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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 $X$.

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    $\begingroup$ It appears correct. Can you complete $dZ(t)$? $\endgroup$ – Gordon Dec 16 '19 at 15:09
  • $\begingroup$ @Gordon no unfortunately I do not know how to handle the cross-variation terms. Any hints? Thank you :) $\endgroup$ – Dreason94 Dec 16 '19 at 17:52
  • $\begingroup$ You have both $X$ and $S$ here. Do you want the dynamics of $X$ or $S$? $\endgroup$ – Gordon Dec 16 '19 at 20:45
  • $\begingroup$ @Gordon I wish to know the dynamics for $X$. However, I do not know how to complete step 2, i.e. find the cross-cariation terms. Appreciate your time man. $\endgroup$ – Dreason94 Dec 16 '19 at 20:53
  • $\begingroup$ Then you need to consider $\frac{X(t) B_f(t)}{B_d(t)}$, rather than $\frac{X(t)S(t)}{B_d(t)}$, and make sure it is a martingale under the domestic risk-neutral measure. $\endgroup$ – Gordon Dec 16 '19 at 21:36

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