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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
    • σ = 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 (σ)

0.4069

Year to maturity (t)

0.5

Risk-free interest rate per annum, rf

4%

Log [P/PV (EX)]

0.02

Log [P/PV (EX)] / σ √t

0.07

σ √t/2

0.14

d1 = log [P/PV (EX)] / σ √t + σ √t/2

0.2134

d2 = d1 – σ √t

-0.0743

N(d1) – Can be calculated by using NORMSDIST (d1) in excel

0.5845

N(d2) – 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 (σ)

0.4069

0.4069

Year to maturity (t)

0.5

0.5

Risk-free interest rate per annum, rf

4%

4%

Log [P/PV (EX)]

0.02

-0.004

Log [P/PV (EX)] / σ √t

0.07

-0.01

σ √t/2

0.14

0.14

d1 = log [P/PV (EX)] / σ √t + σ √t/2

0.2134

0.1309

d2 = d1 – σ √t

-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

 





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