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# Black-Scholes, Warrants, Leasing

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1) Black-Scholes: What are the prices of a call option and a put option with the following characteristics?
Stock price = \$38
Exercise price = \$35
Risk -free rate = \$6% per yr compounded continuously
Maturity = 3 months
Standard deviation = 54% per year

2) Warrant Value: A warrant gives its owner the right to purchase three shares of common stock at an exercise price of \$32 per share. The current market price of the stock is \$39. What is the minimum value of the warrant?

3) Use the following information to work problems 1-3. You work for a nuclear research lab. That is contemplating leasing a diagnostic scanner (leasing is a common practice with expensive high-tech equipment). The scanner costs \$3,000,000, and it would be depreciated straight-line to zero over four years. Because of radiation contamination, it will actually be completely valueless in four years. You can lease it for \$895,000 per year for four years.

1: Assume that the tax rate is 35% you can borrow at 8% before taxes should you lease or buy

2: Leasing cash flows: What are the cash flows from the lease from the lessor's viewpoint? Assume a 35 percent tax bracket.

3: Taxes and Leasing cash flows: Assume that your company does not contemplate paying taxes for the next several years. What are the cash flows from leasing in this case?

4) Setting the lease price: Raymond Rayon corp. wants to expand its manufacturing facilities. Liberty leasing corp. has offered Raymond Rayon the opportunity to lease a machine for \$1,500,000 for six years. The machine will be fully depreciated by the straight -line method. The corporate tax rate for Raymond Rayon is 25 percent, whereas Liberty leasing has a corporate tax rate of 40 percent. Both companies can borrow at 8 percent. Assume lease payments occur at year -end. What IS Raymond's reservation price? What is Liberty's reservation price?

5) Ch 22-2: Understanding Option Quotes: Use the option quote information shown here to answer the questions that follow. The stock is currently selling for \$83.
Option
& NY close Exp Strike price Calls Puts
vol last vol last

March 80 230 2.80 160 0.80

April 80 170 6 127 1.40

July 80 139 8.05 43 3.90

October 80 60 10.20 11 3.65

a) Are the call options in the money? What is the intrinsic value of an RWJ Corp. call option?

b) Are the Put options in the money? What is the intrinsic value of an RWJ Corp Put option?

Two of the options are clearly mispriced. Which ones? At a minimum. What should the mispriced options sell for? Explain how you could profit from the mispricing in each case

#### Solution Preview

1) Black -Scholes : What are the prices of a call option and a put option with the following characteristics
Stock price = \$38
Exercise price = \$35
Risk -free rate = \$6% per yr compounded continuously
Maturity = 3 months
Standard deviation = 54% per year

We will use Black Scholes Pricing Formula
Value of call= S N(d1) - X * e -r(T-t) * N(d2)
Value of put = X * e -r(T-t) * N(-d2) -S N(-d1)
We therefore need to calculate the values of d1 and d2
d1= {ln (S/X) + ( r + ½ s2 ) x (T-t)}/ (s x square root of (T-t))
d2= {ln (S/X) + ( r - ½ s2 ) x (T-t)}/ (s x square root of (T-t)) =d1-s x square root of (T-t)

Inputs

Stock Price= S= \$38.00
Exercise price = X= \$35.00
Time to expiration = T-t= 3 months = 0.25 year
Risk free rate = r = 6.00%
standard deviation = volatility= s = 54.00%

Calculations

ln S/X= 0.0822 = ln (38/35)
( r + ½ s2 ) (T-t)= 0.0515 =(0.06+0.54^2/2) x 0.25
s x square root of (T-t)= 0.2700 =0.54 x square root of 0.25

Substituting the values
d1= 0.4952 =(0.0822+0.0515) / 0.27
d2= 0.2252 =0.4952-0.27
N is the cumulative normal distribution function
N(d1) 0.6898
N(-d1)=1-N(d1)= 0.3102
N(d2) 0.5891
N(-d2)=1-N(d2)= 0.4109
X * e -r(T-t) = 34.4789 =35x e ^ (-0.06x0.25)

