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Quanto options (or cross-currency derivatives) are cash options that have an underlying asset denominated in a "foreign" currency, but settle in a the "domestic" currency at a fixed exchange rate. This limits the foreign exchange exposure for the holder of the option.
For example, an option on the Nikkei stock index held by an investor in the USA. The underlying index and strike would be in JPY but the investor would receive USD based on the fixed exchange rate should the option be in-the-money at expiry.
Option value in domestic currency
Option value in foreign currency
where
S* = Underlying asset price in foreign currency.
X* = Delivery price in foreign currency.
r = Domestic risk free rate.
rf = Foreign risk free rate.
q = Instantaneous proportional dividend payout rate of the underlying asset.
E = Spot exchange rate in units of domestic currency per unit of foreign currency.
E* = Spot exchange rate in units of the foreign currency per unit of domestic currency.
= Volatility of the underlying asset.
= Volatility of the domestic exchange rate.
= Correlation between asset and the domestic exchange rate.
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