Question

In addition to the five factors, dividends also affect the price of an option. The Black–Scholes Option Pricing Model with dividends is:

  

C=S×e−dt×N(d1)−E×e−Rt×N(d2)C=S×e−dt×N(d1)⁢−E×e−Rt×N(d2)

d1=[ln(S/E)+(R−d+σ2/2)×t](σ−t√)d1= [ln(S⁢  /E⁢ ) +(R⁢−d+σ2/2)×t ] (σ−t)

d2=d1−σ×t√d2=d1−σ×t

  

All of the variables are the same as the Black–Scholes model without dividends except for the variable d, which is the continuously compounded dividend yield on the stock.

  

A stock is currently priced at $82 per share, the standard deviation of its return is 48 percent per year, and the risk-free rate is 5 percent per year, compounded continuously. What is the price of a call option with a strike price of $78 and a maturity of six months if the stock has a dividend yield of 3 percent per year? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

  

Answer 1

We use Black-Scholes Model to calculate the value of the call option.

The value of a call option is:

C = (S₀ * e^-qt * N(d₁))   - (Ke^-rt * N(d₂))

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(82 / 78). We input the same formula into Excel, i.e. =LN(82 / 78)
• (r + σ²/2)*t = (0.05 + (0.48²/2)*0.50
• σ√t = 0.48 * √0.50

d₁ = 0.3907

d₂ = 0.0513

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) = 0.6520

N(d₂) = 0.5205

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

C = (S₀ * e^-qt * N(d₁))   - (Ke^-rt * N(d₂)), which is (82 * e^{(-0.03 * 0.50)} * 0.6250) - (78 * e^{(-0.05 * 0.50)} * 0.5205)    ==> $13.07

Value of call option is $13.07