Correlation and FX Options
9.1 PRELIMINARY CONSIDERATIONS
It is not within the scope of this book to extend the analysis to those contracts usually embedded in structured financial products and that consider currencies as an asset class. Such contracts are typically based on baskets of currencies often with averaging (Asian) features, and they are similar to those structures commonly traded in the equity markets. In most cases, a strong dependence on the correlation amongst all the involved pairs marks these kinds of options, and a great deal of theory has been developed for their pricing and (to a lesser extent, to be frank) hedging.
We will not dwell on the subject of the currencies as an asset class, but nevertheless correlation in the FX market deserves special attention for the tight links that can be established amongst pairs, at least those built from a triplet of currencies. As usual, care should be taken because some concepts may be misleading, as we will see in what follows. We start by analysing how correlation can be dealt with in a BS world, and then we expand the results to an economy where the volatility smiles appear in the market.
9.2 CORRELATION IN THE BS SETTING
Assume we are in the BS economy described in Chapter 2, and that three pairs are traded: XXX, YYY and ZZZ. It is possible to build three pairs: XXXYYY, ZZZYYY and XXXZZZ.145 The dynamics are given by the following SDEs:
The single superscripts indicate the currency by ...