# Dynamic Asset Pricing Theory

By Darrell Duffie | Go to book overview

10
Equilibrium

THIS CHAPTER REVIEWS security market equilibrium in a continuous-time setting and derives several implications for security prices and expected returns. These include Breeden’s consumption-based capital asset pricing model (in both complete- and incomplete-market settings) as well as the Cox-Ingersoll-Ross model of the term structure.

A. The Primitives

As usual, we let B = (B1, , Bd) denote a standard Brownian motion in Rd on a probability space (Ω, F, P), and let F = {FT : t0} denote the standard filtration of B. The consumption space is the set L of adapted processes satisfying

for some fixed time horizon T > 0.

There are m agents. Agent i is defined by a nonzero consumption endowment process ei in the set L+ nonnegative processes in L, and by a strictly increasing utility function Ui : L+R.

As in Section 6L, a cumulative-dividend process is a finite-variance process of the form C = Z + V W, where Z = θ dB for some θL(B), and where V and W are increasing adapted right-continuous processes. For any time t, the jump ΔCtCtCt– represents the lump-sum payment at time t. For example, if the security is a unit zero-coupon bond that matures at time τ, then Ct = 0, t < τ, and Ct = 1, tτ. By convention, any dividend process C satisfies C0– = C0 = 0. For example, a dividendrate process δ in L defines the cumulative-dividend process C = VW with

There are N + 1 securities, numbered 0 through N, defined by a cumulative-dividend pro cess D = (D(0), D(1), , D(N)).

Mainly for expositional reasons, we assume that the given dividend processes are measured in nominal units of account, and will shortly

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