# Option Payoff in Different Currencies

In the stackexchange answer Change of numeraire in options with currency exchange features

Pratically speaking, what this expresses is that these two things are the same: Converting the payoff (which is in EUR) to COP at T and then discounting in COP from T to t; Discounting the payoff from T to t in EUR and then converting the discounted value at t from EUR to COP.

Does that hold in general? E.g. if some exotic option payoff is paid in currency GBP at time T, then I can just price it under currency AUD payoff with AUD risk neutral measure then just multiply by TODAY's GBP/AUD exchange rate and arrive at the same price had I proceeded to price under the GBP RN measure?

• Suppose that someone in the emerging markets issues two bonds denominated in local currency. Vanilla bond A just promises to pay 1 unit of local currency in 1 year. Whereas bond B promises that in 1 year, the issuer with observe the currency exchange rate and pay the USD amount equivalent at that time to 1 unit of local currency. I.e. B has an embedded non-delivery forward. What would you need to assume away in order for A and B to be discounted the same? The cross-currency basis is sometimes a lot. Also they'd be discounted for slightly different risks. Jul 24 at 2:10
• These two option prices are the same. Converting an arbitrary payoff into another currency makes it a payoff in that currency. Calculate its PV in that currency and convert it back to the original currency. If that yielded a PV that is different from the original PV we would have an arbitrage. When you let this sink in you will realize that one does not even need math to come to this conclusion. Jul 25 at 8:48