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Lecture Notes in Financial Economicsc by Antonio Mele
London School of Economics & Political Science
May 2011
,Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
I Foundations 14
1 The classic capital asset pricing model 15
1.1 Portfolio selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.1 The wealth constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.2 Portfolio choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.1.3 Without the safe asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.1.4 The market portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2 The CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3 The APT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.1 A first derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.2 The APT with idiosyncratic risk and a large number of assets . . . . . . 25
1.3.3 Empirical evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4 Appendix 1: Some analytical details for portfolio choice . . . . . . . . . . . . . . 27
1.4.1 The primal program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.4.2 The dual program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5 Appendix 2: The market portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.5.1 The tangent portfolio is the market portfolio . . . . . . . . . . . . . . . . 30
1.5.2 Tangency condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.6 Appendix 3: An alternative derivation of the SML . . . . . . . . . . . . . . . . . 32
1.7 Appendix 4: Broader definitions of risk - Rothschild and Stiglitz theory . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2 The CAPM in general equilibrium 36
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
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2.2 The static general equilibrium in a nutshell . . . . . . . . . . . . . . . . . . . . . 36
2.2.1 Walras’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.2 Competitive equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.3 Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 Time and uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Financial assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5 Absence of arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5.1 How to price a financial asset? . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5.2 The Land of Cockaigne . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6 Equivalent martingales and equilibrium . . . . . . . . . . . . . . . . . . . . . . . 49
2.6.1 The rational expectations assumption . . . . . . . . . . . . . . . . . . . . 49
2.6.2 Stochastic discount factors . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6.3 Optimality and equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.7 Consumption-CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7.1 The risk premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7.2 The beta relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.7.3 CCAPM & CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.8 Infinite horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.9 Further topics on incomplete markets . . . . . . . . . . . . . . . . . . . . . . . . 57
2.9.1 Nominal assets and real indeterminacy of the equilibrium . . . . . . . . . 57
2.9.2 Nonneutrality of money . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.10 Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.11 Appendix 2: Proofs of selected results . . . . . . . . . . . . . . . . . . . . . . . . 60
2.12 Appendix 3: The multicommodity case . . . . . . . . . . . . . . . . . . . . . . . 63
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3 Infinite horizon economies 66
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 Consumption-based asset evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.1 Recursive plans: introduction . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.2 The marginalist argument . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.3 Intertemporal elasticity of substitution . . . . . . . . . . . . . . . . . . . 68
3.2.4 Lucas’ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2.5 Arrow-Debreu state prices, the CCAPM and the CAPM . . . . . . . . . 72
3.3 Production: foundational issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3.1 Decentralized economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.2 Centralized economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.4 Stochastic economies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4 Production-based asset pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.4.1 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.4.2 Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.4.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
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3.5 Money, production and asset prices in overlapping generations models . . . . . . 86
3.5.1 Introduction: endowment economies . . . . . . . . . . . . . . . . . . . . . 86
3.5.2 Diamond’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.5.3 Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.5.4 Money in a model with real shocks . . . . . . . . . . . . . . . . . . . . . 93
3.6 Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.6.1 Models with productive capital . . . . . . . . . . . . . . . . . . . . . . . 94
3.6.2 Models with money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.7 Appendix 1: Finite difference equations, with economic applications . . . . . . . 96
3.8 Appendix 2: Neoclassic growth in continuous-time . . . . . . . . . . . . . . . . . 100
3.8.1 Convergence from discrete-time . . . . . . . . . . . . . . . . . . . . . . . 100
3.8.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.9 Appendix 3: Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4 Continuous time models 105
4.1 Lambdas and betas in continuous time . . . . . . . . . . . . . . . . . . . . . . . 105
4.1.1 The pricing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1.2 Expected returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.1.3 Expected returns and risk-adjusted discount rates . . . . . . . . . . . . . 106
4.2 An introduction to continuous time methods in finance . . . . . . . . . . . . . . 108
4.2.1 Partial differential equations and Feynman-Kac probabilistic representa-
tions of the solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2.2 The Girsanov theorem with applications to finance . . . . . . . . . . . . 111
4.3 An introduction to no-arbitrage and equilibrium . . . . . . . . . . . . . . . . . . 113
4.3.1 Self-financed strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3.2 No-arbitrage in Lucas tree . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.3.3 Equilibrium with CRRA . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3.4 Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.3.5 Reflecting barriers and absence of arbitrage . . . . . . . . . . . . . . . . 118
4.4 Martingales and arbitrage in a diffusion model . . . . . . . . . . . . . . . . . . . 119
4.4.1 The information framework . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4.2 Viability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.4.3 Market completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.5 Equilibrium with a representative agent . . . . . . . . . . . . . . . . . . . . . . . 124
4.5.1 Consumption and portfolio choices: martingale approaches . . . . . . . . 124
4.5.2 The older, Merton’s approach: dynamic programming . . . . . . . . . . . 126
4.5.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.5.4 Continuous-time Consumption-CAPM . . . . . . . . . . . . . . . . . . . 128
4.6 Market imperfections and portfolio choice . . . . . . . . . . . . . . . . . . . . . 129
4.7 Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.7.1 Poisson jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.7.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
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