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Principles of Financial Economics

Financial economics plays a far more prominent role in the training of economists than it did even a few years ago. This change is generally attributed to the parallel transformation in capital markets that has occurred in recent years. It is true that trillions of dollars of assets are traded daily in ¯nancial markets|for derivative securities like options and futures, for example|that hardly existed a decade ago. However, it is less obvious how important these changes are. Insofar as derivative securities can be valued by arbitrage, such securities only duplicate primary securities | Principles of Financial Economics Stephen F. LeRoy University of California Santa Barbara and Jan Werner University of Minnesota @ March 10 2000 Stephen F. LeRoy and Jan Werner Contents I Equilibrium and Arbitrage 1 1 Equilibrium in Security Markets 3 1.1 Introduction. 3 1.2 Security Markets. 3 1.3 Agents . 5 1.4 Consumption and Portfolio Choice. 6 1.5 First-Order Conditions. 6 1.6 Left and Right Inverses of X . 7 1.7 General Equilibrium. 8 1.8 Existence and Uniqueness of Equilibrium . 8 1.9 Representative Agent Models. 9 2 Linear Pricing 13 2.1 Introduction . 13 2.2 The Law of One Price . 13 2.3 The Payo Pricing Functional . 13 2.4 Linear Equilibrium Pricing . 14 2.5 State Prices in Complete Markets . 15 2.6 Recasting the Optimization Problem . 16 3 Arbitrage and Positive Pricing 21 3.1 Introduction . 21 3.2 Arbitrage and Strong Arbitrage . 21 3.3 A Diagrammatic Representation . 22 3.4 Positivity of the Payo Pricing Functional . 22 3.5 Positive State Prices . 23 3.6 Arbitrage and Optimal Portfolios . 23 3.7 Positive Equilibrium Pricing . 25 4 Portfolio Restrictions 29 4.1 Introduction. 29 4.2 Short Sales Restrictions . 29 4.3 Portfolio Choice under Short Sales Restrictions. 30 4.4 The Law of One Price. 31 4.5 Limited and Unlimited Arbitrage . 32 4.6 Diagrammatic Representation. 32 4.7 Bid-Ask Spreads . 33 4.8 Bid-Ask Spreads in Equilibrium . 33 i ii CONTENTS II Valuation 39 5 Valuation 41 5.1 Introduction. 41 5.2 The Fundamental Theorem of Finance . 41 5.3 Bounds on the Values of Contingent Claims. 42 5.4 The Extension . 45 5.5 Uniqueness of the Valuation Functional . 46 6 State Prices and Risk-Neutral Probabilities 51 6.1 Introduction . 51 6.2 State Prices . 51 6.3 Farkas-Stiemke Lemma. 53 6.4 Diagrammatic Representation. 54 6.5 State Prices and Value Bounds. 54 6.6 Risk-Free Payoffs. 55 6.7 Risk-Neutral Probabilities . 55 7 Valuation under Portfolio Restrictions 61 7.1 Introduction . 61 7.2 Payoff Pricing under Short Sales Restrictions. 61 7.3 State Prices under