Thus,
Value of call= S N(d1) - X * e -r(T-t) * N(d2)= \$5.9009 =38 x 0.6898 - 34.4789 x 0.5891
Value of put = X * e -r(T-t) * N(-d2) -S N(-d1) = \$2.3798 =34.4789 x 0.4109 - 38 x 0.3102
We can also calculate the value of Put option using put call parity
Put call parity
c+ Xe^-(rt) = p+S
or p=c + Xe^-(rt)-Se^-(q(T-t))= \$2.3798 =5.9009+34.4789-38
which is the same as obtained above

Value of call option= \$5.9009
Value of put option= \$2.3798

2) Warrant Value: A warrant gives its owner the right to purchase three shares of common stock at an exercise price of \$32 per share. The current market price of the stock is \$39. What is the minimum value of the warrant?

Witheach warrant 3 shares can be purchased for 3 x \$32= \$96.00
and sold for 3 x \$39 in the market= \$117.00

Therefore, minimum value = \$117-\$96= \$21.00
This is the minimum value (intrinsic value) as the warrant also has time value before expiration

3)Using the following information to work problems 1-3. You work for a nuclear research lab. That is contemplating leasing a diagnostic scanner (leasing is a common practice with expensive high-tech equipment). The scanner costs \$3,000,000, and it would be depreciated straight -line to zero over four years. Because of radiation contamination, it will actually be completely valueless in four years. You can lease it for \$895,000 per year for four years.

1 :Assume that the tax rate is 35 % you can borrow at 8% before taxes should you lease or buy

Tax rate= 35%
Cost of borrowing= 8%
After tax cost of borrowing = cost of borrowing x (1-Tax rate)= 5.20% =8.% x (1- 0.35)
This is the ...

#### Solution Summary

Answers questions on option pricing using Black-Scholes, warrants, leasing, understanding option quotes.

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## Option Pricing, Warrants, Lease or Buy, Duration of Bond

1. GS, Inc. stock is selling for \$28 a share. A 3-month call on GS stock with a strike price of \$30 is priced at \$1.50. Risk-free assets are currently returning 0.3% per month. What is the price of a 3-month put on GS stock with a strike price of \$30?

2. A firm has 100 shares of stock and 40 warrants outstanding. The warrants are about to expire, and all of them will be exercised. The market value of the firm's assets is \$2,000, and the firm has no debt. Each warrant gives the owner the right to buy 2 shares at \$15 per share. What is the price per share of the stock?

3. What is the value of a 9-month call with a strike price of \$45 given the Black-Scholes Option Pricing Model and the following information?
Stock price \$48
Exercise price \$45
Time to expiration .75
Risk-free rate .05
N(d1) .718891
N(d2) .641713

4. Calculate the duration of a 4-year \$1,000 face value bond, which pays 8% coupons annually throughout maturity and has a yield to maturity of 9%.

5. Turner, Inc. has \$4.2 million in net working capital. The firm has fixed assets with a book value of \$48.6 million and a market value of \$53.4 million. Martin & Sons is buying Turner, Inc. for \$60 million in cash. The acquisition will be recorded using the purchase accounting method. What is the amount of goodwill that Martin & Sons will record on its balance sheet as a result of this acquisition?

6. The Parent CO. has decided to acquire a new electronic milling machine. Parent Co. can purchase the machine for \$87,000 which has an expected life of 8 years and will be depreciated using 7 class MACRS rates of .1428, .2449, .1749, .125, .0892, .0892, .0892 and any remainder in year 8. Miles Leasing has offered to lease the machine to Parent Co. for \$14,000 a year for 8 years. Parent Co. has an 18.64% cost of equity, 12% cost of debt, a 1:1 D/E ratio and faces a 34% marginal tax rate. Should they lease or buy?

7. A stock is currently priced at \$24 per share, the standard deviation of its return is 60% a year, and the risk free rate is 4 percent per year. The firm pays \$0.30 quarterly (\$1.20 a year) dividend per share. What is the price of a call option with a strike price of \$25 and a maturity of three months?

8. From the previous question, what is the price of a put option with the same exercise price and time to expiration as the call option?

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