The Delta of an option will be defined in Chapter 2, where the BS model is presented.
This statement will become clearer with the analysis in Chapter 3.
The definition for each of the options we mention below will be given in the chapters devoted to their analysis.
Vega will be defined in Chapter 2.
This structure is described later on in this chapter.
More details about the definition of bet options can be found in Chapter 6.
It is worth noticing here that in market lore traders refer to options struck at a level implying a 25% Delta, without considering the sign, indistinctly as a 25 or 0.25 or finally 25% Delta call or put.
The approach we describe in this section is based on the works by Harrison and Pliska [34, 35], and we refer to them also for a more formal treatment of the foundations.
Hence we are assuming that the interest rate is a deterministic (possibly time-dependent) variable.
For the sake of clarity we note that, at any time t, one unit of foreign currency equals St units of domestic currency, so that in equation (2.3) interest is instantaneously converted into the domestic currency.
We are assuming O( S, t) is a function sufficiently differentiable for the application of Itô’s lemma.
It is easy to check that the strategy is actually self-financing (see, for example, Bergman ), although the property is also manifest by the choice of α and β.
The evaluation of Φ(x ) can be performed by means of numerical integration or ...