Financial Instrument Pricing Using C++

CHAPTER 1 A Tour of C++ and Environs 1 1.1 Introduction and Objectives 1 1.2 What is C++? 1 1.3 C++ as a Multiparadigm Programming Language 2 1.4 The Structure and Contents of this Book: Overview 4 1.5 A Tour of C++11: Black–Scholes and Environs 6 1.6 Parallel Programming in C++ and Parallel C++ Libraries 12 1.7 Writing C++ Applications; Where and How to Start? 14 1.8 For whom is this Book Intended? 16 1.9 Next-Generation Design and Design Patterns in C++ 16 1.10 Some Useful Guidelines and Developer Folklore 17 1.11 About the Author 18 1.12 The Source Code and Getting the Source Code 19 CHAPTER 2 New and Improved C++ Fundamentals 21 2.1 Introduction and Objectives 21 2.2 The C++ Smart Pointers 21 2.3 Using Smart Pointers in Code 23 2.4 Extended Examples of Smart Pointers Usage 30 2.5 Move Semantics and Rvalue References 34 2.6 Other Bits and Pieces: Usability Enhancements 39 2.7 Summary and Conclusions 52 2.8 Exercises and Projects 52 CHAPTER 3 Modelling Functions in C++ 59 3.1 Introduction and Objectives 59 3.2 Analysing and Classifying Functions 60 3.3 New Functionality in C++: std::function<> 64 3.4 New Functionality in C++: Lambda Functions and Lambda Expressions 65 3.5 Callable Objects 69 3.6 Function Adapters and Binders 70 3.7 Application Areas 75 3.8 An Example: Strategy Pattern New Style 75 3.9 Migrating from Traditional Object-Oriented Solutions: Numerical Quadrature 78 3.10 Summary and Conclusions 81 3.11 Exercises and Projects 82 CHAPTER 4 Advanced C++ Template Programming 89 4.1 Introduction and Objectives 89 4.2 Preliminaries 91 4.3 decltype Specifier 94 4.4 Life Before and After decltype 101 4.5 std::result_of and SFINAE 106 4.6 std::enable_if 108 4.7 Boost enable_if 112 4.8 std::decay()Trait 114 4.9 A Small Application: Quantities and Units 115 4.10 Conclusions and Summary 118 4.11 Exercises and Projects 118 CHAPTER 5 Tuples in C++ and their Applications 123 5.1 Introduction and Objectives 123 5.2 An std:pair Refresher and New Extensions 123 5.3 Mathematical and Computer Science Background 128 5.4 Tuple Fundamentals and Simple Examples 130 5.5 Advanced Tuples 130 5.6 Using Tuples in Code 133 5.7 Other Related Libraries 138 5.8 Tuples and Run-Time Efficiency 140 5.9 Advantages and Applications of Tuples 142 5.10 Summary and Conclusions 143 5.11 Exercises and Projects 143 CHAPTER 6 Type Traits, Advanced Lambdas and Multiparadigm Design in C++ 147 6.1 Introduction and Objectives 147 6.2 Some Building Blocks 149 6.3 C++ Type Traits 150 6.4 Initial Examples of Type Traits 158 6.5 Generic Lambdas 161 6.6 How Useful will Generic Lambda Functions be in the Future? 164 6.7 Generalised Lambda Capture 171 6.7.1 Living Without Generalised Lambda Capture 173 6.8 Application to Stochastic Differential Equations 174 6.9 Emerging Multiparadigm Design Patterns: Summary 178 6.10 Summary and Conclusions 179 6.11 Exercises and Projects 179 CHAPTER 7 Multiparadigm Design in C++ 185 7.1 Introduction and Objectives 185 7.2 Modelling and Design 185 7.3 Low-Level C++ Design of Classes 190 7.4 Shades of Polymorphism 199 7.5 Is there More to Life than Inheritance? 