Hi, is there any layman term/interpretation for explaining why we need to change the risk neutral measure (or drift rate) from foreign to domestic if the stock is a foreign one in the quanto concerned?I read the proof and roughly know mathematically that the drift rate of domestic & foreign measures differs by a "quanto prewashing" term [i.e. correl x stdev(stock) x stdev(FX)]But I would like to know exactly the meaning of this difference... why the drift term are not the same? (they are talking about the same underlying and therefore should expect the same drift rate?)

The easiest way to value quanto style payments is to do the analysis in your base currency and express the quantoed price as a ratio of two values denominated in that currency. E.g. a Japanese stock quantoed into USD can be thought of as the ratio of the dollar value of the stock and the JPY/USD exchange rate. You can now perform the valuation under your usual risk neutral measure. If you squint at the integral you need to perform to take a simple expectation (e.g. for a European option) you may see that the risk neutral density is multiplied by a ratio that can be used as a Radon-Nikodym derivative (resulting in a switch of measure/drift), but feel free to ignore that. In general, your last question can only be meaningfully asked under the original (P) measure -- there is no relationship between what you actually "expect" and probabilities derived under alternative measures.

A more layman intuition type explanation comes from thinking about the hedge for a quanto option. Imagine you have sold a deep in the money call option on a Japanese stock quanto in USD. As it is deep in the money we ignore gamma and vega risk for the purpose of this explanation (not good risk management to do that of course!). Let us imagine that we buy an on market OTC forward on the stock (in JPY) as a delta hedge, and deposit the USD premium of the quanto call. Let us ignore rate risk for simplicity.Let us imagine we have the correct delta hedge, whatever that may be. And we think of our P&L in JPY.Now imagine that the stock jumps up by 5%. In this case our contingent liability in USD has just increased, so we are short USD versus JPY: no problem, we just buy USD and sell JPY in an FX trade (ideally a forward) to hedge.But wait: what if at the same time that the stock jumped up the USD also went up? Oops - it is too late to hedge that simultaneous move: we take a loss.The "oops, two things happened at the same time" loss is a quanto effect caused by correlation. So if the stock price and the USD are positively correlated, we will get little losses building up over time as we do our delta hedging. This is how the stock versus FX correlation comes into the picture.The larger the simultaneous FX and stock move, the bigger the loss - this is how the stock and FX volatilities come into play. The result is a sort of statistical drift in the P&L.And intuitively you would expect that this drift is proportional to the correlation and the volatility of both the stock and the FX rate.Adding in this drift in the option calculation looks a lot like a continuous dividend yield contribution.Hope this helps - apologies if it is too simplistic of an explanation, not sure if that is what you were after.