FX call option is an option to buy a foreign currency at a particular domestic currency price. If we look from the counterparty’s perspective it is also a put option to sell the domestic currency at that price of foreign currency.
Let us take an example of USD/EUR currency option. Suppose the price of 1 unit of Euro is X and strike price is K and c is the price of call option in USD.
On the other side, the price of the USD in Euro is 1/X and the strike will be 1/K to buy 1 unit of USD in Euro. Let the price of this put option will be p_f.
The options c and p_f are similar in nature and hence their price should be identical. But if you calculate it, it won’t be identical. You would be required to make exchange rate adjustments and notional assumptions.
The p_f will have to be multiplied by X to convert the option price in domestic currency.
The resultant value will have to be further multiplied by K because the notional amount in the call is in Euro.
The resultant price will be equal to c.
Summarizing c = p_f * X *K