Notation xix Executive Overview xxiii
0 My First Monte Carlo Application One-Factor Problems 1
0.1 Introduction and objectives 1
0.2 Description of the problem 1
0.3 Ordinary differential equations (ODE) 2
0.4 Stochastic differential equations (SDE) and their solution 3
0.5 Generating uniform and normal random numbers 4
0.6 The Monte Carlo method 8
0.7 Calculating sensitivities 9
0.8 The initial C++ Monte Carlo framework: hierarchy and paths 10
0.9 The initial C++ Monte Carlo framework: calculating option price 19
0.10 The predictor-corrector method: a scheme for all seasons? 23
0.11 The Monte Carlo approach: caveats and nasty surprises 24
0.12 Summary and conclusions 25
0.13 Exercises and projects 25
PART I FUNDAMENTALS
1 Mathematical Preparations for the Monte Carlo Method 31
1.1 Introduction and objectives 31
1.2 Random variables 31
1.3 Discrete and continuous random variables 34
1.4 Multiple random variables 37
1.5 A short history of integration 38
1.6 Ï-algebras, measurable spaces and measurable functions 39
1.7 Probability spaces and stochastic processes 40
1.8 The Ito stochastic integral 41
1.9 Applications of the Lebesgue theory 43
1.10 Some useful inequalities 45
1.11 Some special functions 46
1.12 Convergence of function sequences 48
1.13 Applications to stochastic analysis 49
1.14 Summary and conclusions 50
1.15 Exercises and projects 50
2 The Mathematics of Stochastic Differential Equations (SDE) 53
2.1 Introduction and objectives 53
2.2 A survey of the literature 53
2.3 Mathematical foundations for SDEs 55
2.4 Motivating random (stochastic) processes 59
2.5 An introduction to one-dimensional random processes 59
2.6 Stochastic differential equations in Banach spaces: prologue 62
2.7 Classes of SIEs and properties of their solutions 62
2.8 Existence and uniqueness results 63
2.9 A special SDE: the Ito equation 64
2.10 Numerical approximation of SIEs 66
2.11 Transforming an SDE: the Ito formula 68
2.12 Summary and conclusions 69
2.13 Appendix: proof of the Banach fixed-point theorem and some applications 69
2.14 Exercises and projects 71
3 Alternative SDEs and Toolkit Functionality 73
3.1 Introduction and objectives 73
3.2 Bessel processes 73
3.3 Random variate generation 74
3.4 The exponential distribution 74
3.5 The beta and gamma distributions 75
3.6 The chi-squared, Student and other distributions 79
3.7 Discrete variate generation 79
3.8 The Fokker-Planck equation 80
3.9 The relationship with PDEs 81
3.10 Alternative stochastic processes 84
3.11 Using associative arrays and matrices to model lookup tables and volatility surfaces 93
3.12 Summary and conclusion 96
3.13 Appendix: statistical distributions and special functions in the Boost library 97
3.14 Exercises and projects 102
4 An Introduction to the Finite Difference Method for SDE 107
4.1 Introduction and objectives 107
4.2 An introduction to discrete time simulation, motivation and notation 107
4.3 Foundations of discrete time approximation: ordinary differential equations 109
4.4 Foundations of discrete time approximation: stochastic differential equations 113
4.5 Some common schemes for one-factor SDEs 117
4.6 The Milstein schemes 117
4.7 Predictor-corrector methods 118
4.8 Stiff ordinary and stochastic differential equations 119
4.9 Software design and C++ implementation issues 125
4.10 Computational results 126
4.11 Aside: the characteristic equation of a difference scheme 127
4.12 Summary and conclusions 128
4.13 Exercises and projects 128
5 Design and Implementation of Finite Difference Schemes in Computational Finance 137
5.1 Introduction and objectives 137
5.2 Modelling SDEs and FDM in C++ 137
5.3 Mathematical and numerical tools 138
5.4 The Karhunen-Loeve expansion 143
5.5 Cholesky decomposition 144
5.6 Spread options with stochastic volatility 146
5.7 The Heston stochastic volatility model 153
5.8 Path-dependent options and the Monte Carlo method 160
5.9 A small software framework for pricing options 161
5.10 Summary and conclusions 162
5.11 Exercises and projects 162
6 Advanced Finance Models and Numerical Methods 167
6.1 Introduction and objectives 167
6.2 Processes with jumps 168
6.3 Levy processes 171
6.4 Measuring the order of convergence 172
6.5 Mollifiers, bump functions and function regularisation 176
