Security markets : stochastic models / Darrell Duffie.

Format:

Book

Author/Creator:

Duffie, Darrell.

Series:

Economic theory, econometrics, and mathematical economics

Economic theory, econometrics, and mathematical economics.

Language:

English

Subjects (All):

Securities--Mathematical models.

Securities.

Physical Description:

xx, 358 pages : illustrations ; 24 cm.

Place of Publication:

Boston : Academic Press, [1988]

Summary:

This is a graduate level work covering the economic principles of security markets. Interested readers include students and researchers in economics and finance, as well as financial analysts following the latest theoretical developments in capital asset pricing.

Contents:

A. Market Equilibrium 1

B. Equilibrium under Uncertainty 2

C. Security-Spot Market Equilibrium 2

D. State Pricing Model of Securities 5

E. Binomial Arbitrage Pricing Model of Securities 6

F. Capital Asset Pricing Model 10

G. Stochastic Control Pricing Model 12

H. The Potential-Price Matrix 15

I. Ito's Lemma

A Simple Case 16

J. Continuous-Time Portfolio Control 18

K. Black-Scholes Option Pricing Formula 20

L. Representative Agent Asset Pricing 22

Chapter I. Static Economies 27

1. The Geometry of Choices and Prices 28

A. Vector Spaces 28

B. Normed Spaces 29

C. Convexity and Cones 30

D. Function Spaces 30

E. Topology 30

F. Duality 31

G. Dual Representation 32

2. Preferences 35

A. Preference Relations 35

B. Preference Continuity and Convexity 35

C. Utility Functions 35

D. Utility Representation 36

E. Quasi-Concave Utility 36

F. Monotonicity 36

G. Non-Satiation 37

3. Market Equilibrium 39

A. Primitives of an Economy 39

B. Equilibria 39

C. Exchange and Net Trade Economies 40

D. Production and Exchange Equilibria 41

E. Equilibrium and Efficiency 42

F. Efficiency and Equilibrium 42

G. Existence of Equilibria 44

4. First Probability Concepts 50

A. Probability Spaces 50

B. Random Variables and Distributions 51

C. Measurability, Topology, and Partitions 51

D. Almost Sure Events and Versions 52

E. Expectation and Integration 53

F. Distribution and Density Functions 54

5. Expected Utility 56

A. Von-Neumann-Morgenstern and Savage Models of Preferences 56

B. Expected Utility Representation 56

C. Preferences over Probability Distributions 57

D. Mixture Spaces and the Independence Axiom 57

E. Axioms for Expected Utility 59

6. Special Choice Spaces 61

A. Banach Spaces 61

B. Measurable Function Spaces 61

C. L[superscript q] Spaces 61

D. L[superscript [infinity] Spaces 62

E. Riesz Representation 62

F. Continuity of Positive Linear Functionals 63

G. Hilbert Spaces 63

7. Portfolios 67

A. Span and Vector Subspaces 67

B. Linearly Independent Bases 68

C. Equilibrium on a Subspace 68

D. Security Market Equilibria 69

E. Constrained Efficiency 70

8. Optimization Principles 74

A. First Order Necessary Conditions 74

B. Saddle Point Theorem 76

C. Kuhn-Tucker Theorem 77

D. Superdifferentials and Maxima 78

9. Second Probability Concepts 82

A. Changing Probabilities 82

B. Changing Information 83

C. Conditional Expectation 83

D. Properties of Conditional Expectation 84

E. Expectation in General Spaces 85

F. Jensen's Inequality 85

G. Independence and The Law of Large Numbers 86

10. Risk Aversion 90

A. Defining Risk Aversion 90

B. Risk Aversion and Concave Expected Utility 90

C. Risk Aversion and Second Order Stochastic Dominance 91

11. Equilibrium in Static Markets Under Uncertainty 93

A. Markets for Assets with a Variance 93

B. Beta Models: Mean-Covariance Pricing 93

C. The CAPM and APT Pricing Approaches 94

D. Variance Aversion 95

E. The Capital Asset Pricing Model 95

F. Proper Preferences 96

G. Existence of Equilibria 98

Chapter II. Stochastic Economies 103

12. Event Tree Economies 104

A. Event Trees 104

B. Security and Spot Markets 105

C. Trading Strategies 107

D. Equilibria 107

E. Marketed Subspaces and Tight Markets 108

F. Dynamic and Static Equilibria 109

G. Dynamic Spanning and Complete Markets 109

H. A Security Valuation Operator 111

I. Dynamically Complete Markets Equilibria 111

J. Dynamically Incomplete Markets Equilibria 113

K. Generic Existence of Equilibria with Real Securities 113

L. Arbitrage Security Valuation and State Prices 115

13. A Dynamic Theory of the Firm 118

