Dynamic Asset Pricing Theory

THIS CHAPTER PRESENTS infinite-period analogues of the results of Chapters 2 and 3. Although this setting requires additional technicalities and produces few new insights, it sometimes simplifies results and it serves the large-sample theory of econometrics, which calls for an unbounded number of observations. We start directly with a Markov dynamic programming extension of the finite-horizon results of Chapter 3, and only later consider the implications of no arbitrage or optimality for security prices without using the Markov assumption. Finally, we return to the stationary Markov setting to review briefly the large-sample approach to estimating asset pricing models. Only Sections A and B are essential; the remainder could be skipped on a first reading.

Suppose X = {X₀, X₁X₂, … } is a time-homogeneous Markov chain of shocks valued in a finite set Z = {1, … , k}, defined exactly as in Section C, with the exception that there is an infinite number of time periods. Given a k × k nonnegative matrix q whose rows sum to 1, sources given in the Notes explain the existence of a probability space (Ω, F, P_i), for each initial shock i, satisfying the defining properties P_i(X₀ = i) = 1 and

Click for a larger version

As in Chapter 3, F_t denotes the tribe generated by {X₀, …, X_t}. That is, the source of information is the Markov chain {X_t}. This is the first appearance in the book of a set Ω of states that need not be finite, but because there is only a finite number of events in F_t for each t, most of this chapter can be easily understood without referring to Appendix C for a review of general probability spaces.

-65-