Question

Consider the following call option:

• The current price of the stock on which the call option is written is $32.00;
• The exercise or strike price of the call option is $30.00;
• The maturity of the call option is .25 years;
• The (annualized) variance in the returns of the stock is .16; and
• The risk-free rate of interest is 4 percent.

Use the Black-Scholes option pricing model to estimate the value of the call option.

Answer 1

Given Information

Current Price = S = 32

Strike Price = K = 30

maturity = t = .25 years

Risk free rate = r = 4%

The (annualized) variance in the returns of the stock is .16; and

Standard Deviation = (0.16)^^0.5 = 0.4

Black-Scholes option pricing model =

C SN(d)-Kend2) C Theoretical call premium S Current Stock price t time until option expiratie K opti on striking price rrisk -free interest rate N Cumulative stan dard norm al di stribution e exponential term (2.7183) In(S/K)+(r d2d-s s standard devi ati on of stock returns n natural logari thm

Calculation of variables required (d₁ and d₂)

d₁ = [ln(32/30) + (4% + .16/2)*.25] / [0.4*(.25^^(1/2))]

= [0.0645385211375712 + 0.03] / [0.4*0.5]

= 0.0945385211375712 / 0.2

= 0.472692605687856

d₂ = 0.472692605687856 - [0.4*(.25^^(1/2))]

= 0.472692605687856 - 0.2

= 0.272692605687856

Calculation of value of call option

C = [32 * N(0.472692605687856)] - [30 * e^(-4%*0.25) * N(0.272692605687856)]

C = [32 * 0.681783748622313] - [30 * e^(-4%*0.25) * 0.607455240413025]

C = [21.817079955914] - [30 * 0.990049833749168 * 0.607455240413025]

C = [21.817079955914] - [18.0423287934293]

C = [21.817079955914] - [18.0423287934293]

C = 3.77

Answer : Value of call option = 3.77

Explanation Note :

we can find the standardized normal distribution probability using Microsoft Excel "=NORMSDIST" function and

Natural logarithm using "=LN" function and

Exponential term using "=EXP" function

LN(32/30) 0.064539 =NORMSDIST (0.472692605687856) | =NORMSDIST(0.272692605687856) 0.681784 0.607455 EXP(-4%0.25) 0.99005