CHAPTER 1
Arbitrage and Pricing 1
The Pricing Problem 1
Arbitrage 2
State Prices 2
Present Value as an Expectation of Future Values 4
CHAPTER 2
Fundamentals of Stochastic Calculus 9
Basic Definitions 9
Probability Space 9
Sample Space 10
Filtration and the Revelation of Information 10
Probability Measure 11
Random Variables 12
Stochastic Process 12
Measurable Stochastic Process 12
Adapted Process 13
Conditional Expectation 13
Martingales 13
Wiener Process 13
First Variation of a Differentiable Function 15
First Variation of the Wiener Process 15
Second Variation of a Differentiable Function 15
Second Variation of the Wiener Process 16
Products of Infinitesimal Increments of Wiener Processes 16
Stochastic Integrals 18
Mean Square Limit 18
Ito Integral 19
Properties of the Ito Integral 19
Ito Processes 21
xiv CONTENTS
Multidimensional Processes 22
Multidimensional Wiener Processes 22
Multidimensional Ito Processes 23
Ito’s Lemma 24
Multidimensional Ito’s Lemma 25
Stochastic Differential Equations 27
Moments of SDE Solutions 28
SDE Commonly Used in Finance 29
The Markov Property of Solutions of SDE 30
The Feynman-Kac Theorem 31
Measure Changes 33
Girsanov Theorem 35
Martingale Representation Theorem 36
Processes with Jumps 36
The Poisson Jump Model 37
Defining a Pure Jump Process 37
Defining a Jump-Diffusion Process 38
Ito’s Lemma in the Presence of Jumps 38
CHAPTER 3
Pricing in Continuous Time 41
One-Dimensional Risk Neutral Pricing 42
Multidimensional Market Model 47
Extension to Other Normalizing Assets 51
Deriving Risk-Neutralized Processes 53
The Pricing Equation 56
European Derivatives 57
Hedging Portfolio Approach 58
Feynman-Kac Approach 60
The Pricing Equation in the Presence of Jumps 62
An Application of Jump Processes: Credit Derivatives 63
Defaultable Bonds 65
Full Protection Credit Put 66
American Derivatives 67
Relationship between European and American Derivatives 68
American Options as Dynamic Optimization Problems 69
Conditions at Exercise Boundaries 70
Linear Complementarity Formulation of
American Option Pricing 72
Path Dependency 73
Discrete Sampling of Path Dependency 74
Contents xv
CHAPTER 4
Scenario Generation 77
Scenario Nomenclature 78
Scenario Construction 79
Exact Solution Advancement 80
Sampling from the Joint Distribution of the Random Process 81
Generating Scenarios by Numerical Integration of the
Stochastic Differential Equations 86
Brownian Bridge 93
Brownian Bridge Construction 94
Generating Scenarios with Brownian Bridges 95
Joint Normals by the Choleski Decomposition Approach 100
Quasi-Random Sequences 102
The Concept of Discrepancy 109
Discrepancy and Convergence: The Koksma-Hlawka
Inequality 109
Proper Use of Quasi-Random Sequences 110
Interest Rate Scenarios 113
HJM for Instantaneous Forwards 113
LIBOR Rate Scenarios 115
Principal Component Analysis to Approximate Correlation
Matrices 118
CHAPTER 5
European Pricing with Simulation 121
Roles of Simulation in Finance 121
Monte Carlo in Pricing 122
Monte Carlo in Risk Management 123
The Workflow of Pricing with Monte Carlo 124
Estimators 125
Estimation of the Mean 125
Estimation of the Variance 127
Simulation Efficiency 130
Increasing Simulation Efficiency 131
Antithetic Variates 133
Efficiency of Antithetic Variates 134
Control Variates 135
Efficiency of Control Variates 137
Case Study: Application of Control Variates to Discretely
Sampled Step-Up Barrier Options 137
xvi CONTENTS
Importance Sampling 140
Optimal Importance Density 142
Applying the Girsanov Theorem to Importance Sampling:
European Call Option 143
Importance Sampling by Direct Modeling of the Importance
Density: Credit Put 149
Moment Matching 152
Stratification 155
Stratified Standard Normals in One Dimension 159
Latin Hypercube Sampling 161
Case Study: Latin Hypercube Sampling Applied to Exotic
Basket Option 162
Effect of Discretization on Accuracy and the Emergence of
Computational Barriers 164
Discretization Error for the Log-Normal Process 167
Discretization Error and Computational Barriers
for a European Call 171
CHAPTER 6
Simulation for Early Exercise 177
The Basic Difficulty in Pricing Early Exercise with Simulation 177
Simulation Applied to Early Exercise 179
Dealing with Estimator Bias 180
Path-Bundling Algorithms 183
State Stratification Algorithms 185
Simulated Recombining Lattices 186
Simulated Bushy Trees 187
Least Squares Monte Carlo 188
Least Squares and Conditional Expectation 189
LSMC Algorithm 192
The Moneyness Criterion 196
Implementation Considerations 197
Case Study 1: Bermudan Call on Best-of-Three Assets 198
Specification 198
Basis Functions 199
The Benchmark 200
Numerical Results 201
Case Study 2: Bermudan Swaption 202
Specification 203
Scenario Generation 203
Basis Functions 204
Contents xvii
The Benchmark 205
Numerical Results 206
CHAPTER 7
Pricing with Finite Differences 207
Fundamentals 207
Finite Difference Strategy 210
Constructing Finite Difference Space Discretizations 212
Implementation of Space Discretization 213
The Mechanics of Finite Differences 215
Stability and Accuracy Analysis 217
Analysis of Specific Algorithms 224
Time Advancement and Linear Solvers 228
Direct Solvers 229
Iterative Solvers 231
Finite Difference Approach for Early Exercise 233
The Linear Complementarity Problem 233
Boundary Conditions 237
Implementation of Boundary Conditions 238
Solving Alternative PDEs at Boundaries 240
Barriers 242
Coordinate Transformation Versus Process Transformation 243
Discrete Sampling of Barriers 247
Coordinate Transformations 252
Implementation of Coordinate Transformations 254
Discrete Events and Path Dependency 259
Displacement Shocks 260
Path Dependency and Discrete Sampling 262
Trees, Lattices, and Finite Differences 267
Connection Between the CRR Binomial Tree and Finite
Differences 268
Connection Between the Jarrow and Rudd Binomial Tree and
Finite Differences 270
Implications of the Correspondence Between Trees and Finite
Differences 272
BIBLIOGRAPHY 273
INDEX 277