Valuation of European Currency Options
This chapter discusses the valuation of European currency options. A European currency option is a put or a call on a sum of foreign currency that can be exercised only on the final day of its life. These options are sometimes called vanilla options because they have no exotic features, such as out-barriers. The chapter begins with various arbitrage and parity theorems and then advances to the important Black-Scholes-Merton model for European currency options as adapted by Garman and Kohlhagen (1983).
The following conventions will be used throughout this book:
C is the value of a European currency call option.
P is the value of a European currency put option.
S is the spot exchange rate.
K is the option strike.
Rf is the interest rate on the foreign currency.
Rd is the interest rate on the domestic currency.
For the purpose of presenting theoretical material, it will be assumed that the deliverable underlying asset of a basic put and call is one unit of foreign exchange. Both S and K are denominated in units of domestic currency (i.e., expressed in American spot convention): One unit of foreign currency is worth S units of domestic currency.
In this framework, a call is the right but not the obligation to surrender K units of domestic currency to receive one unit of foreign currency. A put is the right but not the obligation to surrender one unit of foreign currency and receive K units of domestic currency. The current time is ...