Question

Problem 17-30 Black–Scholes and Dividends

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

Answer ) in the current concept , the price of option contract has been calculated by use of Black-Scholes model for eurpean call valuation with dividend yield of stock .

Input data ,

Stock Price now (S)		81
Exercise Price of Option (E)		77
Number of periods to Exercise in years (t)		0.50
Compounded Risk-Free Interest Rate (r)		4.00%
Standard Deviation (annualized s)		50.00%
Devidend yield (d)		2%

The output data with spread sheet formula

Output Data		Spread sheet Formula
Present Value of Exercise Price (PV(K))		E*EXP(-r*t)	75.47529784
s*t^.5		s*t^0.5	0.353553391
d1		(LN(s/e)+(r-d+s*s/2)*t)/(s*t^0.5)	0.348303074
d2		d1-s*t^0.5	-0.005250316
Delta N(d1) Normal Cumulative Density Function		NORMDIST(d1,0,1,TRUE)	0.636193707
N(d2)*PV(K)		NORMDIST(d2,0,1,TRUE)*E*EXP(-r*t)	37.57956111
Value of Call		NORMDIST(d1,0,1,TRUE)*S- NORMDIST(d2,0,1,TRUE)*E*EXP(-r*t)	$13.95