Introduction 1
1 Introduction and Management Summary 1
2 Why We Have Written this Book 2
3 Why You Should Read this Book 3
4 The Audience 3
5 The Structure of this Book 4
6 What this Book Does Not Cover 5
7 Credits 6
8 Code 6
PART I FINANCIAL MARKETS AND POPULAR MODELS
1 Financial Markets – Data, Basics and Derivatives 9
1.1 Introduction and Objectives 9
1.2 Financial Time-Series, Statistical Properties of Market Data and Invariants 10
1.2.1 Real World Distribution 15
1.3 Implied Volatility Surfaces and Volatility Dynamics 17
1.3.1 Is There More than just a Volatility? 19
1.3.2 Implied Volatility 22
1.3.3 Time-Dependent Volatility 22
1.3.4 Stochastic Volatility 23
1.3.5 Volatility from Jumps 23
1.3.6 Traders’ Rule of Thumb 24
1.3.7 The Risk Neutral Density 24
1.4 Applications 26
1.4.1 Asset Allocation 26
1.4.2 Pricing, Hedging and Risk Management 27
1.5 General Remarks on Notation 30
1.6 Summary and Conclusions 31
1.7 Appendix – Quotes 32
2 Diffusion Models 35
2.1 Introduction and Objectives 35
2.2 Local Volatility Models 35
2.2.1 The Bachelier and the Black–Scholes Model 37
2.2.2 The Hull–White Model 40
2.2.3 The Constant Elasticity of Variance Model 46
2.2.4 The Displaced Diffusion Model 50
2.2.5 CEV and DD Models 53
2.3 Stochastic Volatility Models 54
2.3.1 Pricing European Options 55
2.3.2 Risk Neutral Density 56
2.3.3 The Heston Model (and Extensions) 57
2.3.4 The SABR Model 67
2.3.5 SABR – Further Remarks 73
2.4 Stochastic Volatility and Stochastic Rates Models 81
2.4.1 The Heston–Hull–White Model 81
2.5 Summary and Conclusions 90
3 Models with Jumps 93
3.1 Introduction and Objectives 93
3.2 Poisson Processes and Jump Diffusions 94
3.2.1 Poisson Processes 94
3.2.2 The Merton Model 95
3.2.3 The Bates Model 99
3.2.4 The Bates–Hull–White Model 104
3.3 Exponential L´evy Models 105
3.3.1 The Variance Gamma Model 107
3.3.2 The Normal Inverse Gaussian Model 112
3.4 Other Models 118
3.4.1 Exponential L´evy Models with Stochastic Volatility 122
3.4.2 Stochastic Clocks 122
3.5 Martingale Correction 129
3.6 Summary and Conclusions 134
4 Multi-Dimensional Models 137
4.1 Introduction and Objectives 137
4.2 Multi-Dimensional Diffusions 137
4.2.1 GBM Baskets 137
4.2.2 Libor Market Models 139
4.3 Multi-Dimensional Heston and SABR Models 141
4.3.1 Stochastic Volatility Models 141
4.4 Parameter Averaging 143
4.4.1 Applications to CMS Spread Options 144
4.5 Markovian Projection 159
4.5.1 Baskets with Local Volatility 162
4.5.2 Markovian Projection on Local Volatility and Heston Models 162
4.5.3 Markovian Projection onto DD SABR Models 164
4.6 Copulae 172
4.6.1 Measures of Concordance and Dependency 174
4.6.2 Examples 175
4.6.3 Elliptical Copulae 175
4.6.4 Archimedean Copulae 177
4.6.5 Building New Copulae from Given Copulae 179
4.6.6 Asymmetric Copulae 179
4.6.7 Applying Copulae to Option Pricing 180
4.6.8 Applying Copulae to Asset Allocation 180
4.7 Multi-Dimensional Variance Gamma Processes 187
4.8 Summary and Conclusions 193
PART II NUMERICAL METHODS AND RECIPES
5 Option Pricing by Transform Techniques and Direct Integration 197
5.1 Introduction and Objectives 197
5.2 Fourier Transform 197
5.2.1 Discrete Fourier Transform 199
5.2.2 Fast Fourier Transform 200
