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.

**Definition 41.1** (Discrete Market). Let *T* be a positive real number and *N* be a positive integer. Let 0 = *t*_{0} < *t*_{1} < ··· < *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 *n*th period [*t*_{n−1,} *t*_{n}].

For *j* ≥ 1, the *j*th entry *S*^{(j)}_{n} is the price at time *t*_{n} of the *j*th risky asset and follows the following stochastic dynamics:

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