## Black-scholes equation

Hi

Ive been looking at this article http://www.physics.uci.edu/~silverma/bseqn/bs.pdf

and im having issues with simplifying the intergral (equation (20) in the article)

firstly is equation (21) correct, is there an extra 1/(sqrt(4pit') that shouldnt being there

also when completing the square to i get

$\displaystyle $y(z,t^{'}) = \frac{1}{\sqrt{2 \pi }}\int^{\infty}_{\frac{-z}{\sqrt{2 t_{'}}}} E exp\left[ \frac{\frac{1}{2}\sigma^2z}{{\mu} - \frac{1}{2}\sigma^2} - \left[q- \frac{\frac{1}{4}\sigma^2 \sqrt{2t^{'}}}{{\mu} - \frac{1}{2}\sigma^2}\right]^2 + \frac{1}{2}\left[\frac{\frac{1}{2}\sigma^2}{\mu-\frac{1}{2}\sigma^2}\right]^2 t^{'}\right]dx - EN(d_{2})$$

notation differnt to article for r = \mu , v= \sigma , c=E

and so there is a factor of a half that screws everything up . . .??? is this wrong???

thanks for anyone that tackles this cheers