Log N model and GBM

Discussion in 'CM2' started by N_Exam, Jun 26, 2020.

  1. N_Exam

    N_Exam Keen member

    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
  2. Joe Hook

    Joe Hook ActEd Tutor Staff Member

    Hi,

    Q2 - Geometric brownian motion is simply general brownian motion raised as an exponential. General brownian motion may have a non-zero starting value and a positive drift but it still has the potential, as a process, to become negative. One way of avoiding negatives is to use exponentials. Even if the value taken from a general brownian motion was negative it would be positive if we took the exponential of it. Hence, when it comes to trying to model share prices which are positive, geometric brownian motion is the best that we can do (at least in CM2).


    Q1 - We primarily work with changes in the value of a process in CM2 so are looking for expressions that look like dXt. It is the change in standard/general brownian motion that is normally distributed. Hence your expression above St ~ N(mu*t, sigma^2*t) is implicitly St - S0 where S0 = 0. If, as you are suggesting, you had a positive initial value for the share price then you would add this initial value to the mean of the distribution. In that case you would be correct that the distribution would be St St ~ N(S0+mu*t, sigma^2*t).


    Q3/4 - For further explanation as to what's going on here I suggest you take a look at the definition of the continuous time lognormal model in chapter 11 of the course notes. The form of this model is that shown in part (i) to the question. This is subtly different to the result you get when you solve geometric Brownian motion which gives a lognormal model with parameters mu – 0.5*sigma^2 and sigma^2. The result here comes from solving a slightly different SDE -> dSt = (mu+0.5*sigma^2)dt + sigma*dBt. Essentially you construct the drift term of the SDE such that the 0.5*sigma^2 term disappears and we have a more eloquent result in part (i). So with mu and sigma as our parameters for the lognormal distribution we then need only to apply read off results from the tables for part (ii) of the question.

    Hopefully this makes sense. It's not abundantly clear where the lognormal model comes from but hopefully the above reassures you that you can generate the result with the correct choice of parameters. There has historically been little need to use the SDE (mu + 0.5*sigma^2)dt + sigma*dBt. However, quoting the form of the lognormal molel for share prices has been common so it's important that you can do this.

    If anything remains unclear then please shout :)

    Joe
     
  3. N_Exam

    N_Exam Keen member

    Thanks Joe, that clears things up. :)
     

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