Log Normal random walk question

sam · March 21st, 2002, 2:59 pm

Hi, I would very much appreciate some help on this 'brain teaser' that has been making my last hour or so very difficult...

If an asset is log-normally distributed then what distribution do the increments follow?? E.g. if you are at time 0, and interested in the distribution of S(t+1) - S(t) where
dS = u*S*dt + sigma*S*dt

thanks

Sam

Keanu · March 21st, 2002, 4:02 pm

You surely mean:

dS = u*S*dt+sigma*S*dWt

S(t) = S(0)* exp((u-sigma^2/2)*t + sigma W(t))
S(t+1) = S(t)* exp((u-sigma^2/2)*1 + sigma (W(t+1)-W(t))

So S(t+1)-S(t):

S(0)* exp((u-sigma^2/2)*t + sigma W(t))[exp((u-sigma^2/2)*1 + sigma (W(t+1)-W(t)) - 1]

Let X,Y iid N(0,1), and notice that W(t+1)-W(t) is
independent from W(t) then:

S(0)* exp((u-sigma^2/2)*t + sigma sqrt(t) X)* [exp((u-sigma^2/2)*1 + sigma*1*Y) - 1]

I think there is no "elegant" way to express it.

Keanu