Black function is useful to compute prices given by Black-Scholes model.
This function is available in R-code (here). See also its upper level page (here).
Definition 1 A black function
is given by
where
is a normal random variable for some
and
is either
or
.
![\displaystyle Bl(K, S, \nu, w) = \mathbb E [(w(Se^Y - K))^+],](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+Bl%28K%2C+S%2C+%5Cnu%2C+w%29+%3D+%5Cmathbb+E+%5B%28w%28Se%5EY+-+K%29%29%5E%2B%5D%2C&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle Bl(K, S, \nu, w) = \mathbb E [(w(Se^Y - K))^+],")
R-code is written according to the following fact:
Proposition 2 Let
be c.d.f. of standard normal distribution. Then,
where
