Question

In addition to the five factors discussed in the chapter, 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.

  

The put-call parity condition is altered when dividends are paid. The dividend-adjusted put-call parity formula is:

S×e−dt+P=E×e−Rt+CS×e−dt+P=E×e−Rt+C

  

where d is the continuously compounded dividend yield.

   

A stock is currently priced at $84 per share, the standard deviation of its return is 60 percent per year, and the risk-free rate is 5 percent per year, compounded continuously. What is the price of a put option with a strike price of $80 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

As per Black Scholes Model
Value of put option = N(-d2)*K*e^(-r*t)-(S*e^(q*t))*N(-d1)
Where
S = Current price =		84
t = time to expiry =		0.5
K = Strike price =		80
r = Risk free rate =		5.0%
q = Dividend Yield =		3%
σ = Std dev =		60%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(84/80)+(0.05-0.03+0.6^2/2)*0.5)/(0.6*0.5^(1/2))
d1 = 0.350702
d2 = d1-σ*t^(1/2)
d2 =0.350702-0.6*0.5^(1/2)
d2 = -0.073562
N(-d1) = Cumulative standard normal dist. of -d1
N(-d1) =0.362906
N(-d2) = Cumulative standard normal dist. of -d2
N(-d2) =0.529321
Value of put= 0.529321*80*e^(-0.05*0.5)-84*e^(-0.03*0.5)*0.362906
Value of put= 11.27