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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 Ⅱ 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

I 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

I 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