Question

Use the Black–Scholes formula to value the following option: A call option written on

    a stock selling for $60 per share with a $60 exercise price. The stock's standard

    deviation is 6% per month. The option matures in three months. The risk-free

     interest rate is 1% per month.

              What is the value of a put option written on the same stock at the same

           time, with the same exercise price and expiration date.

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))

P = (K * e^-rT)*N(-d₂) - (S₀)*N(-d₁)

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(60 / 60). We input the same formula into Excel, i.e. =LN(60/60)
• (r + σ²/2)*T = (0.01 + (0.06²/2)*0.25
• σ√T = 0.06 * √0.25

d₁ = 0.0983

d₂ = 0.0683

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) = 0.5392

N(d₂) = 0.5272

N(-d₁) = 0.4608

N(-d₂) = 0.4728

Now, we calculate the values of the call and put options as below:

C = (S₀ * N(d₁))   - (Ke^-rT * N(d₂)), which is (60 * 0.5392) - (60 * e^{(-0.01 * 0.25)})*(0.5272)    ==> $0.7946

P = (K * e^-rT)*N(-d₂) - (S₀)*N(-d₁), which is (60 * e^{(-0.01 * 0.25)})*(0.4728) - (60 * (0.4608) ==> $0.6448

Value of call option is $0.7946

Value of put option is $0.6448