第四章

Table of Contents
Preface
I Single-Period Models
1 Utility Functions and Risk Aversion Coefficients
1.1 Uniqueness of Utility Functions
1.2 Concavity and Risk Aversion
1.3 Coefficients of Risk Aversion
1.4 Risk Aversion and Risk Premia
1.5 Constant Absolute Risk Aversion
1.6 Constant Relative Risk Aversion
1.7 Linear Risk Tolerance
1.8 Conditioning and Aversion to Noise
1.9 Notes and References
Exercises
2 Portfolio Choice and Stochastic Discount Factors
2.1 The First-Order Condition
2.2 Stochastic Discount Factors
2.3 A Single Risky Asset
2.4 Linear Risk Tolerance
2.5 Multiple Asset CARA-Normal Example
2.6 Mean-Variance Preferences
2.7 Complete Markets
2.8 Beginning-of-Period Consumption
2.9 Time-Additive Utility
2.10 Notes and References
Exercises
3 Equilibrium and Efficiency
3.1 Pareto Optima
3.2 Social Planner's Problem
3.3 Pareto Optima and Sharing Rules
3.4 Competitive Equilibria
3.5 Complete Markets
3.6 Linear Risk Tolerance
3.7 Beginning-of-Period Consumption 1
3.8 Notes and References
Exercises
4 Arbitrage and Stochastic Discount Factors
4.1 Fundamental Theorem on Existence of SDF's
4.2 Law of One Price and Stochastic Discount Factors
4.3 Risk Neutral Probabilities
4.4 Projecting SDF's onto the Asset Span
4.5 Projecting onto a Constant and the Asset Span
4.6 Hansen-Jagannathan Bound with a Risk-Free Asset
4.7 Hansen-Jagannathan Bound with No Risk-Free Asset
4.8 Hilbert Spaces and Gram-Schmidt Orthogonalization
4.9 Notes and References Exercises
5 Mean-Variance Analysis
5.1 The Calculus Approach
5.2 Two-Fund Spanning
5.3 The Mean-Standard Deviation Trade-Off
5.4 GMV Portfolio and Mean-Variance Efficiency
5.5 Calculus Approach with a Risk-Free Asset
5.6 Two-Fund Spanning Again
5.7 Orthogonal Projections and Frontier Returns
5.8 Risk-Free Return Proxies
5.9 Inefficiency of ~Rp
5.10 Hansen-Jagannathan Bound with a Risk-Free Asset
5.11 Frontier Returns and Stochastic Discount Factors
5.12 Separating Distributions
5.13 Notes and References
Exercises
6 Beta Pricing Models
6.1 Beta Pricing
6.2 Single-Factor Models with Returns as Factors
6.3 The Capital Asset Pricing Model
6.4 Returns and Excess Returns as Factors
6.5 Projecting Factors on Returns and Excess Returns
6.6 Beta Pricing and Stochastic Discount Factors
6.7 Arbitrage Pricing Theory
6.8 Notes and References
Exercises
7 Representative Investors
7.1 Pareto Optimality Implies a Representative Investor
7.2 Linear Risk Tolerance
7.3 Consumption-Based Asset Pricing
7.4 Pricing Options
7.5 Notes and References
Exercises
II Dynamic Models
8 Dynamic Securities Markets
8.1 The Portfolio Choice Problem
8.2 Stochastic Discount Factor Processes
8.3 Self-Financing Wealth Processes
8.4 The Martingale Property
8.5 Transversality Conditions and Ponzi Schemes
8.6 The Euler Equation
8.7 Arbitrage and the Law of One Price
8.8 Risk Neutral Probabilities
8.9 Complete Markets
8.10 Portfolio Choice in Complete Markets
8.11 Competitive Equilibria
8.12 Notes and References
Exercises
9 Portfolio Choice by Dynamic Programming
9.1 Introduction to Dynamic Programming
9.2 Bellman Equation for Portfolio Choice
9.3 The Envelope Condition
9.4 Maximizing CRRA Utility of Terminal Wealth
9.5 CRRA Utility with Intermediate Consumption
9.6 CRRA Utility with an Infinite Horizon
9.7 Notes and References
Exercises
10 Conditional Beta Pricing Models
10.1 From Conditional to Unconditional Models
10.2 The Conditional CAPM
10.3 The Consumption-Based CAPM
10.4 The Intertemporal CAPM
10.5 An Approximate CAPM
10.6 Notes and References
Exercises
11 Some Dynamic Equilibrium Models
11.1 Representative Investors
11.2 Valuing the Market Portfolio
11.3 The Risk-Free Return
11.4 The Equity Premium Puzzle
11.5 The Risk-Free Rate Puzzle
11.6 Uninsurable Idiosyncratic Income Risk
11.7 External Habits
11.8 Notes and References
Exercises
12 Brownian Motion and Stochastic Calculus
12.1 Brownian Motion
12.2 Quadratic Variation
12.3 Ito Integral
12.4 Local Martingales and Doubling Strategies
12.5 Ito Processes
12.6 Asset and Portfolio Returns
12.7 Martingale Representation Theorem
12.8 Ito's Formula: Version I
12.9 Geometric Brownian Motion
12.10 Covariations of Ito Processes
12.11 Ito's Formula: Version II
12.12 Conditional Variances and Covariances
12.13 Transformations of Models
12.14 Notes and References