5.3 The Carr–Madan Method 202
5.3.1 The Optimal α 207
5.4 The Lewis Method 210
5.4.1 Application to Other Payoffs 214
5.5 The Attari Method 215
5.6 The Convolution Method 216
5.7 The Cosine Method 220
5.8 Comparison, Stability and Performance 228
5.8.1 Other Issues 233
5.9 Extending the Methods to Forward Start Options 235
5.9.1 Forward Characteristic Function for L´evy Processes and CIR Time Change 238
5.9.2 Forward Characteristic Function for L´evy Processes and Gamma-OU Time Change 239
5.9.3 Results 242
5.10 Density Recovery 245
5.11 Summary and Conclusions 250
6 Advanced Topics Using Transform Techniques 253
6.1 Introduction and Objectives 253
6.2 Pricing Non-Standard Vanilla Options 253
6.2.1 FFT with Lewis Method 254
6.3 Bermudan and American Options 254
6.3.1 The Convolution Method 257
6.3.2 The Cosine Method 258
6.3.3 Numerical Results 266
6.3.4 The Fourier Space Time-Stepping 270
6.4 The Cosine Method and Barrier Options 277
6.5 Greeks 278
6.6 Summary and Conclusions 287
7 Monte Carlo Simulation and Applications 289
7.1 Introduction and Objectives 289
7.2 Sampling Diffusion Processes 289
7.2.1 The Exact Scheme 290
7.2.2 The Euler Scheme 290
7.2.3 The Predictor-Corrector Scheme 290
7.2.4 The Milstein Scheme 291
7.2.5 Implementation and Results 291
7.3 Special Purpose Schemes 292
7.3.1 Schemes for the Heston Model 294
7.3.2 Unbiased Scheme for the SABR Model 300
7.4 Adding Jumps 313
7.4.1 Jump Models – Poisson Processes 313
7.4.2 Fixed Grid Sampling (FGS) 315
7.4.3 Stochastic Grid Sampling (SGS) 315
7.4.4 Simulation – L´evy Models 322
7.4.5 Schemes for L´evy Models with Stochastic Volatility 330
7.5 Bridge Sampling 339
7.6 Libor Market Model 346
7.7 Multi-Dimensional L´evy Models 351
7.8 Copulae 352
7.8.1 Distributional Sampling Approach (DSA) 353
7.8.2 Conditional Sampling Approach (CSA) 356
7.8.3 Simulation from Other Copulae 358
7.9 Summary and Conclusions 359
8 Monte Carlo Simulation – Advanced Issues 361
8.1 Introduction and Objectives 361
8.2 Monte Carlo and Early Exercise 361
8.2.1 Longstaff–Schwarz Regression 362
8.2.2 Policy Iteration Methods 369
8.2.3 Upper Bounds 374
8.2.4 Problems of the Method 376
8.2.5 Financial Examples and Numerical Results 378
8.3 Greeks with Monte Carlo 382
8.3.1 The Finite Difference Method (FDM) 383
8.3.2 The Pathwise Method 385
8.3.3 The Affine Recursion Problem (ARP) 389
8.3.4 Adjoint Method 391
8.3.5 Bermudan ARPs 393
8.4 Euler Schemes and General Greeks 396
8.4.1 SDE of Diffusions 396
8.4.2 Approximation by Euler Schemes 397
8.4.3 Approximating General Greeks Using ARP 397
8.4.4 Greeks 404
8.5 Application to Trigger Swap 407
8.5.1 Mathematical Modelling 408
8.5.2 Numerical Results 410
8.5.3 The Likelihood Ratio Method (LRM) 413
8.5.4 Likelihood Ratio for Finite Differences – Proxy Simulation 416
8.5.5 Numerical Results 419
8.6 Summary and Conclusions 433
8.7 Appendix – Trees 434
9 Calibration and Optimization 435
9.1 Introduction and Objectives 435
9.2 The Nelder–Mead Method 437
9.2.1 Implementation 442
9.2.2 Calibration Examples 444
9.3 The Levenberg–Marquardt Method 449
9.3.1 Implementation 453
9.3.2 Calibration Examples 455
9.4 The L-BFGS Method 460
9.4.1 Implementation 463
9.4.2 Calibration Examples 464
