Question

Assume you are given the following information for a European call option written on the common stock of a major corporation:

P = $18, X = $18, t = 4 months, risk free interest rate = 4.6%, variance of the rate of return on the stock = 0.25.

Compute the value of this option using the Black-Scholes Option Pricing Model.

Answer 1

We use Black-Scholes Model to calculate the value of the call option

The value of a call option is:

C = (S₀ * N(d₁)) - (Ke^-rT * N(d₂))

where :

S₀ = current spot price

K = strike price

N(x) is the cumulative normal distribution function

r = risk-free interest rate

T is the time to expiry in years

d₁ = (ln(S₀ / K) + (r + σ²/2)*T) / σ√T

d₂ = d₁ - σ√T

σ = standard deviation of underlying stock returns. Standard deviation = variance =  0.25 = 0.50

First, we calculate d₁ and d₂ as below :

• ln(S₀ / K) = ln(18 / 18). We input the same formula into Excel, i.e. =LN(18/18)
• (r + σ²/2)*T = (0.046 + (0.50²/2)*(4/12)
• σ√T = 0.50 * √(4/12)

d₁ = 0.1975

d₂ = -0.0912

N(d₁) and N(d₂) are calculated in Excel using the NORMSDIST function and inputting the value of d₁ and d₂ into the function.

N(d₁) = 0.5783

N(d₂) = 0.4637

Now, we calculate the values of the call option as below:

C = (S₀ * N(d₁))   - (Ke^-rT * N(d₂)), which is (18 * 0.5783) - (18 * e^{(-0.046 * (4/12))})*(0.4637)    ==> $2.1899

Value of call option is $2.1899