Note: Although most of the time, the solution features risk-neutral method in solving binomial model, you should be able to redo the solution entirely with portfolio replicating method.
Chapter 23: Applications of Options to Corporate Finance
23.34The 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 debt’s face value and a time to expiration equal to the debt’s 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 firm’s 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
FV(Debt) = the face value of the firm’s 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 debt’s maturity and the strike price (K) is equal to the face value of outstanding debt.
23.35a.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 debt’s face value and a time to expiration equal to the debt’s time to maturity, the value of Weber’s equity equals a call option with a strike price of $387 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 firm’s possible values at the time of the option’s 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 firm’s value.
The value of Weber today is $421 million. It will either increase to $550 million or decrease to $370 million in one year as a result of its new project. If the firm’s value increases to $550 million, the equityholders will exercise their call option, and they will receive a payoff of $163 million at expiration. However, if the firm’s value decreases to $370 million, the equityholders will not exercise their call option, and they will receive no payoff at expiration.
If the project is successful and Weber’s value rises, the return on Weber over the period is 30.64% [= (550/421) – 1]. If the project is unsuccessful and Weber’s value falls, the return on Weber over the period is –12.11% [= (370/421) –1]. Use the following expression to determine the risk–neutral probability of a rise in the value of Weber:
Risk–Free Rate= (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)
= (ProbabilityRise)(ReturnRise) + (1 – ProbabilityRise)(ReturnFall)
0.08 = (ProbabilityRise)(0.3064) + (1 – ProbabilityRise)(–0.1211)
ProbabilityRise= 0.47044
ProbabilityFall= 1 – ProbabilityRise
= 1 – 0.47
= 0.52956
The risk–neutral probability of a rise in the value of Weber is 47.044%, and the risk–neutral probability of a fall in the value of Weber is 52.956%.
Using these risk–neutral probabilities, determine the expected payoff to the equityholders’ call option at expiration.
Expected Payoff at Expiration = (0.47044)($163,000,000) + (0.52956)($0) = $76,682,440
Since this payoff occurs 1 year from now, it must be discounted at the risk–free rate of 8% in order to find its present value:
PV(Expected Payoff at Expiration) = ($76,682,440 / 1.08) = $71,002,260
A call option with a strike price of $380 million and one year until expiration is worth $71,002,260today.
Therefore, the current value of the firm’s equity is $71,002,260.
The current value of the firm ($421 million) is equal to the value of its equity plus the value of its debt. In order to find the value of Weber’s debt, subtract the value of the firm’s equity from the total value of the firm:
VL = Debt + Equity
$421,000,000 = Debt + $71,002,260
Debt = $ 349,997,740
Therefore, the current value of the firm’s debt is $349,997,740.
b.Since the firm’s equity is worth $71,002,260and there are 510,000 shares outstanding, each share is worth:
Price Per Share = Equity Value / # shares outstanding
= $71,002,260/ 510,000
= $139.22
Therefore, the price of Weber’s equity is $139.22 per share.
c.The market value of the firm’s debt is $349,997,740. The present value of the same face amount of riskless debt is $358,333,333 (=$387,000/ 1.08). The firm’s 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 $358,333,333. Since there is a chance that Weber might not repay its debtholders in full, the debt is worth less than $358,333,333.
d.The value of Weber today is $421 million. It will either increase to $750 million or decrease to $300 million in one year as a result of the new project. If the firm’s value increases to $750 million, the equityholders will exercise their call option, and they will receive a payoff of $329 million at expiration. However, if the firm’s value decreases to $300 million, the equityholders will not exercise their call option, and they will receive no payoff at expiration.
If the project is successful and Weber’s value rises, the return on Weber over the period is 78% [= (750/421) – 1]. If the project is unsuccessful and Weber’s value falls, the return on Weber over the period is –29% [= (300/421) –1]. Use the following expression to determine the risk–neutral probability of a rise in the value of Weber:
Risk–Free Rate= (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)
= (ProbabilityRise)(ReturnRise) + (1 – ProbabilityRise)(ReturnFall)
0.08 = (ProbabilityRise)(0.78) + (1 – ProbabilityRise)(–0.29)
ProbabilityRise= 0.3437
ProbabilityFal= 1 – ProbabilityRise
= 1 – 0.3437
= 0.6563
The risk–neutral probability of a rise in the value of Weber is 34.37%, and the risk–neutral probability of a fall in the value of Weber is 65.63%.
Using these risk–neutral probabilities, determine the expected payoff to the equityholders’ call option at expiration.
Expected Payoff at Expiration = (0.3437)($363,000,000) + (0.6563)($0) = $124,775,000
Since this payoff occurs 1 year from now, it must be discounted at the risk–free rate of 8% in order to find its present value:
PV(Expected Payoff at Expiration) = ($124,775,000/ 1.08) = $115,533,000
A call option with a strike price of $387 million and one year until expiration is worth $115,533,000.