6.6 When Monte Carlo does not work: counterexamples 177
6.7 Approximating SDEs using strong Taylor, explicit and implicit schemes 179
6.8 Summary and conclusions 183
6.9 Exercises and projects 184
7 Foundations of the Monte Carlo Method 189
7.1 Introduction and objectives 189
7.2 Basic probability 189
7.3 The Law of Large Numbers 190
7.4 The Central Limit Theorem 191
7.5 Quasi Monte Carlo methods 194
7.6 Summary and conclusions 198
7.7 Exercises and projects 198
PART II DESIGN PATTERNS
8 Architectures and Frameworks for Monte Carlo Methods: Overview 203
8.1 Goals of Part II of this book 203
8.2 Introduction and objectives of this chapter 203
8.3 The advantages of domain architectures 204
8.4 Software Architectures for the Monte Carlo method 207
8.5 Summary and conclusions 212
8.6 Exercises and projects 213
9 System Decomposition and System Patterns 217
9.1 Introduction and objectives 217
9.2 Software development process; a critique 217
9.3 System decomposition, from concept to code 217
9.4 Decomposition techniques, the process 220
9.5 Whole-part 222
9.6 Whole-part decomposition; the process 223
9.7 Presentation-Abstraction Control (PAC) 226
9.8 Building complex objects and configuring applications 229
9.9 Summary and conclusions 239
9.10 Exercises and projects 239
10 Detailed Design using the GOF Patterns 243
10.1 Introduction and objectives 243
10.2 Discovering which patterns to use 244
10.3 An overview of the GOF patterns 255
10.4 The essential structural patterns 257
10.5 The essential creational patterns 266
10.6 The essential behavioural patterns 270
10.7 Summary and conclusions 276
10.8 Exercises and projects 276
11 Combining Object-Oriented and Generic Programming Models 281
11.1 Introduction and objectives 281
11.2 Using templates to implement components: overview 281
11.3 Templates versus inheritance, run-time versus compile-time 283
11.4 Advanced C++ templates 286
11.5 Traits and policy-based design 294
11.6 Creating templated design patterns 306
11.7 A generic Singleton pattern 307
11.8 Generic composite structures 310
11.9 Summary and conclusions 314
11.10 Exercises and projects 314
12 Data Structures and their Application to the Monte Carlo Method 319
12.1 Introduction and objectives 319
12.2 Arrays, vectors and matrices 319
12.3 Compile-time vectors and matrices 324
12.4 Creating adapters for STL containers 331
12.5 Date and time classes 334
12.6 The class string 339
12.7 Modifying strings 343
12.8 A final look at the generic composite 345
12.9 Summary and conclusions 348
12.10 Exercises and projects 348
13 The Boost Library: An Introduction 353
13.1 Introduction and objectives 353
13.2 A taxonomy of C++ pointer types 353
13.3 Modelling homogeneous and heterogeneous data in Boost 361
13.4 Boost signals: notification and data synchronisation 367
13.5 Input and output 368
13.6 Linear algebra and uBLAS 371
13.7 Date and time 372
13.8 Other libraries 372
13.9 Summary and conclusions 374
13.10 Exercises and projects 374
PART III ADVANCED APPLICATIONS
14 Instruments and Payoffs 379
14.1 Introduction and objectives 379
14.2 Creating a C++ instrument hierarchy 379
14.3 Modelling payoffs in C++ 383
14.4 Summary and conclusions 392
14.5 Exercises and projects 393
15 Path-Dependent Options 395
15.1 Introduction and objectives 395
15.2 Monte Carlo â" a simple general-purpose version 396
15.3 Asian options 401
15.4 Options on the running Max/Min 411
15.5 Barrier options 412
15.6 Lookback options 418
15.7 Cliquet Options 422
15.8 Summary and conclusions 424
15.9 Exercises and projects 424
16 Affine Stochastic Volatility Models 427
16.1 Introduction and objectives 427
16.2 The volatility skew/smile 427
16.3 The Heston model 429
16.4 The Bates/SVJ model 441
16.5 Implementing the Bates model 443
16.6 Numerical results â" European options 444
16.7 Numerical results â" skew-dependent options 446
16.8 XLL â" using DLL within Microsoft Excel 449
16.9 Analytic solutions for affine stochastic volatility models 455
16.10 Summary and conclusions 457
16.11 Exercises and projects 458
17 Multi-Asset Options 461
17.1 Introduction and objectives 461
17.2 Modelling in multiple dimensions 461
17.3 Implementing payoff classes for multi-asset options 465
17.4 Some multi-asset options 466
17.5 Basket options 469
