An Introduction to Financial Mathematics, 2nd Edition

Chapter 9

Martingales in Discrete Time Markets

Martingale theory unifies many concepts in finance and provides techniques for analyzing more general markets. In this chapter we consider those aspects of (discrete-time) martingale theory that bear directly on these issues and show how martingales arise naturally in the binomial model in both the dividend and non-dividend-paying cases. We also show how martingales may be used to price derivatives in general markets. The material in this chapter foreshadows results on continuous-time martingales discussed in later chapters.

9.1    Discrete Time Martingales

Definition and Examples

Let $\left(\Omega ,ℱ,\mathbb{P}\right)$ be a probability space and let (M_n)_n≥0 be a (finite or infinite) stochastic process adapted to a filtration ...