FX derivatives (or foreign exchange derivatives) are financial derivative products whose payoff is a function of the exchange rate of two (or more) currencies. Like derivatives in general, FX derivatives can be grouped in three main categories: futures, swaps, and options. In this chapter, we will only deal with option-type derivatives. We will start with a straightforward generalization of the basic Black-Scholes model, and will show how to price a simple European call or put currency option. Afterwards, we will discuss the pricing of exchange options and quanto options.
Throughout this chapter, we will assume that you have some basic knowledge about derivative pricing, especially the Black-Scholes model and risk-neutral valuation. Occasionally, we will refer to some mathematic relationships often used in quantitative finance (such as Itô's lemma or Girsanov theorem), but a deep understanding of these theorems is not essential for this chapter. However, those interested in the pure mathematical background of this topic can check out Medvegyev (2007).
As we will work with FX rates, it is important to clarify some related terms. Generally, we will denote spot FX rates by S, which measures the price of one currency (called base currency) in terms of another currency (called variable or quote currency). In other words, one unit of the base currency is equivalent to S unit of the variable currency. It is also important to understand how to read FX market quotes. An FX quote on a currency pair is denoted by the abbreviations of the two currencies: a three-letter code for the base currency, followed by another three-letter code for the variable currency. For example, EURUSD=1.25 means that 1 euro is worth 1.25 dollars. This is equivalent to the quote USDEUR=0.8, which means that 1 dollar is worth 0.8 euros. Usually, it depends on historical market conventions that decide which currency is treated as the base currency in a given FX-pair.
In Chapter 4, Big Data – Advanced Analytics, we have already seen how to download currency rates from the Internet, so we can use what we have learned to check this on real data.
This short code plots the EURUSD and USDEUR rates to the same plot window:
library(Quandl) library(xts) EURUSD <- Quandl("QUANDL/EURUSD", start_date="2014-01-01",end_date="2014-07-01", type="xts") USDEUR <- Quandl("QUANDL/USDEUR", start_date="2014-01-01",end_date="2014-07-01", type="xts") dev.new(width = 15, height = 8) par(mfrow = c(1, 2)) plot(USDEUR) plot(EURUSD)
Here, we can see the result in the following image:
We can also check out the first few lines of the data:
USDEUR[1:5,] Rate High (est) Low (est) 2014-01-01 0.725711 0.73392 0.71760 2014-01-02 0.725238 0.73332 0.71725 2014-01-03 0.727714 0.73661 0.71892 2014-01-06 0.733192 0.00000 0.00000 2014-01-07 0.735418 0.00000 0.00000 EURUSD[1:5,] Rate High (est) Low (est) 2014-01-01 1.37791 0.0000 0.0000 2014-01-02 1.37876 1.3949 1.3628 2014-01-03 1.37434 0.0000 0.0000 2014-01-06 1.36346 1.3799 1.3473 2014-01-07 1.35990 1.3753 1.3447
Here, we have to say something about notations. So far, we have denoted FX rates by S. However, the price of the underlying asset in derivatives pricing is generally denoted by S, regardless of whether it is a stock or a currency. On the other hand, FX rates are usually denoted by X or sometimes by E (both come from the word "exchange"). Furthermore, the strike or exercise price of an option is also abbreviated by X or E. Now, as the reader, you may have some idea about how challenging it is to use a consistent notation system in this chapter, where the underlying might be a stock or a currency as well, and where stock prices, FX rates, and strike prices might appear at the same time. We decided to adopt the notations of R-functions as much as possible, so in this chapter, the notations we will follow are as follows:
- The price of the underlying will always be S, but if it is not necessarily a currency, we will use numeric or alphabetic indices such as S1 or SA
- The strike price will always be X
- The expected value operator will be denoted by E
We strongly recommend that you be careful when reading other literature on this topic, because their notation might differ from ours.