# Dynamics of FX rate

I've see a couple of places where a FX rate, denoted $$X$$, such as EURUSD (quoted as "the number of USD needed to buy 1 EUR") is modeled with a diffusion process / Geometric Brownian Motion as follows through its "rate of change":

$$\frac{dX_t}{X_t} = \left[r_d(t)-r_f(t)\right]dt + \sigma(t,X) dW^Q(t),$$

where $$r_d$$ and $$r_f$$ denote the (deterministic) short rate for the domestic (USD) and foreign (EUR) currency. One could also model the same dynamics by multiplying by $$X_t$$.

My question is related to the drift quotient $$r_d(t)-r_f(t)$$: Why is it not $$r_f(t)-r_d(t)$$? We have just seen that the FED (US Cental Bank) hiked interest rate (meaning $$r_d$$ increased). The market response was that EURUSD decreased ($$X$$). [This post]The relation between exchange rate SDE and respective interest rates asks more or less the same question but it does not answer why we cannot model it the other way around - or at least why we should not do so.

• Does this help? In terms of forwards this becomes a no arbitrage realtionship (covered interest rate parity). If you would model it the other way around you would violate the no arbitrage argument. May 24, 2022 at 12:34
• Thanks, I at least understand that this kind of dynamic should be true now. If I understand things correctly, then if $r_d$ jumps (as in the case of the FED hiking rates with 50bp) then the spot rate $X$ will have to immediately decline to avoid arbitrage from investing in USD-deposits financed with EUR-loans. Is this correctly understood? May 24, 2022 at 13:12
• Also, if we consider the recent event of the FED hiking interest rates ($r_d$ increases) and the EURUSD declining ($X$ went from approx. 1.2 to 1.07) then this specify event actually went against the model specified in my question but agreed with the interest rate parity? May 24, 2022 at 13:26
• 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 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). The wikipedia link in my first comment shows why this is a no arbitrage relationship in the graphic on the right hand side. May 24, 2022 at 13:26
• @Landscape . $X_t$ is by all means the spot rate. The following analogy I have not seen in this post (too much to read): we know that the BS model for a stock is similar to that of $X_t$ when you identify $r_d$ with the riskless rate and $r_f$ with the stock's dividend yield. Since $X$ is the price in the domestic currency of a foreign ccy unit that's a way to memorize that the drift is $r_d-r_f$ and not $r_f-r_d$. May 25, 2022 at 11:34

I am answering now instead of commenting. The rate of change in FX is naturally forward looking in this case.

What you confuse is what happened to Spot due to changes in interest rate environments relative to what is expected based on no arbitrage arguments. The recent FED hikes ($$r_d$$ increases) and the EURUSD declining (X went from approx. 1.2 to 1.07) already happened and has nothing to do with this model. Note that this is in fact an appreciation of the USD.

Any news (changes in market data) will be priced in instantaneously. If higher interest rates in the US relative to the EUR make USD investments more attractive, you can expect an appreciation in the USD. However, that already happened (as soon as rates changed).

The above refers to given market data (spot and rates are known). That is also how FX forwards are priced, and shown in the wikipedia article about covered interest rate parity that I referenced in a comment. No matter what you do, returns from investing domestically are equal to the returns from investing abroad. This works because you enter a forward and fix that rate that guarantees no arbitrage.

In terms of expected spot ($$E(S)$$), the story is more nuanced. The so called uncovered interest rate parity is defined like this:

$$(1+i_{\\\})={\frac {E_{t}(S_{{t+k}})}{S_{t}}}(1+i_{c})$$

or rearranged:

$${{S_{t}}}\frac {(1+i_{\\\})}{(1+i_{c})} = E_{t}(S_{{t+k}})$$

If you think of EURUSD now (how many USD per EUR, say 1.2, if US interest rate is 10% and EUR 5% you get (for a year), the value of

$${1.2}*\frac {(1+0.1)}{(1+0.05)} = 1.25714286$$

In other words, you need more USD per EUR - the USD depreciated, EUR appreciated. That said, there exists a widely used strategy called the "carry trade". For the carry trade to work, this cannot be the case (higher interest currencies do not depreciate as much, on average). It is a successful strategy, but very risky because there is a tendency for this depreciation to actually happen - often rapidly during terms of crises. That is why there is the saying that with a carry trade you walk up the stairs (small steady returns) and go down the escalator (large sudden losses). If you wonder why this relationship makes sense, it is mainly due to differences in inflation rates. The image below is from FRED using FredApi in Julia. Turkey has a higher interest than the US (currently 14%) but still, the Lira depreciates steadily against the USD. The reason is higher inflation.