17.6 Min/Max options 471
17.7 Mountain range options 475
17.8 The Heston model in multiple dimensions 480
17.9 Equity interest rate hybrids 482
17.10 Summary and conclusions 486
17.11 Exercises and projects 486
18 Advanced Monte Carlo I â" Computing Greeks 489
18.1 Introduction and objectives 489
18.2 The finite difference method 489
18.3 The pathwise method 492
18.4 The likelihood ratio method 497
18.5 Likelihood ratio for finite differences â" proxy simulation 503
18.6 Summary and conclusions 504
18.7 Exercises and projects 506
19 Advanced Monte Carlo II â" Early Exercise 511
19.1 Introduction and objectives 511
19.2 Description of the problem 511
19.3 Pricing American options by regression 512
19.4 C++ design 513
19.5 Linear least squares regression 516
19.6 Example â" step by step 520
19.7 Analysis of the method and improvements 521
19.8 Upper bounds 525
19.9 Examples 527
19.10 Summary and conclusions 528
19.11 Exercises and projects 528
20 Beyond Brownian Motion 531
20.1 Introduction and objectives 531
20.2 Normal mean variance mixture models 531
20.3 The multi-dimensional case 536
20.4 Summary and conclusions 536
20.5 Exercises and projects 538
PART IV SUPPLEMENTS
21 C++ Application Optimisation and Performance Improvement 543
21.1 Introduction and objectives 543
21.2 Modelling functions in C++: choices and consequences 543
21.3 Performance issues in C++: classifying potential bottlenecks 552
21.4 Temporary objects 560
21.5 Special features in the Boost library 562
21.6 Boost multiarray library 563
21.7 Boost random number library 564
21.8 STL and Boost smart pointers: final remarks 566
21.9 Summary and conclusions 568
21.10 Exercises, projects and guidelines 569
22 Random Number Generation and Distributions 571
22.1 Introduction and objectives 571
22.2 Uniform number generation 571
22.3 The Sobol class 578
22.4 Number generation due to given distributions 580
22.5 Jump processes 588
22.6 The random generator templates 593
22.7 Tests for randomness 596
22.8 Summary and conclusions 596
22.9 Exercises and projects 597
23 Some Mathematical Background 601
23.1 Introduction and objectives 601
23.2 A matrix class 601
23.3 Matrix functions 601
23.4 Functional analysis 608
23.5 Applications to option pricing 610
23.6 Summary and conclusions 614
23.7 Exercises and projects 614
24 An Introduction to Multi-threaded and Parallel Programming 617
24.1 Introduction and objectives 617
24.2 Shared memory models 617
24.3 Sequential, concurrent and parallel programming 101 619
24.4 How fast is fast? Performance analysis of parallel programs 623
24.5 An introduction to processes and threads 625
24.6 What kinds of applications are suitable for multi-threading? 626
24.7 The multi-threaded application lifecycle 627
24.8 Some model architectures 629
24.9 Analysing and designing large software systems: a summary of the steps 633
24.10 Conclusions and summary 634
24.11 Exercises and projects 634
25 An Introduction to OpenMP and its Applications to the Monte Carlo Method 637
25.1 Introduction and objectives 637
25.2 Loop optimisation 637
25.3 An overview of OpenMP 644
25.4 Threads in OpenMP 644
25.5 Loop-level parallelism 646
25.6 Data sharing 646
25.7 Work-sharing and parallel regions 651
25.8 Nested loop optimisation 654
25.9 Scheduling in OpenMP 656
25.10 OpenMP for the Monte Carlo method 657
25.11 Conclusions and summary 661
25.12 Exercises and projects 661
26 A Case Study of Numerical Schemes for the Heston Model 665
26.1 Introduction and objectives 665
26.2 Test scenarios 666
26.3 Numerical approximations for the Heston model 667
26.4 Testing different schemes and scenarios 672
26.5 Results 675
26.6 Lessons learned 678
26.7 Extensions, calibration and more 679
26.8 Other numerical methods for Heston 680
26.9 Summary and conclusions 681
26.10 Exercises and projects 681
27 Excel, C++ and Monte Carlo Integration 685
27.1 Introduction and objectives 685
27.2 Integrating applications and Excel 686
27.3 ATL architecture 686
27.4 Creating my first ATL project: the steps 693
27.5 Creating automation add-ins in Excel 695
27.6 Useful utilities and interoperability projects 696
27.7 Test Case: a COM add-in and complete code 697
27.8 Summary and conclusions 707
27.9 Exercises and projects 707
References 711
Index 719
