For,
Note: In many resources, you can find different symbols for some of these parameters.
For example,
K
(here it is X
).S
(without the zero)T – t
(difference between expiration and now).In the original Black and Scholes paper (The Pricing of Options and Corporate Liabilities, 1973) the parameters were denoted x (underlying price), c (strike price), v (volatility), r (interest rate), and t* – t (time to expiration).
The dividend yield was only added by Merton in Theory of Rational Option Pricing, 1973.
Call option C
and put option P
prices are calculated using the following formulas:
where N(x)
is the standard normal cumulative distribution function.
The formulas for d1
and d2
are:
In the original Black-Scholes model, which doesn’t account for dividends, the equations are the same as above except:
S0
in place of S0 e-qt
q
in the formula for d1
Therefore, if the dividend yield is zero, then e-qt = 1
and the models are identical.
… where T is the number of days per year (calendar or trading days, depending on what you are using).
B1
B2
B3
B4
B5
B6
B7
B8
d1, d2 Calculation
(LN(B1/B2)+(B6-B7+0.5*B5^2)*B8)/(B5*SQRT(B8))
EXP(-(B10^2)/2)/SQRT(2*PI())
B10-B5*SQRT(B8)
NORMSDIST(B12)
Calculation Of Greeks
If You see the above formulas, these are derived directly from those formulas –
(-((B1*B5*EXP(-B7*B8))/(2*SQRT(B8))*(1/(SQRT(2*PI())))*EXP(-(B10*B10)/2))-(B6*B2*EXP(-B6*B8)*NORMSDIST(B12))+(B7*EXP(-B7*B8)*B1*NORMSDIST(B10)))/365
(-((B1*B5*EXP(-B7*B8))/(2*SQRT(B8))*(1/(SQRT(2*PI())))*EXP(-(B10*B10)/2))+(B6*B2*EXP(-B6*B8)*NORMSDIST(-B12))-(B7*EXP(-B7*B8)*B1*NORMSDIST(-B10)))/365
EXP(-B7*B8)*B1*NORMSDIST(B10)-B2*EXP(-B6*B8)*NORMSDIST(B10-B5*SQRT(B8))
B2*EXP(-B6*B8)*NORMSDIST(-B12)-EXP(-B7*B8)*B1*NORMSDIST(-B10)
EXP(-B7*B8)*NORMSDIST(B10)
EXP(-B7*B8)*(NORMSDIST(B10)-1)
(EXP(-B6*B8)/(B1*B5*SQRT(B8)))*(1/(SQRT(2*PI())))*EXP(-(B10*B10)/2)
(EXP(-B6*B8)/(B1*B5*SQRT(B8)))*(1/(SQRT(2*PI())))*EXP(-(B10*B10)/2)
(1/100)*B2*B8*EXP(-B6*B8)*NORMSDIST(B12)
(-1/100)*B2*B8*EXP(-B6*B8)*NORMSDIST(-B12)