Suppose in a BS world I write an ATM option on a Japanese stock that fulfills all of the BS requirements, but I denominate the option in dollars.That is, suppose the yen:$ conversion factor (k) is 100, and the Japanese stock (J) trades at 10,000 yen. Then, S(0) = K = J(0)/k(0) = $100, and S(t) = J(t)/k(t). My question is: what shall be the risk-free rate used in a BS valuation? the prevailing risk-free yen-interest rate or the prevaling risk-free $-rate? Since these differ by ~3% the difference is significant.My answer: The dollar risk-free rate is the correct rate. I can think of the underlying as S(t), and it fulfills the BS assumptions (although it is not a physical asset). S(t) is denominated in dollars, hence, for all intents and purposes, this an option on dollar denominated asset. It just seems counterintuitive to the physical process though.Interested in any and all feedback.

But then you must ask what to use as vol...

Agreed.If I consider the undelying of this option to be S(t) = J(t)/k(t) [the Yen-denominated stock divided by the Yen-$ conversion factor], which unlike J(t) is not a physical quantity, then I must find the vol. of S(t).Clearly, there is no problem if I am comfortable with using historical vol, because J(t) and k(t) are market observables.