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Take for example this passage from "Monte Carlo Methods in Financial Engineering".

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Is this a result of the risk neutral world or is this the real world drift as well? I've never seen the explicit distinction made and I don't see anything in the argument that would imply it only to hold for the risk neutral world. Is this something that can actually be observed in the real world? If it were wouldn't we be seeing consistently rising (falling) FX rates between for example the USD and EUR (do we?) and if so wouldn't we expect the interest rates to start converging to each other as a result (do we?)?

Put in another way, if we were to do a Monte Carlo value at risk simulation of the FX rate to capture the risk of FX derivatives, would we use this drift $\mu = r-r_f$? Or would we only use this drift if simulating the FX rate to, say, value a call option on the FX rate and use zero drift in the Value at Risk simulation (similar as to how we use r as the drift in equity option valuation and zero drift when performing a monte carlo value at risk simulation)?

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Yes, clearly when 2 countries have widely different inflation rates and interest rates we do observe a deterioration of the exchange rate between them over the long term (in the real world, not risk neutral world). For ex. CHF vs USD for last 50 years. In Economics there is a hypothesis called the Uncovered Interest Parity which claims this is generally true for any country, this is more controversial and perhaps not always supported empirically. I am not familiar with the latest empirical studies. Exchange rates change all the time for 1000 reasons so it is sometimes difficult to pin down the role of just one factor like i.r. differences in an empirical study. But the tendency is there, I personally believe.

The Fisher Effect (that the expected local inflation rate is incorporated into local interest rates) and long term PPP (Purchasing Power Parity) also play a role in this phenomenon and may be what drives it.

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  • $\begingroup$ Interesting, thank you. I have heard that Interest Rate Parity doesn't really hold in the real world for various reasons but it does feel intuitive for me that the "tendency", as you say, would be there. Do you agree then that if one were to simulate the FX rate with Geometric Brownian motion for risk purposes (so in the real world measure) you would use the drift as in the question? $\endgroup$
    – Oscar
    Jun 5, 2020 at 12:49
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    $\begingroup$ There is both CIRP and UIRP (covered vs uncovered) and it is true that both have come under criticism (CIRP in recent years and UIRP for a longer time). But yes, in a simulation of FX rate for risk purposes I would use the drift based on interest rates. $\endgroup$
    – nbbo2
    Jun 5, 2020 at 14:39

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