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PART Ⅰ DISCRETE- TIME MODELS1

1 Introduction to State Pricing3

A Arbitrage and State Prices3

B Risk-Neutral Probabilities4

C Optimality and Asset Pricing5

D Efficiency and Complete Markets8

E Optimality and Representative Agents8

F State-Price Beta Models11

Exercises12

Notes17

2 The Basic Multiperiod Model21

A Uncertainty21

B Security Markets22

C Arbitrage,State Prices,and Martingales22

D Individual Agent Optimality24

E Equilibrium and Pareto Optimality26

F Equilibrium Asset Pricing27

G Arbitrage and Martingale Measures28

H Valuation of Redundant Securities30

I American Exercise Policies and Valuation31

J Is Early Exercise Optimal?35

Exercises37

Notes45

3 The Dynamic Programming Approach49

A The Bellman Approach49

B First-Order Bellman Conditions50

C Markov Uncertainty51

D Markov Asset Pricing52

E Security Pricing by Markov Control52

F Markov Arbitrage-Free Valuation55

G Early Exercise and Optimal Stopping56

Exercises58

Notes63

4 The Infinite-Horizon Setting65

A Markov Dynamic Programming65

B Dynamic Programming and Equilibrium69

C Arbitrage and State Prices70

D Optimality and State Prices71

E Method-of-Moments Estimation73

Exercises76

Notes78

PART Ⅱ CONTINUOUS-TIME MODELS81

5 The Black-Scholes Model83

A Trading Gains for Brownian Prices83

B Martingale Trading Gains85

C Ito Prices and Gains86

D Ito's Formula87

E The Black-Scholes Option-Pricing Formula88

F Black-Scholes Formula:First Try90

G The PDE for Arbitrage-Free Prices92

H The Feynman-Kac Solution93

I The Multidimensional Case94

Exercises97

Notes100

6 State Prices and Equivalent Martingale Measures101

A Arbitrage101

B Numeraire Invariance102

C State Prices and Doubling Strategies103

D Expected Rates of Return106

E Equivalent Martingale Measures108

F State Prices and Martingale Measures110

G Girsanov and Market Prices of Risk111

H Black-Scholes Again115

I Complete Markets116

J Redundant Security Pricing119

K Martingale Measures from No Arbitrage120

L Arbitrage Pricing with Dividends123

M Lumpy Dividends and Term Structures125

N Martingale Measures,Infinite Horizon127

Exercises128

Notes131

7 Term-Structure Models135

A The Term Structure136

B One-Factor Term-Structure Models137

C The Gaussian Single-Factor Models139

D The Cox-Ingersoll-Ross Model141

E The Affine Single-Factor Models142

F Term-Structure Derivatives144

G The Fundamental Solution146

H Multifactor Models148

I Affine Term-Structure Models149

J The HJM Model of Forward Rates151

K Markovian Yield Curves and SPDEs154

Exercises155

Notes161

8 Derivative Pricing167

A Martingale Measures in a Black Box167

B Forward Prices169

C Futures and Continuous Resettlement171

D Arbitrage-Free Futures Prices172

E Stochastic Volatility174

F Option Valuation by Transform Analysis178

G American Security Valuation182

H American Exercise Boundaries186

I Lookback Options189

Exercises191

Notes196

9 Portfolio and Consumption Choice203

A Stochastic Control203

B Merton's Problem206

C Solution to Merton's Problem209

D The Infinite-Horizon Case213

E The Martingale Formulation214

F Martingale Solution217

G A Generalization220

H The Utility-Gradient Approach221

Exercises224

Notes232

10 Equilibrium235

A The Primitives235

B Security-Spot Market Equilibrium236

C Arrow-Debreu Equilibrium237

D Implementing Arrow-Debreu Equilibrium238

E Real Security Prices240

F Optimality with Additive Utility241

G Equilibrium with Additive Utility243

H The Consumption-Based CAPM245

I The CIR Term Structure246

J The CCAPM in Incomplete Markets249

Exercises251

Notes255

11 Corporate Securities259

A The Black-Scholes-Merton Model259

B Endogenous Default Timing262

C Example:Brownian Dividend Growth264

D Taxes and Bankruptcy Costs268

E Endogenous Capital Structure269

F Technology Choice271

G Other Market Imperfections272

H Intensity-Based Modeling of Default274

I Risk-Neutral Intensity Process277

J Zero-Recovery Bond Pricing278

K Pricing with Recovery at Default280

L Default-Adjusted Short Rate281

Exercises282

Notes288

12 Numerical Methods293

A Central Limit Theorems293

B Binomial to Black-Scholes294

C Binomial Convergence for Unbounded Derivative Payoffs297

D Discretization of Asset Price Processes297

E Monte Carlo Simulation299

F Efficient SDE Simulation300

G Applying Feynman-Kac302

H Finite-Difference Methods302

I Term-Structure Example306

J Finite-Difference Algorithms with Early Exercise Options309

K The Numerical Solution of State Prices310

L Numerical Solution of the Pricing Semi-Group313

M Fitting the Initial Term Structure314

Exercises316

Notes317

APPENDIXES321

A Finite-State Probability323

B Separating Hyperplanes and Optimality326

C Probability329

D Stochastic Integration334

E SDE,PDE,and Feynman-Kac340

F Ito's Formula with Jumps347

G Utility Gradients351

H Ito's Formula for Complex Functions355

I Counting Processes357

J Finite-Difference Code363

Bibliography373

Symbol Glossary445

Author Index447

Subject Index457