I have been struggling with understanding of the Black-Scholes-Merton model and I have a few confusions that I hope you can kindly enlighten.

1) for d1:

after studying your video, I seem to understand that d1 means "future expected return IN(S/K) divided by future volatility (sigma*square root(t))" and also plus "half of one future volatility(sigma*square root(t))". Assuming it follows a normal distribution, graphically, it represents how far away it sits to the right hand side of the mean (since the "plus" sign means that we are kind of trying to find out how far upward the return can go).

if my understanding is not wrong, why "half" of the future volatility is added, not 0.4, 0.8 or 1? stock return can be higher, right?

2) for d2:

I'm thinking that if my explanation for d1 as above is correct, likewise d2 graphically represents how far away it sits to the left hand side of the mean (since the "minus" sign means that we are kind of trying to find out how far downward the return can go). Then I have the same confusion as for d1: why "half" of the future volatility is added, not 0.4, 0.8 or 1? stock return can be higher, right?

3) in the video-cast titled" Importance of d2 in Black-Scholes to Merton Model in Credit Risk – 10 min screencast" (page (http://www.bionicturtle.com/learn/article/importance_of_d2_in_black_scholes_to_merton_model_in_credit_risk_10_min_scr/ ), the formula you provided for d2 considers In(S/K). I've checked with Hull (page 291)and FRM handbook(5th) (page 144), and found out that Hull's formula is consistent with yours but the FRM formula uses a slight different item, that is "In{S/K*exp(-rt)}". I'm not sure if FRM handbook is making another mistake again.

4) What's more, how can I relate d1 to delta?

Sorry about so many stupid questions. Thank you very much!

Cheers!

Liming

3/11/2009