I need some guidance regarding exchange rate dynamics in currency derivatives.
Following three dynamics are defined below,
$\frac{dS(t)}{S(t)}=\alpha dt+\sigma dW(t)$ ; the stock dynamics in the foreign market
$\frac{dB_d(t)}{B_d(t)}=r_ddt$ ; the domestic money account
$\frac{dX(t)}{X(t)}=\alpha_X dt+\sigma_X dW(t)$ ; the domestic/foreign exchange rate
My goal is to derive the dynamics of $X$ under the equivalent martingale measure where the domestic money account is the numeraire.
My solution so far:
Step 1: I can define the foreign asset in the domestic economy as $S_d(t)=S(t)X(t)$ since we have the domestic/foreign exhange rate. Under the domestic economy and the domestic money account as numeraire I can further define the following martingale (normalized money process); $Z(t):=\frac{S(t)X(t)}{B_d(t)}$
Step 2: Applying Itô's lemma gives me,
$dZ(t)=\frac{X(t)}{B_d(t)}dS(t)+\frac{S(t)}{B_d(t)}dX(t)-\frac{S(t)X(t)}{B_d^2(t)}dB_d(t)+..........$
My first question here is whether I am thinking correctly defining the normalized money process $Z(t)$. The second question is how the cross-variation terms will fair? Should not the cross-variation terms be equal to $0$ since the return rate $r_d $in the domestic money account is deterministic?
Step 3: Should the normalized money process $Z(t)$ be correctly defined and I have solved the first order differentiation of $Z(t)$ i.e. $dZ(t)$ I want to apply a girsanov transform to the p.m. $Q^d$ with the girsanov kernel $\psi$.
Step 4: Insert $dW(t)=\psi dt+dW^{Q^d}(t)$ into the exchange rate $X$.