9.5 The SQP Method 468
9.5.1 The Modified and Globally Convergent SQP Iteration 473
9.5.2 Implementation 475
9.5.3 Calibration Examples 477
9.6 Differential Evolution 482
9.6.1 Implementation 487
9.6.2 Calibration Examples 488
9.7 Simulated Annealing 493
9.7.1 Implementation 497
9.7.2 Calibration Examples 500
9.8 Summary and Conclusions 505
10 Model Risk – Calibration, Pricing and Hedging 507
10.1 Introduction and Objectives 507
10.2 Calibration 508
10.2.1 Similarities – Heston and Bates Models 508
10.2.2 Parameter Stability 511
10.3 Pricing Exotic Options 521
10.3.1 Exotic Options and Different Models 528
10.4 Hedging 528
10.4.1 Hedging – The Basics 531
10.4.2 Hedging in Incomplete Markets 533
10.4.3 Discrete Time Hedging 541
10.4.4 Numerical Examples 544
10.5 Summary and Conclusions 550
PART III IMPLEMENTATION, SOFTWARE DESIGN AND MATHEMATICS
11 Matlab – Basics 553
11.1 Introduction and Objectives 553
11.2 General Remarks 553
11.3 Matrices, Vectors and Cell Arrays 556
11.3.1 Matrices and Vectors 556
11.3.2 Cell Arrays 562
11.4 Functions and Function Handles 564
11.4.1 Functions 564
11.4.2 Function Handles 567
11.5 Toolboxes 570
11.5.1 Financial 570
11.5.2 Financial Derivatives 571
11.5.3 Fixed-Income 571
11.5.4 Optimization 573
11.5.5 Global Optimization 577
11.5.6 Statistics 578
11.5.7 Portfolio Optimization 581
11.6 Useful Functions and Methods 589
11.6.1 FFT 589
11.6.2 Solving Equations and ODE 589
11.6.3 Useful Functions 591
11.7 Plotting 593
11.7.1 Two-Dimensional Plots 593
11.7.2 Three-Dimensional Plots – Surfaces 595
11.8 Summary and Conclusions 597
12 Matlab – Object Oriented Development 599
12.1 Introduction and Objectives 599
12.2 The Matlab OO Model 599
12.2.1 Classes 599
12.2.2 Handling Classes in Matlab 606
12.2.3 Inheritance, Base Classes and Superclasses 607
12.2.4 Handle and Value Classes 609
12.2.5 Overloading 610
12.3 A Model Class Hierarchy 611
12.4 A Pricer Class Hierarchy 613
12.5 An Optimizer Class Hierarchy 618
12.6 Design Patterns 620
12.6.1 The Builder Pattern 621
12.6.2 The Visitor Pattern 624
12.6.3 The Strategy Pattern 626
12.7 Example – Calibration Engine 629
12.7.1 Calibrating a Data Set or a History 631
12.8 Example – The Libor Market Model and Greeks 634
12.8.1 An Abstract Class for LMM Derivatives 634
12.8.2 A Class for Bermudan Swaptions 637
12.8.3 A Class for Trigger Swaps 639
12.9 Summary and Conclusions 641
13 Math Fundamentals 643
13.1 Introduction and Objectives 643
13.2 Probability Theory and Stochastic Processes 643
13.2.1 Probability Spaces 644
13.2.2 Random Variables 644
13.2.3 Important Results 645
13.2.4 Distributions 649
13.2.5 Stochastic Processes 654
13.2.6 L´evy Processes 655
13.2.7 Stochastic Differential Equations 660
13.3 Numerical Methods for Stochastic Processes 665
13.3.1 Random Number Generation 665
13.3.2 Methods for Computing Variates 670
13.4 Basics on Complex Analysis 671
13.4.1 Complex Numbers 671
13.4.2 Complex Differentiation and Integration along Paths 672
13.4.3 The Complex Exponential and Logarithm 673
13.4.4 The Residual Theorem 674
13.5 The Characteristic Function and Fourier Transform 675
13.6 Summary and Conclusions 679
List of Figures 681
List of Tables 691
Bibliography 695
Index 705
