Question

Given: a stock price of $70; an exercise price of $70; 70 days until the expiration of the option; a risk free interest rate of 6%; the annualized volatility of 0.3242. No dividends will be paid before option expires.

Compute the value of the call option.

Use the Black-Scholes Option Pricing Model for the following problem

Answer 1

given annulaized volatility = 0.3242

Annualized Volatality = Standard Deviation * √252

0.3242 = SD * 15.87

SD = 0.3242/15.87

SD = 0.02 or 2%

Variance = (SD^2) = 4%

Particulars	Values
Stock Price or Spot Price	$                                70.00
Strike Price or Exercise Price	$                                70.00
Variance	0.04
Risk free Rate	6.00%
Time period in Years	(70/365) 0.19

Step1:
Ln (S / X )
S - Stock Price
X - Exercise Price
= Ln ( 1 )
= 0

Step2:
d1 ={ [ Ln (S/X) + [ [ ( SD^2 / 2 ) + rf ] * t ] } / [ SD * SQRT ( T ) ]
S - Stock Price
X - Exercise Price
Rf - Risk free Rate per anum
T - Time in Years
= { [ 0 + [ [ ( 0.04 / 2 ) + 0.06 ] * 0.1918 ] } / [ 0.2 * SQRT ( 0.1918 ) ]
= { [ 0 + [ [ 0.02 + 0.06 ] * 0.1918 ] } / [ 0.2 * ( 0.4379 ) ]
= { 0 + [ 0.08 * 0.1918 ] } / [ 0.0876 ]
= { 0 + 0.0153 } / [ 0.0876 ]
= 0.0153 / 0.0876
= 0.1747

Step3 :
d2 = d1 - [ SD * SQRT ( T ) ]
= 0.1747 - [ 0.2 * SQRT ( 0.1918 ) ]
= 0.1747 - [ 0.2 * 0.4379 ]
= 0.1747 - 0.0876
= 0.0871

Step 4 :
NT( d1) = 0.0675
0.0675

Step 5:
NT (d2) = 0.0319
0.03188

Step 6 :
N(d1) = 0.5 + NT(d1)
= 0.5 + 0.0675
= 0.5675

Step7:
N(d2) = 0.5 + NT(d2)
= 0.5 + 0.0319
= 0.5319

Step 8:
e-rt :
= e^-0.06*0.1918
= e^-0.0115
= 0.9886

Step 9:
Value of Call = [ S * N( d1 ) ] - [ X * e^-rt * N ( d2 ) ]
= [ $ 70 * 0.5675 ] - [ $ 70 * 0.9886 * 0.5319 ]
= [ $ 39.7243 ] - [ $ 36.8072 ]
= $ 2.92
Value of the call using the Black scholes model is 2.92. Please let me know if any assistance is required.