QuoteOriginally posted by: eiriamjhQuoteOriginally posted by: quantlearner|ds|=|b||dZ| is definitely wrong. The distribution is not half-normal either. The simplest way is to do a numerical simulation and you should be able to find out in no time what you are looking for (dynamics, distribution etc.).I think | dS/S - \mu dt | is a half normal with mean \sigma \sqrt{2dt/\pi}, so I guess |dS/S| must be a half-normal with a displaced mean... But yes I don't think we can ignore the \mu dt term...you can write informally that distribution |dS| is the same as | mu S dt + sigma S z dt^0.5| = dt^0.5 S |z| [ sigma + mu dt^0.5] where z is N( 0, 1) and sigma can also be assumed positive. Hence the term mu*S dt can be ignored and |dS| approximately in the sense of distribution equal to sigma*S |z| dt^0.5. This derivation would be looking more accurately if differentials are replaced by the correspondent finite differences.

QuoteOriginally posted by: listyou can write informally that distribution |dS| is the same as | mu S dt + sigma S z dt^0.5| = dt^0.5 S |z| [ sigma + mu dt^0.5] where z is N( 0, 1) and sigma can also be assumed positive. Hence the term mu*S dt can be ignored and |dS| approximately in the sense of distribution equal to sigma*S |z| dt^0.5. This derivation would be looking more accurately if differentials are replaced by the correspondent finite differences.following your reasoning, we could do the same approximation without absolute values: dS = dt^0.5 S z [ sigma + mu dt^0.5] ~ dt^0.5 sigma S z, but how good is that approximation really? That seems just like saying we can approximate a GBM with drift by a driftless GBM, and we know that's not really true...

QuoteOriginally posted by: eiriamjhQuoteOriginally posted by: listyou can write informally that distribution |dS| is the same as | mu S dt + sigma S z dt^0.5| = dt^0.5 S |z| [ sigma + mu dt^0.5] where z is N( 0, 1) and sigma can also be assumed positive. Hence the term mu*S dt can be ignored and |dS| approximately in the sense of distribution equal to sigma*S |z| dt^0.5. This derivation would be looking more accurately if differentials are replaced by the correspondent finite differences.following your reasoning, we could do the same approximation without absolute values: dS = dt^0.5 S z [ sigma + mu dt^0.5] ~ dt^0.5 sigma S z, but how good is that approximation really? That seems just like saying we can approximate a GBM with drift by a driftless GBM, and we know that's not really true...of course you right. but to make derivation what you wish you need to use mathematically correct form. |d S| does not make math meaning. unfortunately i do not know a formal presentation of the hedging problem where it comes from.

Eiriamjh and Paul, I also have trememdous interest in this issue. Could it be possible to share your original problem and your discussion? Much appreciated.