Table of Contents

Preface xiii

PART I DISCRETE-TIME MODELS 1

1. Introduction to State Pricing 3

A. Arbitrage and State Prices 3

B. Risk-Neutral Probabilities 4

C. Optimality and Asset Pricing 5

D. Efficiency and Complete Markets 8

E. Optimality and Representative Agents 8

F. State-Price Beta Models 11

Exercises 12

Notes 17

2. The Basic Multiperiod Model 21

A. Uncertainty 21 B Security Markets 22

C. Arbitrage, State Prices, and Martingales 22

D. Individual Agent Optimality 24

E. Equilibrium and Pareto Optimality 26

F. Equilibrium Asset Pricing 27

G. Arbitrage and Martingale Measures 28

H. Valuation of Redundant Securities 30

I. American Exercise Policies and Valuation 31

J. Is Early Exercise Optimal? 35

Exercises 37

Notes 45

3 The Dynamic Programming Approach 49

A. The Bellman Approach 49

B. First-Order Bellman Conditions 50

C. Markov Uncertainty .51

D. Markov Asset Pricing 52

E. Security Pricing by Markov Control 52

F. Markov Arbitrage-Free Valuation 55

G Early Exercise and Optimal Stopping 56

Exercises 58

Notes 63

4. The Infinite-Horizon Setting 65

A. Markov Dynamic Programming .65

B. Dynamic Programming and Equilibrium.69

C. Arbitrage and State Prices 70

D. Optimality and State Prices.71

E. Method-of-Moments Estimation .73

Exercises 76

Notes 78

PART 11 CONTINUOUS-TIME MODELS 81

5. The Black-Scholes Model 83

A. Trading Gains for Brownian Prices 83

B. Martingale Trading Gains 85

C. Ito Prices and Gains 86

D. Ito's Formula 87

E. The Black-Scholes Option-Pricing Formula 88

F. Black-Scholes Formula: First Try 90

G. The PDE for Arbitrage-Free Prices 92

H. The Feynman-Kac Solution 93

I. The Multidimensional Case 94

Exercises 97

Notes 100

6. State Prices and Equivalent Martingale Measures 101

A. Arbitrage 101

B. Numeraire Invariance 102

C. State Prices and Doubling Strategies 103

D. Expected Rates of Return 106

E. Equivalent Martingale Measures 108

F. State Prices and Martingale Measures 110

G. Girsanov and Market Prices of Risk 111

H. Black-Scholes Again 115

I. Complete Markets 116

J. Redundant Security Pricing 119

K. Martingale Measures from No Arbitrage 120

L. Arbitrage Pricing with Dividends 123

M. Lumpy Dividends and Term Structures 125

N. Martingale Measures, Infinite Horizon 127

Exercises 128

Notes 131

7. Term-Structure Models 135

A. The Term Structure 136

B. One-Factor Term-Structure Models 137

C. The Gaussian Single-Factor Models 139

D. The Cox-Ingersoll-Ross Model 141

E. The Affine Single-Factor Models 142

F. Term-Structure Derivatives 144

G. The Fundamental Solution 146

H. Multifactor Models 148

1. Affine Term-Structure Models 149

J. The HJM Model of Forward Rates 151

K. Markovian Yield Curves and SPDEs 154

Exercises 155

Notes 161

8. Derivative Pricing 167

A. Martingale Measures in a Black Box 167

B. Forward Prices 169

C. Futures and Continuous Resettlement 171

D. Arbitrage-Free Futures Prices 172

E. Stochastic Volatility 174

F. Option Valuation by Transform Analysis 178

G. American Security Valuation 182

H. American Exercise Boundaries 186

1. Lookback Options 189

Exercises 191

Notes 196

9. Portfolio and Consumption Choice 203

A. Stochastic Control 203

B. Merton's Problem 206

C. Solution to Merton's Problem 209

D. The Infinite-Horizon Case 213

E. The Martingale Formulation 214

F. Martingale Solution 217

G. A Generalization 220

H. The Utility-Gradient Approach 221

Exercises 224

Notes 232

10. Equilibrium 235

A. The Primitives 235

B. Security-Spot Market Equilibrium 236

C. Arrow-Debreu Equilibrium 237

D. Implementing Arrow-Debreu Equilibrium 238

E. Real Security Prices 240

F. Optimality with Additive Utility 241

G. Equilibrium with Additive Utility 243

H. The Consumption-Based CAPM 245

I. The CIR Term Structure 246

J. The CCAPM in Incomplete Markets 249

Exercises 251

Notes 255

11. Corporate Securities 259

A. The Black-Scholes-Merton Model 259

B. Endogenous Default Timing 262

C. Example: Brownian Dividend Growth 264

D. Taxes and Bankruptcy Costs 268

E. Endogenous Capital Structure 269

F. Technology Choice 271

G. Other Market Imperfections 272

H. Intensity-Based Modeling of Default 274

I. Risk-Neutral Intensity Process 277

J. Zero-Recovery Bond Pricing 278

K. Pricing with Recovery at Default 280

L. Default-Adjusted Short Rate 281

Exercises 282

Notes 288

12. Numerical Methods 293

A. Central Limit Theorems 293

B. Binomial to Black-Scholes 294

C. Binomial Convergence for Unbounded Derivative Payoffs 297

D. Discretization of Asset Price Processes 297

E. Monte Carlo Simulation 299

F. Efficient SDE Simulation 300

G. Applying Feynman-Kac 302

H. Finite-Difference Methods 302

I. Term-Structure Example 306

J. Finite-Difference Algorithms with Early Exercise Options 309

K. The Numerical Solution of State Prices 310

L. Numerical Solution of the Pricing Semi-Group 313

M. Fitting the Initial Term Structure 314

Exercises 316

Notes 317

APPENDIXES 321

A. Finite-State Probability 323

B. Separating Hyperplanes and Optimality 326

C. Probability 329

D. Stochastic Integration 334

E. SDE, PDE, and Feynman-Kac 340

F. Ito's Formula with jumps 347

G. Utility Gradients 351

H. Ito's Formula for Complex Functions 355

I. Counting Processes 357

J. Finite-Difference Code 363

Bibliography 373

Symbol Glossary 445

Author Index 447

Subject Index 457