Did someone choose a wrong previsible procees?April 2005,Q8.

When showing that the discounted share price process is a martingale, the solution uses
y'' = (mu - r + .5 sigma^2 ) where y'' is the previsible process. It goes on to say, In differential form this is written as dW''t = dWt + y''dt which is correct, however I don't think substituting dWt= dW''t - y''dt into the SDE for Z, leads to dZt = sigma*ZtdW''t.

SINCE;

dZt = sigma*Zt*dWt + Zt*y''dt

now substituting dWt= dW''t - y''dt gives

dZt = sigma*Zt*(dW''t - y''dt) + Zt*y''dt
= sigma*Zt*dW''t - sigma*Zty''dt + Zt*y''dt
= cannot be sigma*Zt*dW''t
????

Wasn't the previsible process supposed to be y'' = (mu - r + .5 sigma^2 )/sigma

SO THAT

substituting dWt= dW''t - (mu - r + .5 sigma^2 )/sigma*dt gives

dZt = sigma*Zt*(dW''t - (mu - r + .5 sigma^2 )/sigmadt))+ Zt*(mu - r + .5 sigma^2 )dt
= sigma*Zt*dW''t - Zt*sigma(mu - r + .5 sigma^2 )/sigma*dt+ Zt*(mu - r + .5 sigma^2 )dt
= sigma*Zt*dW''t - (mu - r + .5 sigma^2 )*Ztdt+ (mu - r + .5 sigma^2 )*Ztdt
= sigma*Zt*dW''t

HELP!

 

(I'm assuming you're looking at the Examiners' Report rather than our ASET.)

Yes, you're right, gamma should be the market price of risk, so the function in the ER is missing the sigma denominator, as you point out.

 

Thanks Mike, I was looking at the Examiner's report!