Question

1. A stock sells for $110. A call option on the stock has 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.

1. 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