Black and Scholes Model
 Black and Scholes formula allows for infinitesimally small intervals as well as the need to revise leverage for European options on Non Dividend paying stocks
 The formula is:
 Where,
 Log is the natural log with base e
 N (d) = cumulative normal probability density function
 X = exercise price option;
 T = number of periods to exercise date
 P =present price of stock
 Ïƒ = standard deviation per period of (continuously compounded) rate of return on stock
 Value of Put =
Question: Black and Scholes Model
Calculation of the value of call option:
Price of stock now (P) 
85 
Exercise price (EX) 
85 
Standard deviation of continuously compounded annual returns (Ïƒ) 
0.4069 
Year to maturity (t) 
0.5 
Riskfree interest rate per annum, r_{f} 
4% 
Log [P/PV (EX)] 
0.02 
Log [P/PV (EX)] / Ïƒ âˆšt 
0.07 
Ïƒ âˆšt/2 
0.14 
d1 = log [P/PV (EX)] / Ïƒ âˆšt + Ïƒ âˆšt/2 
0.2134 
d2 = d1 â€“ Ïƒ âˆšt 
0.0743 
N(d1) â€“ Can be calculated by using NORMSDIST (d1) in excel 
0.5845 
N(d2) â€“ Can be calculated by using NORMSDIST (d2) in excel 
0.4704 
PV (EX) = 85 * e^{4%/2} 
83.3169 
Value of call 
10.49 
Question: Black and Scholes Model
For European Options on dividend paying stocks, the present value of expected dividends during the life of the option needs to be reduced from the present price of the stock:
Without Dividend 
With Dividend 

Price of stock now 
85 
85 
Present value of dividend 
0 
1.99 
Price of stock adjustment for dividend (P) 
85 
83.01 
Exercise price (EX) 
85 
85 
Standard deviation of continuously compounded annual returns (Ïƒ) 
0.4069 
0.4069 
Year to maturity (t) 
0.5 
0.5 
Riskfree interest rate per annum, r_{f} 
4% 
4% 
Log [P/PV (EX)] 
0.02 
0.004 
Log [P/PV (EX)] / Ïƒ âˆšt 
0.07 
0.01 
Ïƒ âˆšt/2 
0.14 
0.14 
d1 = log [P/PV (EX)] / Ïƒ âˆšt + Ïƒ âˆšt/2 
0.2134 
0.1309 
d2 = d1 â€“ Ïƒ âˆšt 
0.0743 
0.1568 
N(d1) 
0.5845 
0.5521 
N(d2) 
0.4704 
0.704 
PV (EX) = 85 * e^{4%/2} 
83.316 
83.316 
Value of call 
10.49 
9.36 