206 7.6 An Introduction to Object-Oriented Software Metrics 207 7.7 Summary and Conclusions 210 7.8 Exercises and Projects 210 CHAPTER 8 C++ Numerics, IEEE 754 and Boost C++ Multiprecision 215 8.1 Introduction and Objectives 215 8.2 Floating-Point Decomposition Functions in C++ 219 8.3 A Tour of std::numeric_limits 221 8.4 An Introduction to Error Analysis 223 8.5 Example: Numerical Quadrature 224 8.6 Other Useful Mathematical Functions in C++ 228 8.7 Creating C++ Libraries 231 8.8 Summary and Conclusions 239 8.9 Exercises and Projects 239 CHAPTER 9 An Introduction to Unified Software Design 245 9.1 Introduction and Objectives 245 9.1.1 Future Predictions and Expectations 246 9.2 Background 247 9.3 System Scoping and Initial Decomposition 251 9.4 Checklist and Looking Back 259 9.5 Variants of the Software Process: Policy-Based Design 260 9.6 Using Policy-Based Design for the DVM Problem 268 9.7 Advantages of Uniform Design Approach 273 9.8 Summary and Conclusions 274 9.9 Exercises and Projects 275 CHAPTER 10 New Data Types, Containers and Algorithms in C++ and Boost C++ Libraries 283 10.1 Introduction and Objectives 283 10.2 Overview of New Features 283 10.3 C++ std::bitset and Boost Dynamic Bitset Library 284 10.4 Chrono Library 288 10.5 Boost Date and Time 301 10.6 Forwards Lists and Compile-Time Arrays 306 10.7 Applications of Boost.Array 311 10.8 Boost uBLAS (Matrix Library) 313 10.9 Vectors 316 10.10 Matrices 318 10.11 Applying uBLAS: Solving Linear Systems of Equations 322 10.12 Summary and Conclusions 330 10.13 Exercises and Projects 331 CHAPTER 11 Lattice Models Fundamental Data Structures and Algorithms 333 11.1 Introduction and Objectives 333 11.2 Background and Current Approaches to Lattice Modelling 334 11.3 New Requirements and Use Cases 335 11.4 A New Design Approach: A Layered Approach 335 11.5 Initial ‘101’ Examples of Option Pricing 347 11.6 Advantages of Software Layering 349 11.7 Improving Efficiency and Reliability 352 11.8 Merging Lattices 355 11.9 Summary and Conclusions 357 11.10 Exercises and Projects 357 CHAPTER 12 Lattice Models Applications to Computational Finance 367 12.1 Introduction and Objectives 367 12.2 Stress Testing the Lattice Data Structures 368 12.3 Option Pricing Using Bernoulli Paths 372 12.4 Binomial Model for Assets with Dividends 374 12.5 Computing Option Sensitivities 377 12.6 (Quick) Numerical Analysis of the Binomial Method 379 12.7 Richardson Extrapolation with Binomial Lattices 382 12.8 Two-Dimensional Binomial Method 382 12.9 Trinomial Model of the Asset Price 384 12.10 Stability and Convergence of the Trinomial Method 385 12.11 Explicit Finite Difference Method 386 12.12 Summary and Conclusions 389 12.13 Exercises and Projects 389 CHAPTER 13 Numerical Linear Algebra: Tridiagonal Systems and Applications 395 13.1 Introduction and Objectives 395 13.2 Solving Tridiagonal Matrix Systems 395 13.3 The Crank-Nicolson and Theta Methods 406 13.4 The ADE Method for the Impatient 411 13.5 Cubic Spline Interpolation 415 13.6 Some Handy Utilities 427 13.7 Summary and Conclusions 428 13.8 Exercises and Projects 429 CHAPTER 14 Data Visualisation in Excel 433 14.1 Introduction and Objectives 433 14.2 The Structure of Excel-Related Objects 433 14.3 Sanity Check: Is the Excel Infrastructure Up and Running? 435 14.4 ExcelDriver and Matrices 437 14.5 ExcelDriver and Vectors 444 14.6 Path Generation for Stochastic Differential Equations 448 14.7 Summary and Conclusions 459 14.8 Exercises and Projects 459 14.9 Appendix: COM Architecture Overview 463 14.10 An Example 468 14.11 Virtual Function Tables 471 14.12 Differences between COM and Object-Oriented Paradigm 473 14.13 Initialising the COM Library 474 CHAPTER 15 Univariate Statistical Distributions 475 15.1 Introduction, Goals and Objectives 475 15.2 The Error Function and Its Universality 475 15.3 One-Factor Plain Options 478 15.4 Option Sensitivities and Surfaces 488 15.5 Automating Data Generation 491 15.6 Introduction to Statistical Distributions and Functions 499 15.7 Advanced Distributions 504 15.8 Summary and Conclusions 