Deriving Exchange Rate Dynamics Under the Domestic Risk-Neutral Measure

Summary

The document poses a derivation problem in currency derivatives. It defines stochastic dynamics for a foreign-market stock and the domestic/foreign exchange rate, alongside a deterministic domestic money account. The proposed approach converts the foreign asset price into domestic currency by multiplying it by the exchange rate, then discounts that value by the domestic account. This normalized price is intended to be a martingale under the domestic pricing measure.

The question asks how to apply Itô’s lemma to this product and quotient, whether cross-variation terms vanish when the money account has a deterministic rate, and how a Girsanov change of measure determines the exchange-rate drift. However, the document contains no answer or completed derivation. It therefore identifies the relevant modeling steps and issues but does not establish the resulting dynamics or provide evidence, assumptions about correlation, or boundary conditions. Readers will need to complete the stochastic-calculus and measure-change calculations themselves.

Key ideas

• A foreign asset can be expressed in domestic currency by multiplying its price by the exchange rate.
• Discounting that converted price by the domestic money account gives the candidate normalized process.
• Itô’s lemma introduces a cross-variation term between the stock and exchange rate when they share stochastic drivers.
• A deterministic money account has no stochastic cross-variation with the other processes.
• The domestic risk-neutral measure can be characterized by requiring the normalized asset price to be a martingale.

Tags

# How to determine exchange rate dynamics in currency derivatives


# 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 $X$.

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