Deltas and Market Conventions
While the Black–Scholes model introduced in Chapter 2 is the industry benchmark and an important market reference, it has several deficiencies when it comes to describing realistic FX markets. The assumption of constant interest rates in foreign and domestic currencies is inadequate, as is the assumption of a single volatility σ sufficient to price options of different maturities and strikes correctly. We have already shown how Black–Scholes term structure models can remedy some of these deficiencies, but the inability of a Black–Scholes model equipped solely with a single volatility to price options of various strikes cannot be remedied by the imposition of a term structure.
However, Black–Scholes prices provide a natural starting point for a fuller and more accurate description of the volatility surfaces encountered in the FX markets, which can be used in developing more advanced models, such as the ones discussed in this work.
Given a Black–Scholes price for an option, one can calculate the change in that price for infinitesimal changes in the underlying spot rate (or forward rate for that matter). This gives us the notion of the option delta, which is normally thought of as the instantaneous sensitivity of the price to infinitesimal changes in the price of the underlying asset. In most asset classes other than FX, the delta is perfectly straightforward. We shall see that this is not true in FX – the explanation is simple, even though ...