511 15.9 Exercises and Projects 511 CHAPTER 16 Bivariate Statistical Distributions and Two-Asset Option Pricing 515 16.1 Introduction and Objectives 515 16.2 Computing Integrals Using PDEs 516 16.3 The Drezner Algorithm 521 16.4 The Genz Algorithm and the West/Quantlib Implementations 521 16.5 Abramowitz and Stegun Approximation 525 16.6 Performance Testing 528 16.7 Gauss–Legendre Integration 529 16.8 Applications to Two-Asset Pricing 531 16.9 Trivariate Normal Distribution 536 16.10 Chooser Options 543 16.11 Conclusions and Summary 545 16.12 Exercises and Projects 546 CHAPTER 17 STL Algorithms in Detail 551 17.1 Introduction and Objectives 551 17.2 Binders and std::bind 554 17.3 Non-modifying Algorithms 557 17.4 Modifying Algorithms 567 17.5 Compile-Time Arrays 575 17.6 Summary and Conclusions 576 17.7 Exercises and Projects 576 17.8 Appendix: Review of STL Containers and Complexity Analysis 583 CHAPTER 18 STL Algorithms Part II 589 18.1 Introduction and Objectives 589 18.2 Mutating Algorithms 589 18.3 Numeric Algorithms 597 18.4 Sorting Algorithms 601 18.5 Sorted-Range Algorithms 604 18.5.5 Merging 608 18.6 Auxiliary Iterator Functions 609 18.7 Needle in a Haystack: Finding the Right STL Algorithm 612 18.8 Applications to Computational Finance 613 18.9 Advantages of STL Algorithms 613 18.10 Summary and Conclusions 614 18.11 Exercises and Projects 614 CHAPTER 19 An Introduction to Optimisation and the Solution of Nonlinear Equations 617 19.1 Introduction and Objectives 617 19.2 Mathematical and Numerical Background 618 19.3 Sequential Search Methods 619 19.4 Solutions of Nonlinear Equations 620 19.5 Fixed-Point Iteration 622 19.6 Aitken’s Acceleration Process 623 19.7 Software Framework 623 19.8 Implied Volatility 632 19.9 Solvers in the Boost C++ Libraries 632 19.10 Summary and Conclusions 633 19.11 Exercises and Projects 633 19.12 Appendix: The Banach Fixed-Point Theorem 636 CHAPTER 20 The Finite Difference Method for PDEs: Mathematical Background 641 20.1 Introduction and Objectives 641 20.2 General Convection–Diffusion–Reaction Equations and Black–Scholes PDE 641 20.3 PDE Preprocessing 64520.3.2 Reduction of PDE to Conservative Form 646 20.4 Maximum Principles for Parabolic PDEs 649 20.5 The Fichera Theory 650 20.6 Finite Difference Schemes: Properties and Requirements 654 20.7 Example: A Linear Two-Point Boundary Value Problem 655 20.8 Exponentially Fitted Schemes for Time-Dependent PDEs 659 20.9 Richardson Extrapolation 663 20.10 Summary and Conclusions 665 20.11 Exercises and Projects 666 CHAPTER 21 Software Framework for One-Factor Option Models 669 21.1 Introduction and Objectives 669 21.2 A Software Framework: Architecture and Context 669 21.3 Modelling PDEs and Finite Difference Schemes: What is Supported? 670 21.4 Several Versions of Alternating Direction Explicit 671 21.5 A Software Framework: Detailed Design and Implementation 673 21.6 C++ Code for PDE Classes 674 21.7 C++ Code for FDM Classes 679 21.8 Examples and Test Cases 690 21.9 Summary and Conclusions 693 21.10 Exercises and Projects 694 CHAPTER 22 Extending the Software Framework 701 22.1 Introduction and Objectives 701 22.2 Spline Interpolation of Option Values 701 22.3 Numerical Differentiation Foundations 704 22.4 Numerical Greeks 710 22.5 Constant Elasticity of Variance Model 715 22.6 Using Software Design (GOF) Patterns 715 22.7 Multiparadigm Design Patterns 720 22.8 Summary and Conclusions 721 22.9 Exercises and Projects 721 CHAPTER 23A PDE Software Framework in C++11 for a Class of Path-Dependent Options 727 23.1 Introduction and Objectives 727 23.2 Modelling PDEs and