Changing Measure for Foreign Interest Rates in FX Models

Summary

This question asks how to express a foreign short-rate model under the domestic risk-neutral measure so both domestic and foreign rates can be simulated consistently. It introduces correlated Brownian motions under different measures and a spot FX process whose drift is the domestic rate minus the foreign rate, then asks how the foreign-rate dynamics change under the measure change.

The document frames the modeling issue but provides no derivation or answer. It does not establish the required Brownian correlations, market price of risk, or resulting drift adjustment. Those details depend on the FX volatility and its correlation with the rate factors, as well as on the precise numeraire conventions. It is therefore a useful statement of a cross-currency modeling problem, rather than a worked technique or a validated simulation recipe.

Key ideas

• The question concerns simulating domestic and foreign short rates under one probability measure.
• Changing numeraire changes Brownian drift terms while preserving instantaneous covariance structure.
• The FX rate links domestic and foreign numeraires and requires an explicitly specified volatility and correlation model.
• The document poses the problem but does not supply the foreign-rate drift adjustment.

Tags

# Change of Numeraire technique (Cross-currency models)


# Change of Numeraire technique (Cross-currency models)












Hey I have problem with understanding change of numeraire technique. For example we have

$dr^d(t)=\kappa_1(\theta_1(t)-r^d(t))dt+\sigma_1 dW_1$ (under measure $Q^1$ associated with domestic bank account)

$dr^f(t)=\kappa_2(\theta_2(t)-r^f(t))dt+\sigma_2 dW_2$ (under measure $Q^2$ associated with domestic bank account)

where $W_1$ - Wiener process under $Q^1$, $W_2$ - Wiener process under $Q^2$ and $dW_1dW_2=\rho dt$ (here the first question - these two Wiener processes are written under different measures but we can write this correlation because when we write one of them under another measure then only drift term will change and it doesn't have any impact on correlation?)

And now I want to write $r^f$ process under $Q^1$ measure because I want to simulate these processes. How to do it? I know that we must to define FX rate process. Can we write this process directly under $Q^1$ measure in the form (1 unit of foreign currency = $X$ times unit of domestic currency): $$dX(t)=[r^d(t)-r^f(t)]dt+vdW^X(t)$$ where now $dW^XdW_1=\rho_{X,d}dt$ and $dW^XdW_2=\rho_{X,f}dt$.

If so far everything is OK, how can we find dynamics of $r^f$ under $Q^1$? Maybe there exists books/papers where everything is calculated step by step?

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