# Black function

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 ${Bl}$ is given by

$\displaystyle Bl(K, S, \nu, w) = \mathbb E [(w(Se^Y - K))^+],$

where ${Y\sim \mathcal{N}(-\nu^2/2, \nu^2)}$ is a normal random variable for some ${\nu>0}$ and ${w}$ is either ${+1}$ or ${-1}$.

R-code is written according to the following fact:

Proposition 2 Let ${\Phi(\cdot)}$ be c.d.f. of standard normal distribution. Then,

$\displaystyle Bl(K, S, \nu, w) = w S \Phi(w d_{1}) - K w \Phi(wd_{2}),$

where

$\displaystyle d_{1} = \frac{\frac 1 2 \nu^{2} - \ln \frac K S}{\nu}, \quad d_{2} = \frac{-\frac 1 2 \nu^{2} - \ln \frac K S}{\nu}.$