Hi again Nicola
Em 5 ? where did that come from its not true sure its not , we can get 5 from 3 by divide across by S zero then taking logs. (and a bit of fooling around with the times t , t zero.)
likewise for 6 can be got from 4 in the same fashion.
So you can in fact always replace mu with r and z with zTilda.
But whats that i hear you ask , why oh why , is the variances both sigma squared dt in 5 and 6 ????
well this is cause we are changing the drift but not the volatility of our Brownian motion when we shift to the EMM q .
So our variance in both cases is the same sigma squared dt
how do we work out this variance
well it comes from the fact that
::EDIT :: integral sigma dBs
is a random variable that is normal distributed with mean zero and variance given by
integral sigma squared dt
this is the variance that appears in 5 and 6.
Hope this is understandable and the correct way to think about it cause I am by no means a pro at this .
Good luck in your exam.
Last edited by a moderator: Apr 25, 2010