This first part of the book takes place in a discrete-time setting with a discrete set of states. This should ease the development of intuition for the models to be found in Part II. The three pillars of the theory, arbitrage, optimality, and equilibrium, are developed repeatedly in different settings. Chapter 1 is the basic single-period model. Chapter 2 extends the results of Chapter 1 to many periods. Chapter 3 specializes Chapter 2 to a Markov setting and illustrates dynamic programming as an alternate solution technique. The Ho-and-Lee and Black-Derman-Toy term-structure models are included as exercises. Chapter 4 is an infinite-horizon counterpart to Chapter 3 that has become known as the Lucas model.
The focus of the theory is the notion of state prices, which specify the price of any security as the state-price weighted sum or expectation of the security’s state-contingent dividends. In a finite-dimensional setting, there exist state prices if and only if there is no arbitrage. The same fact is true in infinite-dimensional settings under mild technical regularity conditions. Given an agent’s optimal portfolio choice, a state-price vector is given by that agent’s utility gradient. In an equilibrium with Pareto optimality, a state-price vector is likewise given by a representative agent’s utility gradient at the economy’s aggregate consumption process.