# 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 : t ≥ 0} 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 ΔCtCt – Ct– 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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Dynamic Asset Pricing Theory

• Title Page iii
• Contents vii
• Preface xiii
• I - Discrete-Time Models 1
• 1 - Introduction to State Pricing 3
• 2 - The Basic Multiperiod Model 21
• 3 - The Dynamic Programming Approach 49
• 4 - The Infinite-Horizon Setting 65
• II - Continuous-Time Models 81
• 5 - The Black-Scholes Model 83
• 6 - State Prices and Equivalent Martingale Measures 101
• 7 - Term-Structure Models 135
• 8 - Derivative Pricing 167
• 9 - Portfolio and Consumption Choice 203
• 10 - Equilibrium 235
• 11 - Corporate Securities 259
• 12 - Numerical Methods 293
• Appendixes 321
• Bibliography 373
• Symbol Glossary 445
• Author Index 447
• Subject Index 457
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