Question

Use the Black-Scholes formula for the following stock:

Time to expiration	6 months
Standard deviation	41% per year
Exercise price	$42
Stock price	$42
Annual interest rate	7%
Dividend	0

Recalculate the value of the call with the following changes:

a.	Time to expiration	3 months
b.	Standard deviation	20% per year
c.	Exercise price	$50
d.	Stock price	$50
e.	Interest rate	9%

Calculate each scenario independently. (Round your answers to 2 decimal places.)

C rises or falls for a, b c d e ?

Value of the call option for a b c d e ?

Answer 1

Solution A>

The value of the option is $5.51. All the formula used are shown in the below excel.

Type of Option	Call Option	Formulas
Stock Price (S0)	$          42.00
Exercise (Strike) Price (K)	$          42.00
Time to Maturity (in years) (t)	0.50
Annual Risk-Free Rate (r)	7.00%
Annualized Volatility (σ)	41.00%
Option Price	$            5.51	=$42*0.605-$42*0.96561*0.490
Additional Calculation Parameters
ln(S0/K)	-
(r+σ2/2)t	0.077	=(7%+(41%^2)/2)*0.5
σ√t	0.290	=41%*SQRT(0.5)
d1	0.266	=0+0.077/0.290
d2	(0.024)	=0.266-0.290
N(d1)	0.605	=NORM.S.DIST( 0.266)
N(d2)	0.490	=NORM.S.DIST(-0.024)
N(-d1)	0.395	=NORM.S.DIST(- 0.266)
N(-d2)	0.510	=NORM.S.DIST(-0.024)
e-rt	0.96561	=EXP(-7%*0.5)

Solution B>

The value of the call falls in Scenario 2.

The value of the option is $2.58. All the formula used are shown in the below excel.

Type of Option	Call Option
Stock Price (S0)	$          50.00
Exercise (Strike) Price (K)	$          50.00
Time to Maturity (in years) (t)	0.25
Annual Risk-Free Rate (r)	9.00%
Annualized Volatility (σ)	20.00%
Option Price	$            2.58	=$50*0.608-$50*0.97775*0.569
Additional Calculation Parameters
ln(S0/K)	-
(r+σ2/2)t	0.028	=(9%+(20%^2)/2)*0.25
σ√t	0.100	=20%*SQRT(0.25)
d1	0.275	=0+0.028/0.100
d2	0.175	=0.275-0.100
N(d1)	0.608	=NORM.S.DIST(0.275)
N(d2)	0.569	=NORM.S.DIST(0.175)
N(-d1)	0.392	=NORM.S.DIST(-0.275)
N(-d2)	0.431	=NORM.S.DIST(-0.175)
e-rt	0.97775	=EXP(-9%*0.25)

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