**Please can people help with my understanding on the ****Log N model and General Brownian Motion and how** securities are priced using these. Thank You Lots to all the replies .

Log N model for a security S has distribution St~ LogN( Ln(So) + (mu -0.5 sigma^2)*t, sigma^2*t)

and General Brownian Motion (GBM) has St ~ N(mu*t, sigma^2*t).

**Q1) For GBM**

GBM has a Wo that can be Wo >0. So why don't we say St ~ N(Wo + mu*t, sigma^2*t).

**Q2) Differences of the Standard, General and Geometric (Log N model) Brownian Motion**

I understand how we go from Standard Brownian Motion to the General Brownian Motion(GBM).

However, I am confused on how we go from a GBM to a Geometric BM? How are these two related and different? So, do we consider these 3 motions, Standard GM, GBM and Geo. BM, as distinct?

**Q3.1) CT8, Apr 2008**

A security is modeled by a log N model. Is the log return of the security a GBM? I ask because

CT8, Apr 2008, Q8i)a) considers a log N security with parameters mu and sigma. The Log return of the security price is Ln(ST) - Ln(St) ~ N(mu(T-t), sigma^2*(T-t))

Why is the above like a GBM and not like a log N model, say " ~ N((mu - 0.5*Sigma^2)*(T-t), sigma^2*(T-t)) " ?

**Q3.2) ****CT8, Apr 2008**

For Q8i)b) The expected value of an investment at future time is E[ST|Ft] = St*exp((mu + 0.5*sigma^2)(T-t)).

Again, this is Log N model and not the return in Q8i)a)? I am confused how you can switch from answer in 8)i)a) to this answer which uses the Log N models "(mu - 0.5*Sigma^2)" term?

Last edited: Jun 28, 2020