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Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

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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 < ··· < tN = T. Let (Ω, , P) be a probability space, where Ω is finite, = 2Ω, and P{ω} > 0 for every ω Ω. A discrete market is a (d + 1)-dimensional stochastic process

equation

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

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

equation

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

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

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