Therefore, the current value of the firm’s equity is $115,533,000.
The current value of the firm ($421 million) is equal to the value of its equity plus the value of its debt. In order to find the value of Weber’s debt, subtract the value of the firm’s equity from the total value of the firm:
VL = Debt + Equity
$421,000,000 = Debt + $115,533,000
Debt = $305,467,000
Therefore, the current value of the firm’s debt is $305,467,000.
The riskier project increases the value of the firm’s equity and decreases the value of the firm’s debt. If Weber 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 firm’s debt decreases.
Bondholders would prefer Weber to undertake the more conservative project.
23.36Since the firm has 700 bonds outstanding, each with a face value of $1,000, the total face value of the firm’s 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 debt’s face value and a time to expiration equal to the debt’s time to maturity, the value of this firm’s equity equals a call option with a strike price of $700,000 and six months until expiration. Use the Black–Scholes formula to calculate the price of this call option.
After identifying the inputs, solve for d1 and d2:
d1 = [ln(S/K) + (r + ½2)(t) ] / (2t)1/2
= [ln(1,100,000 / 700,000) + {0.14 + ½(0.14)}(0.50) ] / (0.14*0.50)1/2
= 2.1052
d2 = d1 – (2t)1/2
= 2.1– (0.14*0.50)1/2
= 1.8406
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(2.1052) = 0.9824
N(d2) = N(1.8406) = 0.9672
According to the Black–Scholes formula, the price of a European call option (C) on a non–dividend paying common stock is:
C = SN(d1) – Ke–rtN(d2)
= (1,100,000)(0.9824) – (700,000)e–(0.14)(0.50) (0.9672)
= $449,356
The Black–Scholes Price of the call option is $449,356.
Therefore, the current value of the firm’s equity is $449,356.
The current value of the firm ($1.1 million) is equal to the value of its equity plus the value of its debt. In order to find the value of the firm’s debt, subtract the value of the firm’s equity from the total value of the firm:
VL = Debt + Equity
$1,100,000= Debt + $449,356
Debt = $650,644
Therefore, the current value of the firm’s debt is $650,644.
Chapter 24: Options and Corporate Finance: Extensions and Applications
24.1a.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 (2), and the continuously–compounded risk–free interest rate (r).
Mr. Levin has been granted 25,000 European call options on Mountainbrook’s stock with 4 years until expiration. Since these options were granted at–the–money, the strike price of each option is equal to the current value of one share, or $55.
After identifying the inputs, solve for d1 and d2:
d1= [ln(S/K) + (r + ½2)(t) ] / (2t)1/2
d1= [ln(55/55) + {0.054 + ½(0.422)}(4) ] / (0.422*4)1/2= 0.677
d2= 0.677– (0.422*4)1/2= –0.1628
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(0.677) =0.7508
N(d2) = N(–0.1628) = 0.4353
According to the Black–Scholes formula, the price of a European call option (C) on a non–dividend paying common stock is:
C = SN(d1) – Ke–rtN(d2)
C = (55)(0.7508) – (55)e–(0.054)(4) (0.4353)
= $22.005
The Black–Scholes Price of one call option is $22.005.
Since Mr. Levin was granted 25,000 options, the current value of his options package is $550,125(= 25,000 * $22.005).
b.Because Mr. Levin is risk–neutral, you should recommend the alternative with the highest net present value. Since the expected value of the stock option package is worth more than $550,000, Mr. Levin would prefer to be compensated with the options rather than with the immediate bonus.
c.If Mr. Levin is risk–averse, he may or may not prefer the stock option package to the immediate bonus. Even though the stock option package has a higher net present value, he may not prefer it because it is undiversified. The fact that he cannot sell his options prematurely makes it much more risky than the immediate bonus. Therefore, we cannot say which alternative he would prefer.
If Mr. Levin is reasonably risk averser, then he would prefer the immediate bonus because the NPV difference is really small, but the risk is very different.
24.5When solving a question dealing with real options, begin by identifying the option–like features of
the situation. First, since the company will only choose to drill and excavate if the price of oil rises,
the right to drill on the land can be viewed as a call option. Second, since the land contains 125,000
barrels of oil and the current price of oil is $55 per barrel, the current price of the underlying asset
(S) to be used in the Black–Scholes model is:
“Stock” price = 125,000($55)
“Stock” price = $6,875,000
Third, since the company will not drill unless the price of oil in one year will compensate its
excavation costs, these costs can be viewed as the real option’s strike price (K). Finally, since the
winner of the auction has the right to drill for oil in one year, the real option can be viewed as having
a time to expiration (t) of one year. Using the Black–Scholes model to determine the value of the
option, we find:
d1= [ln(S/K) + (r + σ2/2)(t) ] / (σ 2t)1/2
d1 = [ln($6,875,000/$10,000,000) + (0.065 + 0.502/2) (1)] / (0.50)(√1 ) = –0.3694
d2 = –0.3694 – (0.50)(√1 ) = –0.8694
Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative
infinity to d2, respectively. Doing so:
N(d1) = N(–0.3694) = 0.3559
N(d2) = N(–0.8694) = 0.1923
Now we can find the value of call option, which will be:
C = SN(d1) – Ke—rtN(d2)
C = $6,875,000(0.3559) – ($10,000,000e–.065(1))(0.1923)
C = $644,800.53
This is the maximum bid the company should be willing to make at auction.
