# The relation between exchange rate SDE and respective interest rates

The exchange rate between a domestic currency money market and a foreign currency money market can be expressed as $$dQ(t) = (r_d - r_f)Q(t)dt + \sigma Q(t)d\tilde{W}(t)$$ where $r_d$ is the interest rate for the domestic market, and $r_f$ for foreign.

In my head, I believe that the exchange rate should decrease when the domestic interest rate goes up, indicating the domestic currency is strengthening. For example if the Fed were to increase rates, then $EUR/USD$ should decrease, given that the ECB doesn't do much. So, if $EUR/USD$ was 1.14 yesterday, it should be below 1.14 today.

To my understanding, the SDE for $Q(t)$ doesn't seem to reflect this fact. It seems that $Q(t)$ would increase if $r_d$ were to go up. I would like to resolve this contradiction, so any help would be appreciated.

• This problem is also explained also here. May 27, 2022 at 12:16

I think you are right. The SDE does not attempt to describe the dynamics of the spot exchange rate with respect to random changes in interest rates. Rather, it describes the evolution of the FX rate as a drift term proportional to the rate differential, plus a random term. Specifically, it says that if domestic rates go up, the rate at which the foreign currency strengthens in the forward market, relative to spot, goes up. It doesn't say anything about what happens to the spot FX rate in this situation.

• I've been thinking about this all day and I think your approach to $Q(t)$ is correct. Taking into consideration the interest rate parity, it is indeed the case that a higher Fed rate, or $r_d$, implies an expected appreciation of the foreign currency at a rate of $r_d - r_f$. It turns out I was confused between the spot rate and the $Q(t)$ dynamics, which isn't necessarily the spot. Thanks for clarifying that. May 2, 2016 at 1:03
• Higher (relative) rates may cause spot to increase (in terms of EURUSD, if USD rate is higher, the USD should appreciate from say 1.14 to 1.12) but as correctly pointed, this relationship is not about spot, but describes the evolution of the FX rate (hence the forward). If the US interest rate is higher, in line with (un)covered interest rate parity, there is an expected depreciation in the USD relative to EUR (higher exchange rate - say from 1.14 to 1.16). May 24, 2022 at 12:42

The dynamics for the exchange rate $Q$ that converts one unit foreign currency to units of domestic currency is given by \begin{align*} dQ(t) = Q(t)\big[(r_d-r_f)dt + \sigma dW_t \big], \end{align*} where $r_d$ and $r_f$ are, respectively, the domestic and foreign interest rates.

In your example, the exchange rate EUR/USD is to convert on unit EUR to units of USD. Here, EUR is the foreign currency, and USD is the domestic currency. If the US Fed raises the USD interest rate, $r_d$, while there is no change in ECB for the EUR interest rate $r_f$, then $Q(t)$ will increase. There is no contradiction.

You confusion may be caused by the notation EUR/USD, where you might have treated the US Fed rate as $r_f$.

• Thanks for the input. However, I am confused to see how I may confused the notation EUR/USD. I have correctly treated the Fed rate as $r_d$, for which a hike would cause $r_d - r_f$ and $Q(t)$ to increase as well. This seems to imply that $EUR/USD$ would increase. But in reality, the dollar would strengthen, meaning that 1 Euro would cost less than what it cost before, say 1.12 dollars per Euro. Since the exchange rate dropped from 1.14 dollars to 1.12 dollars, this would imply that $Q(t)$ decreased, which seems like a contradiction to me. May 1, 2016 at 20:40
• The value of EURUSD Spot will initially decline (say from 1.14 to 1.12, which is an appreciation in USD relative to EUR, you need less USD now for 1 EUR). However, this relation here takes spot as given (and rates). Insofar spot already has the (new) fair value. The dynamics concern the evolution of spot (the foward). If the USD rate is higher, the EURUSD exchange rate increases from say 1.14 to 1.16. (which is in fact a depreciation of the USD). See for example (un)covered interest rate parity. May 24, 2022 at 13:31
• also here for the problem at hand May 27, 2022 at 12:16

The term rd - rf results from the non arbitrage condition rather than an explicit modelling of the fx rate in terms of interest rates. In the extreme case of a fixed exchange rate (sigma=0), the term rd - rf simply states that investing in the cash account of the foreign currency is equivalent to investing in the cash account of the local currency, modulo the exchange rate.

The fx rate itself is driftless similar to a future, ie it doesn't provide any return.

The fact that the fx rate goes up or down depending on an interest rate is an empirical observation that needs to be explicitly modelled, ie it doesn't magically derive from the basic stochastic equation.