# Dynamic Asset Pricing Theory

By Darrell Duffie | Go to book overview

1
Introduction to State Pricing

THIS CHAPTER INTRODUCES the basic ideas in a finite-state one-period setting. In many basic senses, each subsequent chapter merely repeats this one from a new perspective. The objective is a characterization of security prices in terms of “state prices,” one for each state of the world. The price of a given security is simply the state-price weighted sum of its payoffs in the different states. One can treat a state price as the “shadow price,” or Lagrange multiplier, for wealth contingent on a given state of the world. We obtain a characterization of state prices, first based on the absence of arbitrage, then based on the first-order conditions for optimal portfolio choice of a given agent, and finally from the first-order conditions for Pareto optimality in an equilibrium with complete markets. State prices are connected with the “beta” model for excess expected returns, a special case of which is the Capital Asset Pricing Model (CAPM). Many readers will find this chapter to be a review of standard results. In most cases, here and throughout, technical conditions are imposed that give up much generality so as to simplify the exposition.

A. Arbitrage and State Prices

Uncertainty is represented here by a finite set {1, … , S} of states, one of which will be revealed as true. The N securities are given by an N × S matrix D, with Dij denoting the number of units of account paid by security i in state j. The security prices are given by some q in RN· A portfolio θ ∈ RN has market value q · θ and payoff Dθ in RS. An arbitrage is a portfolio θ in RN with q · θ ≤ 0 and Dθ > 0, or q · θ < 0 and Dθ ≥ 0. An arbitrage is therefore, in effect, a portfolio offering “something for nothing.” Not surprisingly, it will later be shown that an arbitrage is naturally ruled out, and this gives a characterization of security prices as follows. A

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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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