24.6 When solving a question dealing with real options, begin by identifying the option–like features of
the situation. First, since Sardano will only choose to manufacture the steel rods if the price of steel
falls, the lease, which gives the firm the ability to manufacture steel, can be viewed as a put option.
Second, since the firm will receive a fixed amount of money if it chooses to manufacture the rods:
Amount received = 4,800 steel rods($360 – $120)
Amount received = $1,152,000
The amount received can be viewed as the put option’s strike price (K). Third, since the project
requires Sardano to purchase 400 tons of steel and the current price of steel is $3,600 per ton, the
current price of the underlying asset (S) to be used in the Black–Scholes formula is:
“Stock” price = 400 tons($3,600 per ton)
“Stock” price = $1,440,000
Finally, since Sardano must decide whether to purchase the steel or not in six months, the firm’s real
option to manufacture steel rods can be viewed as having a time to expiration (t) of six months. In
order to calculate the value of this real put option, we can use the Black–Scholes model to determine
the value of an otherwise identical call option then infer the value of the put using put–call parity.
Using the Black–Scholes model to determine the value of the option, we find:
d1= [ln(S/K) + (r + σ2/2)(t) ] / (σ2t)1/2
d1 = [ln($1,440,000/$1,152,000) + (0.045 + 0.452/2) (6/12)] / (0.45 √6/12 ) = 0.9311
d2 = 0.9311 – (0.45 √6/12 ) = 0.6129
Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative
infinity to d2, respectively. Doing so:
N(d1) = N(0.9311) = 0.8241
N(d2) = N(0.6129) = 0.7300
Now we can find the value of call option, which will be:
C = SN(d1) – K e—rt N(d2)
C = $1,440,000(0.8241) – ($1,152,000e–0.045(6/12))(0.7300)
C = $364,419.87
Now we can use put–call parity to find the price of the put option, which is:
C = P + S – K e–rt
$364,419.87 = P + $1,440,000 – $1,152,000e–0.045(6/12)
P = $50,789.29
This is the most the company should be willing to pay for the lease.
24.7When solving a question dealing with real options, begin by identifying the option–like features of the situation. First, since Webber will exercise its option to build if the value of an office building rises, the right to build the office building is similar to a call option. Second, an office building in downtown Sacramento would be worth $10 million today. This amount can be viewed as the current price of the underlying asset (S). Third, it will cost Webber $10.5 million to construct such an office building. This amount can be viewed as the strike price of a call option (K), since it is the amount that the firm must pay in order to ‘exercise’ its right to erect an office building. Finally, since the firm’s right to build on the land lasts only 1 year, the time to expiration (t) of the real option is one year. The Webber Company can use a Two–State model to value its option to build on the land.
If demand increases and the value of the building rises, the return on the value of the building over the period is 25% [= (12.5/10) – 1]. If demand decreases and the value of the building falls, the return on the value of the building over the period is –20% [= (8/10) –1]. Use the following expression to determine the risk–neutral probability of a rise in the value of the building:
Risk–Free Rate= (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)
= (ProbabilityRise)(ReturnRise) + (1 – ProbabilityRise)(ReturnFall)
0.021= (ProbabilityRise)(0.25) + (1 – ProbabilityRise)(–0.20)
ProbabilityRise= 0.4911
ProbabilityFall= 1 – 0.49= 0.5089
The risk–neutral probability of a rise in the value of the building is 49.11%, and the risk–neutral probability of a fall in the value of the building is 50.89%.
Using these risk–neutral probabilities, determine the expected payoff of Webber’s real option at expiration.
Expected Payoff at Expiration = (.4911)($2,000,000) + (.5089)($0) = $982,222
Since this payoff will occur 1 year from now, it must be discounted at the risk–free rate of 2.1% in order to find its present value:
PV(Expected Payoff at Expiration) = ($982,222 / 1.021) = $962,020
A call option with a strike price of $10.5 million and 1 year until expiration is worth $962,020.
Therefore, the right to build on office building in downtown Sacramento over the next year is worth $962,020today.
Since $750,000 is less than the value of the real option to build, Webber should not accept the offer from his competitor. Instead, Webber should retain the right to erect an office building on the land.
Answers to End-of-Chapter ProblemsB-1