# How to determine exchange rate dynamics in currency derivatives

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 $$X$$.

• It appears correct. Can you complete $dZ(t)$? – Gordon Dec 16 '19 at 15:09
• @Gordon no unfortunately I do not know how to handle the cross-variation terms. Any hints? Thank you :) – Dreason94 Dec 16 '19 at 17:52
• You have both $X$ and $S$ here. Do you want the dynamics of $X$ or $S$? – Gordon Dec 16 '19 at 20:45
• @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. – Dreason94 Dec 16 '19 at 20:53
• 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. – Gordon Dec 16 '19 at 21:36