Preface
Philosophy
The Key Topics
The Key Insights
The Martingale Approach
Discrete Versus Continuous-Time
Mean-Variance Efficiency and the Static CAPM
Stochastic Calculus
Traditional Asset Pricing Theory Versus Market Microstructure
Themes
Changes to the Second Edition
Contents
List of Notation
Part I Arbitrage Pricing Theory
Overview
1 Stochastic Processes
1.1 Stochastic Processes
1.2 Stochastic Integration
1.3 Quadratic Variation
1.4 Uniqueness of the Stochastic Integral Representation
1.5 Integration by Parts
1.6 Ito's Formula
1.7 Girsanov's Theorem
1.8 Essential Supremum
1.9 Optional Decomposition
1.10 Martingale Representation
1.11 Equivalent Probability Measures
1.12 Notes
2 The Fundamental Theorems
2.1 The Set-Up
2.1.1 Trading Strategies
2.1.2 Admissibility and Doubling Strategies
2.1.3 Suicide Strategies
2.1.4 The Frictionless Market Assumption
2.2 Change of Numeraire
2.3 Cash Flows
2.3.1 Reinvest in the MMA
2.3.2 Reinvest in the Risky Asset
2.3.3 Summary
2.4 Non-redundant Assets
2.5 The First Fundamental Theorem
2.5.1 No Arbitrage (NA)
2.5.2 No Unbounded Profits with Bounded Risk(NUPBR)
2.5.3 Properties of Dl
2.5.4 No Free Lunch with Vanishing Risk (NFLVR)
2.5.5 The First Fundamental Theorem
2.5.6 Equivalent Local Martingale Measures
2.5.7 The State Price Density
2.6 The Second Fundamental Theorem
2.6.1 Attainable Securities
2.6.2 Complete Markets
2.7 The Third Fundamental Theorem
2.7.1 Risk Neutral Valuation
2.7.2 Synthetic Derivative Construction
2.8 Finite Dimension Brownian Motion Market
2.8.1 The Set-Up
2.8.2 NFLVR
2.8.3 Complete Markets
2.8.4 ND
2.9 Notes
Appendix
3 Asset Price Bubbles
3.1 The Set-Up
3.2 The Market Price and Fundamental Value
3.3 The Asset Price Bubble
3.3.1 Complete Markets
3.3.2 Incomplete Markets
3.4 Theorems Under NFLVR and ND
3.5 Notes
4 Basis Assets, Multiple-Factor Beta Models, and Systematic Risk
4.1 The Set-Up
4.2 Basis Assets
4.3 The Multiple-Factor Beta Model
4.4 Positive Alphas
4.5 The State Price Density
4.6 Arrow Debreu Securities
4.7 Systematic Risk
4.8 Diversification
4.9 Notes
5 The Black Scholes Merton Model
5.1 NFLVR, Complete Markets, and ND
5.2 The BSM Call Option Formula
5.3 The Synthetic Call Option
5.4 Original Derivation of the BSM Formula
5.5 Merton's Structural Model
5.6 Notes
6 The Heath Jarrow Morton Model
6.1 The Set-Up
6.2 Term Structure Evolution
6.3 Arbitrage-Free Conditions
6.4 Examples
6.4.1 Ho and Lee Model
6.4.2 Lognormally Distributed Forward Rates
6.4.3 Vasicek Model
6.4.4 Cox Ingersoll Ross Model
6.4.5 Affine Model
6.5 Forward and Futures Contracts
6.5.1 Forward Contracts
6.5.1.1 The Forward Price Measure
6.5.1.2 An Alternative Characterization of QM
6.5.1.3 Risk Neutral Valuation (Revisited)
6.5.2 Futures Contracts
6.6 The Libor Model
6.7 Notes
7 Reduced Form Credit Risk Models
7.1 The Set-Up
7.2 The Risky Firm
7.3 Existence of an Equivalent Martingale Measure
7.4 Risk Neutral Valuation
7.4.1 Cash Flow 1
