The Black-Scholes Formulas
c  S 0 e  qT N (d1 )  K e  rT N (d 2 )
p  K e  rT N (d 2 )  S 0 e  qT N ( d1 )
ln( S 0 / K )  (r  q   2 / 2)T
where d1 
 T
ln( S 0 / K )  (r  q   2 / 2)T
d2 
 d1   T
 T
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The Black-Scholes Formulas
where
N(x) is cumulative probability distribution function for
a standardized normal distribution. It is the
probability that a variable with a standard normal
distribution will be less than x.
= annualized standard deviation (volatility) of the
continuously compounded return on the stock
r = continuously compounded risk-free rate
q = continuously compounded dividend yield
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N(d2) is recognized as the probability under the risk neutral
measure Q that the call expires in-the-money, so Xe¡r¿N(d2)
represents the present value of the risk neutral expectation of
payment paid by the option holder at expiry.
² SN(d1) is the discounted risk neutral expectation of the terminal
asset price conditional on the call being in-the-money at expiry.
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