Empirically, FX is also more volatile than this relationship suggests, which is why "overshooting models" were developed. These are part of the stock approach to FX modelling and consist of flexible and sticky price monetary models which combine capital markets, goods markets and money markets. Sticky price monetary models are also known as overshooting models, initially designed by Dornbusch (1976).

In terms of forecasting (what might to be your ultimate goal?), neither of these models perform well or can be used. Kenneth Rogoff and Richard Meese received an incredulous reaction to their now-famous paper showing that random-walk (RW) forecasts outperform economic models of exchange rates. Reactions were along the line of “You just cannot possibly have done it right” or "the results are obviously garbage". Turned out they were correct. Rogoff makes an interesting point in some later paper. If money supplies are hard to predict, then one should not blame the models if exchange rates are hard to predict. It is unforeseen news that matters. However, as Rogoff further stated, their finding was even more extreme. They tested predicting the exchange rate in one year, given the information about what money supplies, interest rates, and outputs are going to be in one year. However, even in this case, no economic model beat(s) the RW.

The use of the SDE is mainly for option pricing. These models should be internally consistent, to the extent that the covered interest parity should also hold. That is why usually many pricing engines (e.g Bloomberg) imply one of the 4 inputs (spot, forward, two rates), by default the least liquid interest rate.

This also answers why it cannot be modelled the other way around, because it would create arbitrage opportunities.

• Thanks for the answer - it is very helpful. I believe that you are right about my confusion between the what happened to spot, $X_t$, due to changes in interest rate environments relative to what is expected based on no arbitrage arguments. My current experience with currency-modeling is limited to a single chapter in my textbook by Björk. Reviewing his arguments I can tell that the no arbitrage arguments are derived similar to the dynamics of the Black Scholes stock dynamics with continous dividends. May 25, 2022 at 18:32
• However, the interest rate parity is new to me - although it makes intuitive sense. So is it working with "empirical" arguments that cannot be explained directly by the model (as when FED hikes rates). May 25, 2022 at 18:32

Here is a way to demonstrate the logic of standard FX rate models. The key takeaway is that these models assume that uncovered interest parity (UIP) holds but still imply that a US rate hike leads to dollar appreciation. This is confusing to many since it would appear that UIP would rather predict USD depreciation.

Like in standard models assume investors are risk neutral. The continuous time version of the uncovered interest rate parity (UIP) condition states (for log-exchange rates):

$$\mathbb{E}_t[ds(t)]=(r_d(t)-r_f(t))dt$$

It is a statement about the expected FX rate change. The well-known logic is that risk neutral investors must be indifferent between investing in the two currencies so lower rates are compensated by expected appreciation. This has the forward solution.

$$s_t=\mathbb{E}_t\int_t^{\infty}(r_f(\tau)-r_d(\tau))d\tau+\underset{T \rightarrow \infty}{\lim}\mathbb{E}_t[s_T]$$

Note that the sign of the interest rate differential gets reversed.

To demonstrate the logic of standard FX rate models assume the long run exchange rate $$\underset{T \rightarrow \infty}{\lim}\mathbb{E}_t[s_T]$$ is constant.

Now an increase in US interest rates leads to a jump in the integral term and hence to an immediate dollar appreciation (smaller $$s_t$$). However, by the first equation it implies that $$\mathbb{E}_t[ds(t)]$$, decreases and hence the dollar is from that point on expected to depreciate.

The logic is that in order for USD to be expected to depreciate after the rate hike and still return to some long run value, it must initially appreciate so the initial reaction goes against the UIP prediction. The Dornbusch model adds an additional twist as the shocks change the long-run exchange rate but the logic is still the same.