CT8 September 2013 - Q5(i)(a)

[JuppWeber]

Hi,
How is Ito's Lemma being used to calculate (dlogS_t) - when looking at page 46 of the tables I can't quite see why the final term outside of the brackets which depends on (dz) is not within the answer ...
Would someone be able to shed some light on this?
Thanks in advance, James

 

[John Potter]

James,

My advice is to give Ito's lemma a miss and just use Taylor's formula (which is effectively using Ito's lemma). My advice is completely analogous to solving quadratics by completing the square rather than using the formula (the formula comes from completing the square).

log St is a function of St, so we use
d(log St) = df(St)
= f'(St) dSt + f''(St) (dSt)^2
= (1/St) (0.4St dt + 0.5St dBt) + (-1/St^2) 0.5^2 (St)^2 dt

Collect all your dt terms together and collect all your dBt terms together and you will be happy.

There is no partial derivative w.r.t time since f(St) is not explicitly dependent on t. This will be one of your terms on page 46 missing for a start. But the dz term on page 46 is the dBt term here. To avoid the headache of copying formulas and trying to work out which bit is which (I think in this case missing that dz is dBt ??) just use Taylor's formula,

John

 

[JuppWeber]

Great, thanks John.
Is there a 2! missing in the expansion of the f''(St) term in your final line above?
Otherwise I can't get the dt terms to checkout ...

 

[John Potter]

Great, thanks John.
Is there a 2! missing in the expansion of the f''(St) term in your final line above?
Otherwise I can't get the dt terms to checkout ...

Yes, sorry about that - good spot!