Monte Carlo Frameworks - Building Customisable High-Performance C++ Applications

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 -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 - 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 - European options 444 16.7 Numerical results - skew-dependent options 446 16.8 XLL - 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 - 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 - proxy simulation 503 18.6 Summary and conclusions 504 18.7 Exercises and projects 506 19 Advanced Monte Carlo II - 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 - 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