A currency option is a contract that gives the buying party the right, but not the obligation, to buy (call) or sell (put) a foreign currency at a pre-defined exchange rate (strike).
For a given currency pair (e.g. EURJPY), the call or put designation of the option contract always refers to the base currency; therefore, the foreign currency is the base currency and the domestic currency is the quoted currency. For example, a Japanese company expects to make a payment of 1,000,000 Euros in six months; they can hedge this exposure (while also participating in favorable price movements) by purchasing a call option on EUR/JPY. Similarly, if they expected to receive Euros, they'd enter into a put option on EUR/JPY.
Because of the nature of FX transactions, every currency option is technically two options: a call and a put. So a call on EUR/JPY is a call (buy/receive) on EUR and a put (sell/give) on JPY for a specified exchange rate (the strike) and a put on EUR/JPY is a put (sell/give) on EUR and a call (buy/receive) on JPY.
Exercise. Sometimes a currency pair is not actively quoted. For example, if the domestic currency is EUR and the foreign currency is JPY, the pair underlying a given currency contract would be JPY/EUR, which is not actively quoted. Of course, one could simply invert this currency pair and the above discussion applies with this exchange rate. However, show that for currencies C1 and C2 that a call (put) on C1/C2 is a equivalent to a put (call) on C2/C1. So if the European company (domestic currency) wants to buy a put on JPY/EUR (they want to sell JPY that they expect to receive sometime in the future for EUR), then they can buy an (equivalent) call on EUR/JPY.As a final note, each currency has an associated risk free rate $r_{d}$ and $r_{f}$ (domestic and floating, respectively).
II. Valuation Theory
The Black-Scholes model on a dividend paying equity is easily adapted to the present situation by regarding the currency pair exchange rate as a price with an associated volatility and the foreign riskless rate as a dividend yield. The justification for the ladder follows the same cost of carry argument as in the case of a dividend.
The usual formulas are
$$c(S,t)=Se^{-r_{f}(T-t)}\mathbb{N}(d_{1})-Ke^{-r_{d}(T-t)}\mathbb{N}(d_{2})$$
and
$$p(S,t)=Ke^{-r_{d}(T-t)}\mathbb{N}(-d_{2})-Se^{-r_{f}(T-t)}\mathbb{N}(-d_{1})$$
where of course
$$d_{1}=\frac{\log(S/K)+(r_{d}-r_{f}+\sigma^{2}/2)(T-t)}{\sigma\sqrt{T-t}}$$
and
$$d_{2}=d_{1}-\sigma\sqrt{T-t}.$$
Because $F_{t}=S_{t}e^{-(r_{d}r_{f})(T-t)}$, the forward price of the currency pair, the formulas above can be simplified to
Thus one only needs the discount rate in the domestic currency.
III. Example
Consider a call on
