Question

The stock price of a company is currently $75 per share. A call option on the company’s stock has an exercise price of $80 and six months to expiration. The continuous riskfree rate is 5% per year and the stock's volatility is 28% per year.

A.) Use the Black-Scholes formula to find the value of the call option.

B.) Calculate the hedge ratio for the call option.

Answer 1

a]

We use Black-Scholes Model to calculate the value of the call 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 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(75 / 80). We input the same formula into Excel, i.e. =LN (75 / 80)
• (r + σ²/2)*T = (0.05 + (0.28²/2)*0.50
• σ√T = 0.28 * √0.50

d₁ = -0.1007

d₂ = -0.2987

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) = 0.4599

N(d₂) = 0.3826

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

C = (S₀ * N(d₁))   - (Ke^-rt * N(d₂)), which is (75 * 0.4599) - (80 * e^{(-0.05 * 0.50)})*(0.3826)    ==> $4.6407

Value of call option is $4.6407

b]

Hedge ratio = 1 / delta of call option

delta of call option = N(d₁)

Hedge ratio = 1 / 0.4599 = 2.17