Question

A stock sells for $110.A call option on the stock has an exercise price of $105 and expires in 43 days.Assume that the interest rate is 0.11 and the standard deviation of the stock's return is 0.25

Required:

What is the call price according to the Black Scholes model?

Answer 1

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

The value of a call and put option are:

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 maturity 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(110 / 105). We input the same formula into Excel, i.e. =LN (110 / 105)
• (r + σ²/2)*T = (0.11 + (0.25²/2)*(43 / 365)
• σ√T = 0.48 * √(43 / 365)

d₁ = 0.7361

d₂ = 0.6503

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) = 0.7692

N(d₂) = 0.7422

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

C = (S₀ * N(d₁))   - (Ke^-rt * N(d₂)),

which is (110 * 0.7692) - (105 * e^{(-0.11 * (43 / 365))})*(0.7422)    ==> $7.68

Value of call option is $7.68