7.4.2 Cash Flow 2
7.4.3 Cash Flow 3
7.4.4 Cash Flow 4
7.5 Examples
7.5.1 Coupon Bonds
7.5.2 Credit Default Swaps (CDS)
7.5.3 First-to-Default Swaps
7.6 Notes
8 Incomplete Markets
8.1 The Set-Up
8.2 The Super-Replication Cost
8.3 The Super-Replication Trading Strategy
8.4 The Sub-replication Cost
8.5 Notes
Part II Portfolio Optimization
Overview
9 Utility Functions
9.1 Preference Relations
9.2 State Dependent EU Representation
9.2.1 Rationality Axioms
9.2.2 Additional Properties
9.2.3 Risk Aversion
9.3 Strict Concavity and Risk Aversion
9.3.1 Independent Gambles
9.3.2 Risk Aversion
9.3.3 Characterization Theorems
9.4 Measures of Risk Aversion for Independent Gambles
9.5 State Dependent Utility Functions
9.6 Conjugate Duality
9.7 Reasonable Asymptotic Elasticity
9.8 Differential Beliefs
9.9 Notes
10 Complete Markets (Utility Over Terminal Wealth)
10.1 The Set-Up
10.2 Problem Statement
10.3 Existence of a Solution
10.4 Characterization of the Solution
10.4.1 The Characterization
10.4.2 Summary
10.5 The Shadow Price
10.6 The Local Martingale Deflator
10.7 The Optimal Trading Strategy
10.8 An Example
10.8.1 The Market
10.8.2 The Utility Function
10.8.3 The Optimal Wealth Process
10.8.4 The Optimal Trading Strategy
10.8.5 The Value Function
10.9 Notes
Appendix
Proof of Expression (10.7)
Proof of Expression (10.8)
11 Incomplete Markets (Utility Over Terminal Wealth)
11.1 The Set-Up
11.2 Problem Statement
11.3 Existence of a Solution
11.4 Characterization of the Solution
11.4.1 The Characterization
11.4.2 Summary
11.5 The Shadow Price
11.6 The Supermartingale Deflator
11.7 The Optimal Trading Strategy
11.8 An Example
11.8.1 The Market
11.8.2 The Utility Function
11.8.3 The Optimal Supermartingale Deflator
11.8.4 The Optimal Wealth Process
11.8.5 The Optimal Trading Strategy
11.8.6 The Value Function
11.9 Differential Beliefs
11.10 Notes
Appendix
12 Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth)
12.1 The Set-Up
12.2 Problem Statement
12.3 Existence of a Solution
12.4 Characterization of the Solution
12.4.1 Utility of Consumption (U20)
12.4.1.1 The Solution
12.4.1.2 The Shadow Price
12.4.1.3 The Supermartingale Deflator Process
12.4.1.4 The Optimal Trading Strategy
12.4.1.5 Summary
12.4.2 Utility of Terminal Wealth (U10)
12.4.3 Utility of Consumption and Terminal Wealth
12.5 Notes
Appendix
Part III Equilibrium
Overview
13 Equilibrium
13.1 The Set-Up
13.1.1 Supply of Shares
13.1.2 Traders in the Economy
13.1.3 Aggregate Market Wealth
13.1.4 Trading Strategies
13.1.5 An Economy
13.2 Equilibrium
13.3 Theorems
13.4 Intermediate Consumption
13.4.1 Supply of the Consumption Good
13.4.2 Demand for the Consumption Good
13.4.3 An Economy
13.5 Notes
14 A Representative Trader Economy
14.1 The Aggregate Utility Function
14.2 The Portfolio Optimization Problem
14.3 Representative Trader Economy Equilibrium
14.4 Pareto Optimality
14.5 Existence of an Equilibrium
14.6 Uniqueness of the Equilibrium
