Question

1. The stock of the McCall Corporation is currently trading at $42 per share. The stock’s volatility as measured by its standard deviation is 20%. If the strike (exercise) price for a certain set of options on McCall stock carry a strike price of $40, and the options run for 6 months (180 days), determine the Black-Scholes model values for:

N (d1), N (d2), N (- d1), and N (- d2). (Assume the risk-free rate is 10% and that the stock pays no dividends.)

2. Given the information in question one (1) above and your calculated values for N (d1), N (d2), N (- d1), and N (- d2), determine: a. The price (premium) one must pay for a call option (Ct) on McCall stock using the Black-Scholes model.

b. How much of the call’s value is composed of intrinsic value? How much of the call’s value is composed of a time premium? (Explain how you determined these values.)

3. Given the information in question one (1) above and your calculated values for N (d1), N (d2), N (- d1), and N (- d2), determine:

a. The price (premium) of a put option (Pt) for McCall stock using the BlackScholes model. (Use the put/call parity model to confirm your answer.)

b. How much of the put’s value (price) is composed of intrinsic value? How much of the put’s value is composed of a time premium? (Explain how you determined these values.)

These are the answers below, how do I get to them?

1. N (d1) = 0.7792 (where d1 = 0.7694) N (d2) = 0.7350 (where d2 = 0.6280) N (- d1) = 0.2208 N (- d2) = 0.2650

2. a. $4.76 (call price) b.

3. a. $0.81 (put price) b.

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 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(42 / 40). We input the same formula into Excel, i.e. =LN (42 / 40)
• (r + σ²/2)*T = (0.10 + (0.20²/2)*0.50
• σ√T = 0.20 * √0.50

d₁ = 0.7693

d₂ = 0.6278

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) = 0.7791

N(d₂) = 0.7349

N(-d₁) = 0.2209

N(-d₂) = 0.2651

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

C = (S₀ * N(d₁))   - (Ke^-rt * N(d₂)), which is (42 * 0.7791) - (40 * e^{(-0.10 * 0.50)})*(0.7349)    ==> $4.76

P = (K * e^-rt)*N(-d₂) - (S₀)*N(-d₁), which is (40 * e^{(-0.10 * 0.50)})*(0.2651) - (42 * (0.2209) ==> $0.81

Value of call option is $4.76

Value of put option is $0.81

Intrinsic value of in-the-money call option = current stock price - strike price = $42 - $40 = $2

time value = option value - intrinsic value = $4.76 - $2 = $2.76

Intrinsic value of out-of-money put option = 0

time value = option value - intrinsic value = $0.81 - $0 = $0.81