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.