Chapter 2
Mathematical Preliminaries
2.1 THE BLACK–SCHOLES MODEL
We require a model for FX spot rates that allows them to experience stochastic behaviour and strict positivity. These are the same requirements as for equities, in which the Black–Scholes model is applicable. We therefore follow Black and Scholes (1973) and the associated work of Garman and Kohlhagen (1983) as applied to foreign currency options, and describe the spot rate by a geometric Brownian motion
Note that this is far from the only choice for a simple one-factor asset price process – Bachelier (1900) used an arithmetic Brownian motion
to derive closed-form pricing formulae for European options. Though of extremely limited practical use as it doesn’t impose positivity of asset prices,1 Crack (2001) poses this as a three-star exercise (Question 3.20) and provides a comprehensive worked solution in Appendix B.
Other more complex models for the FX spot process can be introduced. We will see more of these in Section 2.10 and subsequent chapters. For now, we introduce the Black–Scholes model.
2.1.1 Assumptions of the Black–Scholes model
The analysis of this chapter presupposes the standard Black–Scholes assumptions, as stated in Section 11.4 of Hull (1997):
1. The spot price St (in domestic currency) of one ...
