Asset Pricing Theory

Costis Skiadas is the Harold L. Stuart Professor of Finance at Northwestern University's Kellogg School of Management.

• Preface xi Notation and Conventions xv PART ONE: SINGLE-PERIOD ANALYSIS CHAPTER ONE: Financial Market and Arbitrage 3 1.1 Market and Arbitrage 3 1.2 Present Value and State Prices 6 1.3 Market Completeness and Dominant Choice 9 1.4 Probabilistic Representations of Value 12 1.5 Financial Contracts and Portfolios 15 1.6 Returns 17 1.7 Trading Constraints 19 1.8 Exercises 22 1.9 Notes 27 CHAPTER TWO: Mean-Variance Analysis 29 2.1 Market and Inner Product Structure 29 2.2 Minimum-Variance Cash Flows 32 2.3 Minimum-Variance Returns 35 2.4 Beta Pricing 37 2.5 Sharpe Ratios 40 2.6 Mean-Variance Efficiency 43 2.7 Factor Pricing 46 2.8 Exercises 49 2.9 Notes 54 CHAPTER THREE: Optimality and Equilibrium 55 3.1 Preferences, Optimality and State Prices 55 3.2 Equilibrium 58 3.3 Effective Market Completeness 62 3.4 Representative-Agent Pricing 65 3.4.1 Aggregation Based on Scale Invariance 66 3.4.2 Aggregation Based on Translation Invariance 69 3.5 Utility 71 3.5.1 Compensation Function Construction of Utilities 72 3.5.2 Additive Utilities 76 3.6 Utility and Individual Optimality 79 3.7 Utility and Allocational Optimality 83 3.8 Exercises 87 3.9 Notes 91 CHAPTER FOUR: Risk Aversion 94 4.1 Absolute and Comparative Risk Aversion 94 4.2 Expected Utility 99 4.3 Expected Utility and Risk Aversion 103 4.3.1 Comparative Risk Aversion 103 4.3.2 Absolute Risk Aversion 105 4.4 Risk Aversion and Simple Portfolio Choice 109 4.5 Coefficients of Risk Aversion 112 4.6 Simple Portfolio Choice for Small Risks 116 4.7 Stochastic Dominance 120 4.8 Exercises 124 4.9 Notes 129 PART TWO: DISCRETE DYNAMICS CHAPTER FIVE: Dynamic Arbitrage Pricing 135 5.1 Dynamic Market and Present Value 135 5.1.1 Time-Zero Market and Present-Value Functions 135 5.1.2 Dynamic Market and Present-Value Functions 138 5.2 Financial Contracts 142 5.2.1 Basic Arbitrage Restrictions and Trading Strategies 142 5.2.2 Budget Equations and Synthetic Contracts 146 5.3 Probabilistic Representations of Value 150 5.3.1 State-Price Densities 150 5.3.2 Equivalent Martingale Measures 154 5.4 Dominant Choice and Option Pricing 159 5.4.1 Dominant Choice 160 5.4.2 Recursive Value Maximization 164 5.4.3 Arbitrage Pricing of Options 166 5.5 State-Price Dynamics 170 5.6 Market Implementation 174 5.7 Markovian Pricing 178 5.8 Exercises 183 5.9 Notes 193 CHAPTER SIX: Dynamic Optimality and Equilibrium 195 6.1 Dynamic Utility 195 6.2 Expected Discounted Utility 199 6.3 Recursive Utility 202 6.4 Basic Properties of Recursive Utility 206 6.4.1 Comparative Risk Aversion 206 6.4.2 Utility Gradient Density 208 6.4.3 Concavity 211 6.5 Scale/Translation Invariance 213 6.5.1 Scale-Invariant Kreps-Porteus Utility 213 6.5.2 Translation-Invariant Kreps-Porteus Utility 217 6.6 Equilibrium Pricing 219 6.6.1 Intertemporal Marginal Rate of Substitution 220 6.6.2 State Pricing with SI Kreps-Porteus Utility 221 6.6.3 State Pricing with TI Kreps-Porteus Utility 227 6.7 Optimal Consumption and Portfolio Choice 229 6.7.1 Generalities 230 6.7.2 Scale-Invariant Formulation 232 6.7.3 Translation-Invariant Formulation 240 6.8 Exercises 248 6.9 Notes 252 PART THREE: MATHEMATICAL BACKGROUND APPENDIX A: Optimization Principles 259 A.1 Vector Space 259 A.2 Inner Product 261 A.3 Norm 264 A.4 Continuity 266 A.5 Compactness 268 A.6 Projections 270 A.7 Supporting Hyperplanes 274 A.8 Global Optimality Conditions 276 A.9 Local Optimality Conditions 278 A.10 Exercises 281 A.11 Notes 284 APPENDIX B: Discrete Stochastic Analysis 285 B.1 Events, Random Variables, Expectation 285 B.2 Algebras and Measurability 289 B.3 Conditional Expectation 292 B.4 Stochastic Independence 296 B.5 Filtration, Stopping Times and Stochastic Processes 299 B.6 Martingales 304 B.7 Predictable Martingale Representation 308 B.8 Change of Measure and Martingales 312 B.9 Markov Processes 317 B.10 Exercises 320 B.11 Notes 324 Bibliography 327 Index 341

"Costis Skiadas has hit a grand-slam with Asset Pricing Theory which fills a great void. It will speak to you in a well-designed, and thoughtful manner encouraging you to read a high-level and rigorous development of the subject regardless of your age, profession or position as economists, mathematicians, financial engineers, and physicists. I am adding it to my 'must read list' for my students and associates. I predict that Asset Pricing Theory will establish itself as a standard reference for many years to come, and this is the quality I admire--a quality that can only be born from experience. Read this book if you want to lead an organization, or lead the way."--Current Engineering Practice "I am sure any ambitious student who has read it will be drawn into the field immediately... I like the book very much and would recommend it for use in any serious asset pricing theory subject."--Qi Zeng, Economic Record