Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

CHAPTER 41

DISCRETE-TIME MODELS

In discrete-time models, it is assumed that trading takes place at discrete time points. In this chapter, we present discrete-time models of financial markets. In particular, we will introduce the binomial model.

41.1 Basic Concepts and Facts

Definition 41.1 (Discrete Market). Let T be a positive real number and N be a positive integer. Let 0 = t₀ < t₁ < ··· < t_N = T. Let (Ω, , P) be a probability space, where Ω is finite, = 2^Ω, and P{ω} > 0 for every ω Ω. A discrete market is a (d + 1)-dimensional stochastic process

defined on the probability space (Ω, , P).

The first entry S⁽⁰⁾_n is the price at time t_n of a riskless asset and is deterministic, that is,

where r_n > − 1 is the risk-free interest rate in the nth period [t_n−1, t_n].

For j ≥ 1, the jth entry S^(j)_n is the price at time t_n of the jth risky asset and follows the following stochastic dynamics: