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}.

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