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

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

Answer:
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

Answer: $21.00

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 ...