Question

For 3-month 55-strike European options on a stock, you are given:

(1) The stock's price follows the Black-Scholes framework.

(2) The stock's price is 52.

(3) The stock's volatility is 0.5.

(5)The stock's continuous dividend rate is 3%.

(6) The continuously compounded risk-free interest rate is 7%.

Calculate the premiums for call and put options.

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 * e^-qt * N(d₁))   - (Ke^-rt * N(d₂))

P = (K * e^-rt * N(-d₂)) -    (S₀ * e^-qt * N(-d₁))

where :

S₀ = current spot price

K = strike price

N(x) is the cumulative normal distribution function

q = dividend yield

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(52 / 55). We input the same formula into Excel, i.e. =LN(52 / 55)
• (r + σ²/2)*t = (0.07 + (0.50²/2)*0.25
• σ√t = 0.50* √0.25

d₁ = -0.0294

d₂ = -0.2794

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) = 0.4883

N(d₂) = 0.3900

N(-d₁) = 0.5117

N(-d₂) = 0.6110

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

C = (S₀ * e^-qt * N(d₁))   - (Ke^-rt * N(d₂)), which is (52 * e^{(-0.03 * 0.25)} * 0.4883) - (55 * e^{(-0.07 * 0.25)} * 0.3900)    ==> $4.1243

P = (K * e^-rt * N(-d₂))   -    (S₀ * e^-qt * N(-d₁)), which is (55 * e^{(-0.07 * 0.25)} * 0.6100) - (52 * e^{(-0.03 * 0.25)} * 0.5117) ==> $6.5587

Call premium = $4.1243

Put premium = $6.5587

Value of call option is $6.1987

Value of put option is $6.5134