Ito's Lemma - Geometric Brownian Motion

[RedCoat]

Hi, please could somebody help with applying Ito's Lemma to Geometric Brownian Motion? Apologies for it being horrendous to read, I couldn't seem to get the maths notation to display in my post. I will use pd to show partial derivatives.

Suppose we have dSt = St (mu dt + sigma dZt) where Zt is a Wiener process.

Then to apply Ito's Lemma we let dQt = dSt / St = mu dt + sigma dZt, so this is now in a form to apply Ito's Lemma.

Ito's Lemma says that if dXt = mu dt + sigma dWt then df(Xt) = (mu pdf/pdXt + 0.5 * sigma^2 pd^2f/pdXt^2) dt + sigma pdf/pdXt dWt

Substituting,

df(Qt) = (mu * pdf/pdQt + 0.5 * sigma^2 * pd^2f/pdQt^2) dt + sigma pdf/pdQt dZt

Let f(Qt) = log(Qt)

Then pdf/pdQt = 1/Qt and pd^2f/pdQt^2 = -1/Qt^2
Substituting again

= df(Qt) = (mu/Qt - 0.5sigma^2 / Qt^2) dt + sigma/Qt dZt

Now, looking at (for example) the solution to Q3 on the April 2017 exam, I think my Qts should = 1 here, but I'm not sure why! Any help would be hugely appreciated. Think I'm making a huge mess of this.

 

[Joe Hook]

I think you have indeed overcomplicated this.

Noting the form we need for applying Ito's lemma in the Tables: a(X,t)dt + b(X,t)dz

Our SDE has a dt term and a dz term: mu*St*dt + sigma*St*dZt so all we need to do is set a = mu*St and b=sigma*St and then, with our log function as G, Ito's lemma will take care of the rest.

Hopefully a little simpler than you were making it out to be!

Joe