A. Stock Market Equilibria 118

B. An Example 119

C. Security Trading by Firms 121

D. Invariance of Stock Values to Security Trading by Firms 123

E. Modigliani-Miller Theorem 123

F. Invariance of Firm's Total Market Value Process 124

G. Firms Issue and Retire Securities 124

H. Tautology of Complete Information Models 126

I. The Goal of the Firm 127

14. Stochastic Processes 130

A. The Information Filtration 130

B. Informationally Adapted Processes 131

C. Information Generated by Processes and Event Trees 133

D. Technical Continuity Conditions 134

E. Martingales 135

F. Brownian Motion and Poisson Processes 135

G. Stopping Times, Local Martingales, and Semimartingales 136

15. Stochastic Integrals and Gains From Security Trade 138

A. Discrete-Time Stochastic Integrals 138

B. Continuous-Time Primitives 140

C. Simple Continuous-Time Integration 141

D. The Stochastic Integral 142

E. General Stochastic Integrals 144

F. Martingale Multiplicity 146

G. Stochastic Integrals and Changes of Probability 146

16. Stochastic Equilibria 148

A. Stochastic Economies 148

B. Dynamic Spanning 150

C. Existence of Equilibria 151

17. Transformations to Martingale Gains from Trade 155

A. Introduction: The Finite-Dimensional Case 155

B. Dividend and Price Processes 156

C. Self-Financing Trading Strategies 157

D. Representation of Implicit Market Values 157

E. Equivalent Martingale Measures 159

F. Choice of Numeraire 162

G. A Technicality 163

H. Generalization to Many Goods 164

I. Generalization to Consumption Through Time 165

Chapter III. Discrete-Time Asset Pricing 169

18. Markov Processes and Markov Asset Valuation 170

A. Markov Chains 170

B. Transition Matrices 170

C. Metric and Borel Spaces 171

D. Conditional and Marginal Distributions 173

E. Markov Transition 173

F. Transition Operators 175

G. Chapman-Kolmogorov Equation 175

H. Sub-Markov Transition 176

I. Markov Arbitrage Valuation 177

J. Abstract Markov Process 179

19. Discrete-Time Markov Control 182

A. Robinson Crusoe Example 183

B. Dynamic Programming with a Finite State Space 184

C. Borel-Markov Control Models 187

D. Existence of Stationary Markov Optimal Control 189

E. Measurable Selection of Maxima 190

F. Bellman Operator 190

G. Contraction Mapping and Fixed Points 191

H. Bellman Equation 192

I. Finite Horizon Markov Control 194

J. Stochastic Consumption and Investment Control 195

20. Discrete-Time Equilibrium Pricing 202

A. Markov Exchange Economies 202

B. Optimal Portfolio and Consumption Policies 203

C. Conversion to a Borel-Markov Control Problem 204

D. Markov Equilibrium Security Prices 205

E. Relaxation of Short-Sales Constraints 208

F. Markov Production Economies 210

G. A Central Planning Stochastic Production Problem 210

H. Market Decentralization of a Growth Economy 211

I. Markov Stock Market Equilibrium 213

Chapter IV. Continuous-Time Asset Pricing 221

21. An Overview of the Ito Calculus 222

A. Ito Processes and Integrals 222

B. Ito's Lemma 223

C. Stochastic Differential Equations 224

D. Feynman-Kac Formula 225

E. Girsanov's Theorem: Change of Probability and Drift 228

22. The Black-Scholes Model of Security Valuation 232

A. Binomial Pricing Model 233

B. Black-Scholes Framework 235

C. Reduction to a Partial Differential Equation 237

D. The Black-Scholes Option Pricing Formula 239

E. An Application of the Feynman-Kac Formula 239

F. An Extension 240

G. Central Limit Theorems 243

H. Limiting Binomial Formula 245

I. Uniform Integrability 247

J. An Application of Donsker's Theorem 248

K. An Application of Girsanov's Theorem 253

23. An Introduction to the Control of Ito Processes 266

A. Sketch of Bellman's Equation 266

B. Regularity Requirements 269

C. Formal Statement of Bellman's Equation 269

24. Portfolio Choice with I.I.D.

Returns 274

A. The Portfolio Control Problem 274

B. The Solution 276

25. Continuous-Time Equilibrium Asset Pricing 291

A. The Setting 292

B. Definition of Equilibrium 293

C. Regularity Conditions 294

D. Equilibrium Theorem 295

E. Conversion to Consumption Numeraire 297

F. Equilibrium Interest Rates 298

G. The Consumption-Based Capital Asset Pricing Model 299

H. The Cox-Ingersoll-Ross Term Structure Model 301.

Notes:

Includes indexes.

Bibliography: pages 323-344.

ISBN:

012223345X

OCLC:

17209521