Exercises
13 Continuous-Time Securities Markets and SDF Processes
13.1 Dividend-Reinvested Asset Prices
13.2 Securities Markets
13.3 Self-Financing Wealth Processes
13.4 Conditional Mean-Variance Frontier
13.5 Stochastic Discount Factor Processes
13.6 Properties of SDF Processes
13.7 Sufficient Conditions for MW to be a Martingale
13.8 Valuing Consumption Streams
13.9 Risk Neutral Probabilities
13.10 Complete Markets
13.11 SDF Processes without a Risk-Free Asset
13.12 Inflation and Foreign Exchange
13.13 Notes and References
Exercises
14 Continuous-Time Portfolio Choice and Beta Pricing
14.1 The Static Budget Constraint
14.2 Complete Markets
12 CONTENTS
14.3 Constant Capital Market Line
14.4 Dynamic Programming Example
14.5 General Markovian Portfolio Choice
14.6 The CCAPM
14.7 The ICAPM
14.8 The CAPM
14.9 Infinite-Horizon Dynamic Programming
14.10 Dynamic Programming with CRRA Utility
14.11 Verification Theorem
14.12 Notes and References
Exercises
III Derivative Securities
15 Option Pricing
15.1 Introduction to Options
15.2 Put-Call Parity and Option Bounds
15.3 SDF Processes
15.4 Changes of Measure
15.5 Market Completeness
15.6 The Black-Scholes Formula
15.7 Delta Hedging
15.8 The Fundamental PDE
15.9 American Options
15.10 Smooth Pasting
15.11 European Options on Dividend-Paying Assets
15.12 Notes and References
Exercises
16 Forwards, Futures, and More Option Pricing
16.1 Forward Measures
16.2 Forward Contracts
16.3 Futures Contracts
16.4 Exchange Options
16.5 Options on Forwards and Futures
16.6 Dividends and Random Interest Rates
16.7 Implied Volatilities and Local Volatilities
16.8 Stochastic Volatility
16.9 Notes and References
Exercises
17 Term Structure Models
17.1 Vasicek Model
17.2 Cox-Ingersoll-Ross Model
17.3 Multi-Factor CIR Models
17.4 Affine Models
17.5 Completely Affine Models
17.6 Quadratic Models
17.7 Forward Rates
17.8 Fitting the Yield Curve
17.9 Heath-Jarrow-Morton Models
17.10 Notes and References
Exercises
IV Topics
18 Heterogeneous Priors
18.1 State-Dependent Utility Formulation
18.2 Representative Investors in Complete Single-Period Markets
18.3 Representative Investors in Complete Dynamic Markets
18.4 Short Sales Constraints and Biased Prices
18.5 Speculative Trade
18.6 Notes and References
Exercises
19 Asymmetric Information
19.1 The No-Trade Theorem
19.2 Normal-Normal Updating
19.3 A Fully Revealing Equilibrium
19.4 Noise Trading and Partially Revealing Equilibria
19.5 A Model with a Large Number of Investors
19.6 The Kyle Model
19.7 The Kyle Model in Continuous Time
19.8 Notes and References
Exercises
20 Alternative Preferences in Single-Period Models
20.1 The Ellsberg Paradox
20.2 The Sure Thing Principle
20.3 Multiple Priors and Max-Min Utility
20.4 Non-Additive Set Functions
20.5 The Allais Paradox
20.6 The Independence Axiom
20.7 Betweenness Preferences
20.8 Rank-Dependent Preferences
20.9 First-Order Risk Aversion
20.10 Framing and Loss Aversion
20.11 Prospect Theory
20.12 Notes and References
Exercises
21 Alternative Preferences in Dynamic Models
21.1 Recursive Preferences
21.2 Portfolio Choice with Epstein-Zin-Weil Utility
21.3 A Representative Investor with Epstein-Zin-Weil Utility
21.4 Internal Habits
21.5 Linear Internal Habits in Complete Markets
21.6 A Representative Investor with an Internal Habit
21.7 Keeping/Catching Up with the Joneses
21.8 Ambiguity Aversion in Dynamic Models
21.9 Notes and References
Exercises
22 Production Models
22.1 Discrete-Time Model
22.2 Marginal q
22.3 Costly Reversibility
22.4 Project Risk and Firm Risk
22.5 Irreversibility and Options
22.6 Irreversibility and Perfect Competition
22.7 Irreversibility and Risk
22.8 Irreversibility and Perfect Competition: An Example
22.9 Notes and References
Exercises
Appendices
A Some Probability and Stochastic Process Theory
A.1 Random Variables
A.2 Probabilities
A.3 Distribution Functions and Densities
A.4 Expectations
A.5 Convergence of Expectations
A.6 Interchange of Differentiation and Expectation
A.7 Random Vectors
A.8 Conditioning
A.9 Independence
A.10 Equivalent Probability Measures
A.11 Filtrations, Martingales, and Stopping Times
A.12 Martingales under Equivalent Measures
A.13 Local Martingales
A.14 The Usual Conditions

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