# 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.

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.

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.

• I have some doubts, because the expectation of the inverse is not the inverse of the expectation (when the exchange rate is stochastic). AFAIK Siegel's Paradox is alive and well. Apr 4, 2017 at 14:30
• Siegel's Paradox case deals when the exchange rate fluctuates to either 1:X or X:1. Here, in this problem, the case is different. Here, in the question it seemed that different RN probabilities are observed from different countries, but when no. of periods is increased, the probabilities converge Apr 7, 2017 at 18:10
• I agree with @noob. The risk-neutral probabilities in your trees cannot converge as number of steps increases, if you have fixed the possible values of the fx rate at the various steps.
– dm63
Apr 8, 2017 at 10:34
• One more thing. Siegel"s paradox seems to say that the real world probabilities must be the same for all investors. Hence the foreign and domestic investors can both have positive expected returns in their own currencies. ( but since they are taking risk, there's no contradiction)
– dm63
Apr 8, 2017 at 10:39
• I think there is some confusion about this question. To rephrase the solution to the final part of the question: consider a RN person, say Hari, who has interests in both INR and USD. For this person, the RN probability of FX moving up wrt INR (equivalent is moving down wrt USD) is unique. There cannot be two values for a single RN probability. Apr 9, 2017 at 13:34