Currency Conversion and Numeraire Choice for Option Payoffs

Summary

The document asks whether a currency-conversion identity used in pricing an option with a currency exchange feature holds generally. It describes two valuation routes for a payoff denominated in one currency: convert the payoff at maturity and discount in the other currency, or discount in the payoff currency and convert the resulting value at the valuation date. It then asks whether an exotic option paid in GBP can be valued under an AUD risk-neutral measure and converted using today’s GBP/AUD exchange rate.

This raises a central change-of-numeraire issue in multicurrency derivatives pricing: valuation should be invariant to the chosen numeraire when the associated measure and exchange-rate dynamics are handled consistently. The document itself gives no answer or derivation, so it does not establish that multiplying by today’s spot rate is sufficient in every case. Payoff currency, conversion terms, discounting, and the measure change all matter; the question leaves these conditions unspecified.

Key ideas

• The question compares converting a foreign-currency payoff at maturity with converting its discounted value today.
• Change of numeraire can support equivalent valuations when the measure and currency dynamics are handled consistently.
• The document asks whether today’s exchange rate alone converts a valuation between payoff currencies.
• It provides no derivation or answer establishing when that shortcut is valid.

Tags

# Option Payoff in Different Currencies


# 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?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.