Question 1

What is the value of a 6-month put with a strike price of $27.50 given the Black-Scholes option pricing model and the following information?

Accepted Answer

P = ($27.50 F1F1F1S1F1F1F10 2.71828^-0.035F1F1F1S1^F1F1F10^0.5) + $1.46106 - $21.10 = $7.38

Question 2

A stock is currently selling for $36 a share. The risk-free rate is 3.8 percent and the standard deviation is 27 percent. What is the value of d₁ of a 9-month call option with a strike price of $40?

Accepted Answer

Using the Black-Scholes formula: d1 = [ln(S0/K) + (r + 0.5 * sigma^2) * t] / (sigma * sqrt(t)) d2 = d1 - sigma * sqrt(t) N(d1) = 0.4091 N(d2) = 0.3171 where S0 = 36, K = 40, r = 0.038, t = 0.75 (9 months), and sigma = 0.27. Using the formula for the call option premium: C = S0 * N(d1) - K * exp(-r * t) * N(d2) C = $2.37 Using the formula for delta: Delta = N(d1) Delta = 0.4091 Using the formula for d₁: d₁ = Delta * S0 / C d₁ = -0.21179

Question 3

Explain why financial mergers tend to benefit bondholders more than shareholders.

Accepted Answer

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Question 4

What is the price of a put option given the following information?

Accepted Answer

d₁ = [ln ($81/$88) + (0.04 + 0.64²/2) F1F1F1S1F1F1F10 0.5]/[0.64 F1F1F1S1F1F1F10 (0.5^1/2)] = 0.0873d₂ = 0.0873 - [0.64 F1F1F1S1F1F1F10 (0.5^1/2)] = -0.3652N(d₁) = 0.5348N(d₂) = 0.3575C = $81(0.5348) - ($88e^-0.04(0.5)) (0.3575) = $12.48P = $88e^-0.04(0.5) + $12.48 - $81 = $17.74

Question 5

The current market value of the assets of Nano Tek is $16 million. The market value of the equity is $7.5 million. The risk-free rate is 4.5 percent and the outstanding debt matures in 5 years. What is the market value of the firm's debt?

Accepted Answer

Market value of debt = $16m - $7.5m = $8.5m

Question 6

What is the value of a 6-month call with a strike price of $25 given the Black-Scholes option pricing model and the following information?

Accepted Answer

C = ($17.20 F1F1F1S1F1F1F10 0.26016) - ($25 F1F1F1S1F1F1F10 2.71828^-0.0375F1F1F1S1^F1F1F10^0.5 F1F1F1S1F1F1F10 0.14456) = $0.93

Question 7

The current market value of the assets of Smethwell, Inc. is $56 million, with a standard deviation of 16 percent per year. The firm has zero-coupon bonds outstanding with a total face value of $40 million. These bonds mature in 2 years. The risk-free rate is 4.5 percent per year compounded continuously. What is the value of d₁?

Accepted Answer

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Question 8

A put option that expires in eight months with an exercise price of $57 sells for $3.85. The stock is currently priced at $59, and the risk-free rate is 3.1 percent per year, compounded continuously. What is the price of a call option with the same exercise price and expiration date?

Accepted Answer

$59 + $3.85 = $57 e^{-(0.031)(2/3)} + C; C = $7.02

Question 9

A stock is selling for $60 per share. A call option with an exercise price of $67 sells for $3.31 and expires in 4 months. The risk-free rate of interest is 2.8 percent per year, compounded continuously. What is the price of a put option with the same exercise price and expiration date?

Accepted Answer

$60 + P = $67e^{-(0.028)(1/3)} + $3.31; P = $9.69

Question 10

You need $12,000 in 6 years. How much will you need to deposit today if you can earn 11 percent per year, compounded continuously? Assume this is the only deposit you make.

Accepted Answer

PV = $12,000 F1F1F1S1F1F1F10 e^-0.11(6) = $6,202.22

Question 11

Explain how an increase in T-bill rates will affect the value of an American call and an American put.

Accepted Answer

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Question 12

The delta of a call option on a firm's assets is 0.767. This means that a $50,000 project will increase the value of equity by:

Accepted Answer

Increase in equity value = $50,000 F1F1F1S1F1F1F10 0.767 = $38,350

Question 13

What is the delta of a put option given the following information?

Accepted Answer

d₁ = [ln ($90/$85) + (0.07 + 0.5²/2) F1F1F1S1F1F1F10 (10/12)]/[0.5 F1F1F1S1F1F1F10 (10/12)^1/2] = 0.4812N(d₁) = 0.685Put delta = 0.685 - 1 = -0.315

Question 14

Give an example of a protective put and explain how this strategy reduces investor risk.

Accepted Answer

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Question 15

The current market value of the assets of Cristopherson Supply is $46.5 million. The market value of the equity is $28.7 million. The risk-free rate is 4.75 percent and the outstanding debt matures in 4 years. What is the market value of the firm's debt?

Accepted Answer

Market value of debt = $46.5m - $28.7m = $17.8m

Question 16

A stock is currently selling for $55 a share. The risk-free rate is 4 percent and the standard deviation is 18 percent. What is the value of d₁ of a 9-month call option with a strike price of $57.50?

Accepted Answer

Using the Black-Scholes option pricing model: d1 = [ln(55/57.50) + (0.04 + 0.18^2/2)(0.75)] / (0.18√0.75) = -0.2291 d2 = -0.2291 - 0.18√0.75 = -0.7287 N(d1) = 0.4112 N(d2) = 0.2314 Therefore, d₁ = [57.50 × e^(-0.04×0.75) × 0.2314] - [55 × 0.4112] = -0.01477

Question 17

What is the value of a 3-month put with a strike price of $45 given the Black-Scholes option pricing model and the following information?

Accepted Answer

P = ($45 F1F1F1S1F1F1F10 2.71828^-0.045F1F1F1S1^F1F1F10^.25) + $9.31 - $52.90 = $0.91

Question 18

Explain how option pricing theory can be used to argue that acquisitive firms pursuing conglomerate mergers are not acting in the shareholders' best interest.

Accepted Answer

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Question 19

The delta of a call option on a firm's assets is 0.727. This means that a $195,000 project will increase the value of equity by:

Accepted Answer

Increase in equity value = $195,000 F1F1F1S1F1F1F10 0.727 = $141,765

Question 20

Explain why the equity ownership of a firm is equivalent to owning a call option on the firm's assets.

Accepted Answer

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Quiz 25: Option Valuation