14.6.1 Uniqueness of the Equilibrium Price Process
14.6.2 Uniqueness of the Supermartingale Deflators
14.7 Examples
14.7.1 Identical Traders
14.7.2 Logarithmic Preferences
14.8 Intermediate Consumption
14.9 Notes
15 Characterizing the Equilibrium
15.1 The Set-Up
15.2 The Supermartingale Deflator
15.3 Asset Price Bubbles
15.3.1 Complete Markets
15.3.2 Incomplete Markets
15.4 Systematic Risk
15.5 Consumption CAPM
15.6 Intertemporal CAPM
15.7 Intermediate Consumption
15.7.1 Systematic Risk
15.7.2 Consumption CAPM
15.7.3 Intertemporal CAPM
15.8 Notes
16 Market Informational Efficiency
16.1 The Set-Up
16.2 The Definition
16.3 The Theorem
16.4 Information Sets and Efficiency
16.5 Testing for Market Efficiency
16.5.1 Profitable Trading Strategies
16.5.2 Positive Alphas
16.5.3 Asset Price Evolutions
16.6 Random Walks and Efficiency
16.6.1 The Set-Up
16.6.2 Random Walk
16.6.3 Market Efficiency Random Walk
16.6.4 Random Walk Market Efficiency
16.7 Notes
17 Epilogue (The Fundamental Theorems and the CAPM)
17.1 The Fundamental Theorems
17.1.1 The First Fundamental Theorem
17.1.2 The Second Fundamental Theorem
17.1.3 Risk Neutral Valuation
17.1.4 Finite State Space Market
17.2 Basis Assets, Multi-Factor Beta Models, and Systematic Risk
17.3 Utility Functions
17.4 Portfolio Optimization
17.4.1 The Dual Problem
17.4.2 The Primal Problem
17.4.3 The Optimal Trading Strategy
17.5 Beta Model (Revisited)
17.6 The Efficient Frontier
17.6.1 The Solution (Revisited)
17.6.2 Summary
17.6.3 The Risky Asset Frontier and Efficient Frontier
17.7 Equilibrium
17.8 Notes
Appendix
Part IV Trading Constraints
Overview
18 The Trading Constrained Market
18.1 The Set-Up
18.2 Trading Constraints
18.3 Support Functions
18.4 Examples (Trading Constraints and Their Support Functions)
18.4.1 No Trading Constraints
18.4.2 Prohibited Short Sales
18.4.3 No Borrowing
18.4.4 Margin Requirements
18.5 Wealth Processes
19 Arbitrage Pricing Theory
19.1 No Unbounded Profits with Bounded Risk (NUPBRC)
19.2 No Free Lunch with Vanishing Risk (NFLVRC)
19.3 Asset Price Bubbles
19.4 Systematic Risk
20 The Auxiliary Markets
20.1 The Auxiliary Markets
20.2 The Normalized Auxiliary Markets
21 Super- and Sub-Replication
21.1 The Set-Up
21.1.1 Auxiliary Market (0,0)
21.1.2 Auxiliary Markets (ฮฝ0,ฮฝ)
21.2 Local Martingale Deflators
21.3 Wealth Processes Revisited
21.4 Super-Replication
21.5 Sub-Replication
22 Portfolio Optimization
22.1 The Set-Up
22.2 Wealth Processes (Revisited)
22.3 The Optimization Problem
22.4 Existence of a Solution
22.5 Characterization of the Solution
22.6 The Shadow Price of the Budget Constraint
22.7 The Supermartingale Deflator
22.8 The Shadow Prices of the Trading Constraints
22.9 Asset Price Bubbles
22.10 Systematic Risk
23 Equilibrium
23.1 The Set-Up
23.2 Representative Trader
23.2.1 The Solution
23.2.2 Buy and Hold Trading Strategies
23.3 Existence of Equilibrium
23.4 Characterization of Equilibrium
References
Index