April 2005 Q5

[laura_mils]

Hi,

I am pretty stuck on this question, I am not sure if its because the notation is different.
I can see that it is like a chapter 10 question, trying to solve with Taylor's theorem/ Ito's

But I am quite stuck on this and not sure how they obtained the solution at all.

Thanks for the help

 

[Joe Hook]

Hi Laura,

I apologise for the late reply.

I agree that the examiner's report solution is not particularly helpful or detailed but you are right that the required approach is either Taylor's theorem or Ito's lemma. However, the maths is pretty challenging...

Finding the partial derivatives of the function given in the question:

(1) 1st partial derivative w.r.t t = f(t,rt) * (rt + 0.5*sigma^2*(T-t)^2)

(2) 1st partial derivative w.r.t rt = f(t,rt) * -(T-t)

(3) 2nd partial derivative w.r.t. rt = f(t,rt) * (T-t)^2

If you're able to obtain these partial derivatives then subbing them into Taylor's formula:

(1) * dt + (2) * drt + 0.5 * (3) * (drt)^2 with:

drt = mu*rt*dt + sigma*dZt
(drt)^2 = sigma^2 * dt using the multiplication grid

...yields the solution in the examiners report.

I hope this helps. Let me know if there's anything outstanding on a second look.

Joe

 

[laura_mils]

awesome, thanks will take another look at this one with your help

Laura