Martingales and Financial Mathematics in Discrete Time

7American Options

We will now look at a second family of options called American options. The chief difference between these and European options lies in the date of exercise: American options can be exercised at any point of time between the initial date and the maturity date. Section 7.1 defines American options and provides a recursive formula to calculate their value. Section 7.2 offers a generic study of the recursive problem posed earlier. Section 7.3 applies the general theory to specific case of American options. Next, section 7.4 examines the particular case of the Cox, Ross and Rubinstein model. Finally, sections 7.5 and 7.6 offer exercises and practical work, respectively.

It is still assumed, throughout this chapter, that Ω has a finite cardinal and that ℙ(ω) > 0 for any ω ∈ Ω. Let N be a time horizon and let (_n)_0≤n≤N be a filtration such that ₀ = {∅, Ω} and _N = . In all that follows, it is assumed that the market is viable and complete and ℙ^∗ denotes the unique, risk-neutral probability.

7.1. Definition

Let us start by defining an American option.

DEFINITION 7.1.– An