Im slightly confused about this section of the booklet regarding option prices byIain J. Clark.
1) Regarding the part of obtaining a martingale property we require that the last exponential term disappears for Z, why is this? Is this because we dont want the expectation to depend on any previous results before time t, and the interest rates will? 2) Why is that particular drift change required? 3) And im completely lost with Radon-Nikodym.
I realise I am basically struggling with the whole of this question however any help would be greatly appreciated.
Thank you in advance,
James
REVISED QUESTION:
Maybe we can go back to the equities one for a simpler argument on how to get the Wd and apply R-N: Please see the photos below:
Using Ito's lemma we can see the condition for $Z_{t}$ to be martingale in Q2.7.2 as follows $$ dZ_{t} = \left(\frac{\partial Z_{t}}{\partial t} + \frac{1}{2}\frac{\partial^{2} Z_{t}}{\partial W_{t}^{2}}\right)dt + \frac{\partial Z_{t}}{\partial W_{t}}dW_{t} \implies \\ dZ_{t} = \left[\left(\mu + r^{f}-r^{d} - \frac{\sigma^{2}}{2} + \frac{\sigma^{2}}{2}\right)dt + \sigma dW_{t}\right]Z_{t} $$ So for $Z_t$ to be martingale the drift term must vanish i.e. coefficient of dt identically zero. or $$ \mu + r^{f}-r^{d} - \frac{\sigma^{2}}{2} + \frac{\sigma^{2}}{2} = 0, $$ that is, $$ \mu = r^{d} - r^{f} $$
For the next bit who to get the "change in variable" we utilise that we can transform a process with $$ W^{d}_{t} = \tilde{W_{t}} - \int \gamma_{t}dt $$
(I can't remember the appropriate name for the above transform) Though you have to satisfy a bunch of conditions, namely that $\gamma_{t}$ must be bounded.
So using Ito on the above we arrive at $$ dW^{d}_{t} = dW_{t} - \gamma dt $$ Here I have dropped the subscript $t$ for $\gamma$ as the drift is constant wrt $t$. so $$ dS_{t} = \sigma S_{t} dW_{t} + \left(r^{d} - r^{f}\right)S_{t}dt,\\ \,\,\,\,= \sigma S_{t}\left(dW_{t} - \gamma dt \right) + \left(r^{d} - r^{f}\right)S_{t}dt,\\ =\sigma S_{t} dW_{t} + \left(r^{d} - r^{f} - \gamma \sigma\right)S_{t}dt. $$ Now for $S_t$ to become martingale under this change we require the drift to be zero again. Therefore $$ \left(r^{d} - r^{f} - \gamma \sigma\right) = 0, $$ that is, $$ \gamma = \frac{r^{d} - r^{f}}{\sigma} $$ or $$ W_{t}^{d} = W_{t} + \frac{r^{f} - r^{d}}{\sigma}t. $$
Now I may of made a mistake, but this is how i followed through.
Please note that this is not complete and does not correspond to the Original question result, so highlight where I went wrong and help the OP and I will remove/edit where necessary.