Dear all,if I wanted to derive the characteristic function (CF) for the distribution of a stock following the GBM (or in other words, I guess one could say the CF for the BS model), how would that be? can you point me to a source where i find the derivation or just show the crucial steps?Thanks to the search function of the forum, I found a representation of the BS CF in http://ssrn.com/abstract=921336 , page 5, equation no. 26. - but, again, I don't know how they come up with that.I apologize if the question is so trivial that it isn't worth posing in some people's opinion but I'm a beginner with CFs, so I have no choice but asking .Thank you in advance for any useful comment.Steve06

Look for Characterstic Function of Normal distribution.Brownian Motion follws Normal distribution.

Look for Characterstic Function of Normal distribution.Brownian Motion follws Normal distribution.

Thank you for the comment.It confuses me a bit. Isn't it the case that the Geometric BM is lognormally distributed? And if it is, I read on wikipedia that there doesn't exist an explicit CF for the Lognormal density function.

The paper you refer to deals with the value of options such as max(exp(X) - K,0) where the characteristic function of X is known.In the case of the BS model, X is an arithmetic Brownian motion, and exp(X) is a geometric Brownian motion. Hence, we need toknow the characteristic function of an arithmetic Brownian motion, or of a normal random variable. You find this by calculatingE[exp(i*u*X)] where X is a normal random variable.An explicit CF of a lognormal random variable does indeed not exist.