# Dynamic Asset Pricing Theory

By Darrell Duffie | Go to book overview

2
The Basic Multiperiod Model

THIS CHAPTER EXTENDS the results of Chapter 1 on arbitrage, optimality, and equilibrium to a multiperiod setting. A connection is drawn between state prices and martingales for the purpose of representing security prices. The exercises include the consumption-based capital asset pricing model and the multiperiod “binomial” option pricing model.

A. Uncertainty

As in Chapter 1, there is some finite set, say Ω, of states. In order to handle multiperiod issues, however, we will treat uncertainty a bit more formally as a probability space (Ω, F, P), with F denoting the tribe of subsets of Ω that are events (and can therefore be assigned a probability), and with P a probability measure assigning to any event B in F its probability P(B). Those not familiar with the definition of a probability space can consult Appendix A. The terms σ-algebra” and σ-field,” among others, are often used in place of the word “tribe.”

There are T + 1 dates: 0, 1, , T. At each of these, a tribe FtF denotes the set of events corresponding to the information available at time t. In effect, an event B in Ft is known at time t to be true or false. (A definition of tribes in terms of “partitions” of Ω is given in Exercise 2.11.) We adopt the usual convention that FtFs whenever t s, meaning that events are never “forgotten.” For simplicity, we also take it that every event in F0 has probability 0 or 1, meaning roughly that there is no information at time t = 0. Taken altogether, the filtration F = {F0, , FT} represents how information is revealed through time. For any random variable Y, we let Et (Y) = E (Y | Ft) denote the conditional expectation of Y given Ft. (Appendix A provides definitions of random variables and of conditional expectation.) An adapted process is a sequence X = {X0, , XT} such that

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