View Full Version : Generic Lognormal question (MFE)

April 29th 2007, 03:33 AM
Can anyone explain why the BS formula uses (r - d + 1/2 sigma ^2) instead of (r - d - 1/2 sigma^2) ?

McDonald's text for MFE includes a development of this topic, but it is VERY confusing about this point.

For example, on page 599 he uses the expression

'Prob( St < K ) = N (-d2) <-- 18.24
where d2 is the standard Black-Scholes argument (see equation (12.1)) with the risk-free rate, r, replace with the actual expected return on the stock, a <alpha>.'

<note: in the text, the d has a hat>

However, the equation that McDonald references is NOT the d2 in the BS formula, which has "+ .5 sigma ^2", it has the "- .5 sigma ^2" term.

On page 604, the "+ .5 sigma ^2" term seems to magically appear just prior to explicit treatment of BS formula.

Please help, I'm confused! The only thing I can figure is that the BS formula decides alpha = r + sigma^2 , but ???

I don't think chap 18 is on the syllabus, but this certainly underlies MFE material.

May 10th 2007, 10:08 AM
I can't say I have reviewed it enough to understand the derivation of the black scholes formula, but it looks like the +.5 sigma^2 comes from solving the black scholes PDE.

There's a discussion on pg 597 that explians why we have -.5 sigma^2 in the expected value of the stock price.

I know I didn't really answer your question, but hopefully this helps a little.

May 10th 2007, 10:12 PM
I'm just going to let it go...maybe at some point I'll re-investigate...I'm at peace with the situation now. Thanks for the help.