($1 million) is equal to the value of its equity plus the value of its debt. In order to find the value of the firms debt, subtract the value of the firms equity from the total value of the firm: VL = Debt + Equity $1,000,000= Debt + $335,824 Debt = $664,176 Therefore, the current value of the firms debt is $664,176.
Answers to End-of-Chapter Problems
B-366