Question

A stock trades for ​$47 per share. A call option on that stock has a strike price of ​$53 and an expiration date six months in the future. The volatility of the​ stock's returns is 32​%, and the​ risk-free rate is 5​%. What is the Black and Scholes value of this​ option? The Black and Scholes value of this call option is ​$ ________. ​(Round to the nearest​ cent.)

Answer 1

We use Black-Scholes Model to calculate the value of the call and put options.

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(47 / 53). We input the same formula into Excel, i.e. =LN(47/53)
• (r + σ²/2)*T = (0.05 + (0.32²/2)*0.50
• σ√T = 0.32 * √0.50

d₁ = =0.3073

d₂ = -0.5336

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

N(d₁) = 0.3793

N(d₂) = 0.2968

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

C = (S₀ * N(d₁))   - (Ke^-rT * N(d₂)), which is (47 * 0.3793) - (53 * e^{(-0.05 * 0.50)})*(0.2968)    ==> $2.4845

Value of call option is $2.4845