Martingales and Financial Mathematics in Discrete Time

5Financial Markets

In this chapter, we begin to discuss financial mathematics properly. We define the basic vocabulary and the objects that will be of use in the following chapters. The main references for this chapter and the following two chapters are [VIN 04, HUL 18, SHR 03, SHR 04]. As stated in the Introduction, most texts focus largely or even exclusively on continuous-time models. However, we will be focusing on discrete-time models.

Section 5.1 introduces the concept of a financial asset, section 5.2 introduces the concept of an investment strategy, and section 5.3 introduces the concept of arbitration. This key concept allows us to establish a relationship with the martingale theory, presented in Chapter 4. Section 5.4 introduces a discrete-time financial market model: the Cox, Ross and Rubinstein model. This model will be used to illustrate all the concepts, defined in a generic manner for any discrete financial markets. The chapter concludes with exercises in section 5.5 and practical work in section 5.6, which presents simulations and portfolio optimization using the Cox, Ross and Rubinstein model.

It is assumed here, and in the following, that Ω has a finite cardinal and that ℙ(ω) > 0 for any ω ∈ Ω. These conditions are not strictly necessary; however, they make it possible to considerably simplify all questions of measurability, integrability and the existence of an optimum.

Let N be a time horizon and let (_n)_0≤n≤N be a filtration such that ₀ = {∅, Ω} and F_N = ...