Question

What is the price of a European call option on a non- dividend-paying stock when the stock price is $51, the strike price is $50, the risk-free interest rate is 10% per annum, the volatility is 30% per annum, and the time to maturity is three months?

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

First, we calculate d₁ and d₂ as below :

• ln(S₀ / K) = ln(51 / 50). We input the same formula into Excel, i.e. =LN (51 / 50)
• (r + σ²/2)*T = (0.1 + (0.30²/2)*0.25
• σ√T = 0.30 * √0.25

d₁ = 0.3737

d₂ = 0.2237

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) = 0.6457

N(d₂) = 0.5885

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

C = (S₀ * N(d₁))   - (Ke^-rT * N(d₂)), which is (51 * 0.6457) - (50 * e^{(-0.10 * 0.25)})*(0.5885)    ==> $4.2313

Value of call option is $4.2313