Possible typo in first passage probability formula?
The first passage probability given by Bunch and Johnson is given as
$$Z = {1 \over \sigma }\left[ {\log {S \over {Sc}} + (r - {1 \over 2}{\sigma ^2})t} \right]$$
after reading some literature should it not be the case that the first passage probability should be
$$Z = {1 \over {\sigma \sqrt t }}\left[ {\log {S \over {Sc}} + (r - {1 \over 2}{\sigma ^2})t} \right]$$
It was said before (3) that "Under the standard assumptions, Z is normal with a zero mean and a variance of t" , ie it looks Z does not normalized by sq root of t. If you look through (3-12) you could not find where Z is used. More interesting is an assumption $$\gamma = \frac{r}{\sigma^2} = 1$$ and the next statement that "Equation (1) therefore requires an expression for the first-passage time of a (standardized) Brownian motion to the “position” a. It was not written but it is very possible that $$dS_t = r S_t dt + \sigma S_t dw ( t )$$
though it is not a fact because this is underlying of the option that guarantees perfect hedging. If perfect hedging does not mention then it might be real stock with drift [$]\mu[$]. Why does assumption [$]\gamma = 1 [$] is sufficient that S is a Brownian process does not clear it looks that under this condition S is log-Brownian. Though, who knows but they have a proof. If this problem is clarified they do not use Z for P at least up to(12)