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Why FX Risk-Neutral Probabilities Depend on the Investor’s Currency

Article Quant Q&A · Author: honeybadger
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Summary

The discussion considers why a binomial model for an exchange rate can produce different risk-neutral up probabilities when constructed from the domestic and foreign currency perspectives. It clarifies that risk and returns are measured in the investor’s chosen currency, so the two viewpoints need not define identical finite-step trees. A second explanation focuses on approximation: a coarse binomial tree only approximates the continuous lognormal model, and its implied probabilities can differ depending on the currency perspective.

The questioner reports that increasing the number of time steps makes the two estimates converge to the probability associated with the lognormal model. The post says this limit can be examined using l’Hôpital’s rule, but does not include the derivation itself. Thus the discussion offers intuition and a reported convergence observation, rather than a full proof or numerical study. Its conclusions apply to the described binomial approximation setup and do not establish that risk-neutral probabilities are physical forecasts of currency appreciation or depreciation.

Key ideas

  • Risk-neutral valuation measures payoffs in a selected numeraire currency.
  • Changing the investor’s currency perspective changes how risk and returns are expressed.
  • Finite-step binomial trees approximate the lognormal model, so their implied probabilities may differ across currency perspectives.
  • The questioner reports convergence toward the lognormal-model probability as the number of steps increases.
  • The post mentions a limiting argument but does not provide its complete derivation.

Tags

ForexStatisticsDerivatives pricing
Full text
# Risk neutral probabilities for foreign currency exchange rate


# Risk neutral probabilities for foreign currency exchange rate












Suppose that there are two currencies INR(domestic) and USD(foreign). Let the for exchange rate be S_inr. Using historical data, one can find out the volatility. For example, assume that, S_inr=60,σ=0.2,T=1,r_inr=0.1,r_usd=0 (the usual notation); I constructed the tree and found out Risk Neutral probability(RN1).

I also constructed the tree from an American investor perspective and found out the Risk Neutral probability(RN2).

RN1 and RN2 are not the same. I understand that it gives mathematically inconsistent trees when we use the same risk-neutral probability "p" for an American Investor and for an Indian Investor. However, I fail to comprehend the following: Why is that the risk neutral(RN) probabilities change depending on whether we consider an Indian or an American perspective? RN probability is simply the probability, as anticipated by a Risk Neutral investor, on whether the exchange rate moves in a certain way. In other words, it is the probability expected by an RN investor that the currency appreciates or depreciates. So, it should not matter whether we consider USD to INR or INR to USD.

I am sure that there is something I am missing.

## Answer by honeybadger (score 0, accepted)

https://quant.stackexchange.com/a/33477

Actually, I ran the model with higher number of time periods and RN probabilities from both, USD and INR, perspectives seem to converge to the real RN probability associated with a lognormal model. So, the apparent paradox is because binomial approximation to Lognormal model works only when the duration of each time period is very small(or high number of time periods. When we use fewer number of time periods, the binomial model is at most a good approximation. So, the approximations are bound to differ from the real RN probability when viewed from INR or USD perspectives.

Thanks everyone for trying it out and helping me.

For others, just an update: Here is the mathematics behind the convergence of RN probability when you increase the number of time periods. The final value remains the same whether you approach from USD or INR perspective.

When n tends to infinity, use L-hospitols rule for finding the limit

You can checkout this link as well: PDF

The same can be repeated and verified from USD perspective.

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/33407

The explanation that works for me is that what constitutes risk for a US investor (i.e. Making or losing money measured in dollars) is different from what constitutes risk for an Indian investor ( i.e. Making or losing money measured in INR). Hence there is no paradox.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.