How do you trade a call on a stock which is quoted on two exchanges in two different currencies. Assume that the price in the two currencies is watched and arbitraged fairly efficiently against the FX. Also you can short/long options in both countries. An example would be BHP listed in London and in Australia. If you think about it long a call in one currency versus short a call in the other currency has some pretty wacky behaivour dependent on the FX rate as well as the underlying stock price. I have tried seaching for info on this type of play and the only thing I could find was some cr@p about quanto's

Stop wasting your time. Like you said--------"the markets are arbitraged fairly". Therefore there's no "free money" to be earned. If you're focused on currency movements, concentrate on trading currency instruments instead.

You'd have to hedge FX going forward, which would kill any IA? As a more general question, though, why does the underlying sometimes diverge so much for so long? Can't imagine it's all FX risk, reg risk? eg Wipro, quotes from Yahoo: WIT(ADR) 19.6USD vs 507685.BO (? not sure), INR723/(45INR/USD) = 16USD). While I appreciate that I could not arb that, why does'nt the Depository Trust (BNY?)?

Assuming there is not arb and the asset is fungible (bigger question) it's possible to express implied volatility in two currencies on the same asset as implied volatility on the exchange rate. In it's no different then the regular currency symmetry.

I assume you are talking about a VolFX^2=VolA^2+VolB^2+2*VolA*VolB If you think about it whats the likelyhood that all pairs will give the same volatility number for the FX. If you think about it the FX arb argument comes from the delta of each option in each currency but the delta is dependent on the volatility you plug into the BS formula for each stock. Theory suggests that if you trade the delta of each stock versus the FX you should end up with the risk free rate right. All very well except the delta of each option not going to be the 'true" delta. I am pretty sure that you will end up with some pretty strange effects because of jumps in the underlying stock and jumps in the FX rate. For example if the FX rate gaps, which stock moves or do they both move?