Question

Use the Black-Scholes formula for the following stock:

Time to expiration 6 months

Standard deviation 44%

per year Exercise price $46

Stock price $45

Annual interest rate 4%

Dividend 0

Calculate the value of a call option.

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(45 / 46). We input the same formula into Excel, i.e. = LN(45/46)
• (r + σ²/2)*T = (0.04 + (0.44²/2)*0.50
• σ√T = 0.44 * √0.50

d₁ = 0.1492

d₂ = -0.1619

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) = 0.5593

N(d₂) = 0.4357

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

C = (S₀ * N(d₁))   - (Ke^-rt * N(d₂)), which is (45 * 0.5593) - (46 * e^{(-0.04 * 0.50)})*(0.4357)    ==> $5.5241

Value of call option is $5.5241