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.
  
A stock is currently priced at $81 per share, the standard deviation of its return is 50 percent per year, and the risk-free rate is 4 percent per year, compounded continuously. What is the price of a call option with a strike price of $77 and a maturity of six months if the stock has a dividend yield of 2 percent per year? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)
  
Price of call option  

Answer 1

S' = Stock price =	81
D = Dividend yield =	4.00%
S = Stock price ajusted = S'*exp(D*T) = 81*exp(-2%*0.5) =	80.194037
K = Strike price =	77
r = rate =	4%
e = exponential value = exp(.) =	2.71828183
t = time =	0.5
s = standard deviation or volatility =	50%
* N(d1) is Normal distribution probability value
* N(d2) is Normal distribution probability value
Use normal distribution table

d1 = (Ln(S/(K*exp(-r*t))+0.5*s^2*t)/(s*t^0.5)                                                                            

=(LN(80.194037/((77*EXP(-0.04*0.5))))+0.5*50%^2*0.5)/(50%*0.5^0.5)                                                                       

d1 =       0.348303            Hence, N(d1) =0.6361937                       

                                                                                     

d2 = d1 - (s*t^0.5)                                                                                  

d2 = 0.348303-(50%*0.5^0.5)                                                                            

d2 =       -0.005250391    Hence, N(d2) =0.497905407                   

                                                                                     

C = S*N(d1)-K*exp(-r*t)*N(d2)                                                                         

=80.194037*0.6361937-77*exp(-4%*0.5)*0.497905                                                                               

C = 13.44                                                                                                                                                           

Value of call option = 13.44