Initial Boundary Value Problems in the Functional Programming Style 728 23.3 PDE Preprocessing 731 23.4 The Anchoring PDE 732 23.5 ADE for Anchoring PDE 739 23.6 Useful Utilities 746 23.7 Accuracy and Performance 748 23.8 Summary and Conclusions 750 23.9 Exercises and Projects 751 CHAPTER 24 Ordinary Differential Equations and their Numerical Approximation 755 24.1 Introduction and Objectives 755 24.2 What is an ODE? 755 24.3 Classifying ODEs 756 24.4 A Palette of Model ODEs 757 24.5 Existence and Uniqueness Results 760 24.6 Overview of Numerical Methods for ODEs: The Big Picture 763 24.7 Creating ODE Solvers in C++ 770 24.8 Summary and Conclusions 776 24.9 Exercises and Projects 776 24.10 Appendix 778 CHAPTER 25 Advanced Ordinary Differential Equations and Method of Lines 781 25.1 Introduction and Objectives 781 25.2 An Introduction to the Boost Odeint Library 782 25.3 Systems of Stiff and Non-stiff Equations 791 25.4 Matrix Differential Equations 796 25.5 The Method of Lines: What is it and what are its Advantages? 799 25.6 Initial Foray in Computational Finance: MOL for One-Factor Black-Scholes PDE 801 25.7 Barrier Options 806 25.8 Using Exponential Fitting of Barrier Options 808 25.9 Summary and Conclusions 808 25.10 Exercises and Projects 809 CHAPTER 26 Random Number Generation and Distributions 819 26.1 Introduction and Objectives 819 26.2 What is a Random Number Generator? 820 26.3 What is a Distribution? 821 26.4 Some Initial Examples 825 26.5 Engines in Detail 827 26.6 Distributions in C++: The List 830 26.7 Back to the Future: C-Style Pseudo-Random Number Generation 831 26.8 Cryptographic Generators 833 26.9 Matrix Decomposition Methods 833 26.10 Generating Random Numbers 845 26.11 Summary and Conclusions 848 26.12 Exercises and Projects 849 CHAPTER 27 Microsoft .Net, C# and C++11 Interoperability 853 27.1 Introduction and Objectives 853 27.2 The Big Picture 854 27.3 Types 858 27.4 Memory Management 859 27.5 An Introduction to Native Classes 861 27.6 Interfaces and Abstract Classes 861 27.7 Use Case: C++/CLI as ‘Main Language’ 862 27.8 Use Case: Creating Proxies, Adapters and Wrappers for Legacy C++ Applications 864 27.8.1 Alternative: SWIG (Simplified Wrapper and Interface Generator) 871 27.9 ‘Back to the Future’ Use Case: Calling C# Code from C++11 872 27.10 Modelling Event-Driven Applications with Delegates 876 27.11 Use Case: Interfacing with Legacy Code 886 27.12 Assemblies and Namespaces for C++/CLI 889 27.13 Summary and Conclusions 895 27.14 Exercises and Projects 896 CHAPTER 28 C++ Concurrency, Part I Threads 899 28.1 Introduction and Objectives 899 28.2 Thread Fundamentals 900 28.3 Six Ways to Create a Thread 903 28.4 Intermezzo: Parallelising the Binomial Method 909 28.5 Atomics 916 28.6 Smart Pointers and the Thread-Safe Pointer Interface 924 28.7 Thread Synchronisation 926 28.8 When should we use Threads? 929 28.9 Summary and Conclusions 929 28.10 Exercises and Projects 930 CHAPTER 29 C++ Concurrency, Part II Tasks 935 29.1 Introduction and Objectives 935 29.2 Finding Concurrency: Motivation 936 29.3 Tasks and Task Decomposition 937 29.4 Futures and Promises 941 29.5 Shared Futures 945 29.6 Waiting on Tasks to Complete 948 29.7 Continuations and Futures in Boost 950 29.8 Pure Functions 952 29.9 Tasks versus Threads 953 29.10 Parallel Design Patterns 953 29.11 Summary and Conclusions 955 29.12 Quizzes, Exercises and Projects 955 CHAPTER 30 Parallel Patterns Language (PPL) 961 30.1 Introduction and Objectives 961 30.2 Parallel Algorithms 962 30.3 Partitioning Work 967 30.4 The Aggregation/Reduction Pattern in PPL 971 30.5 Concurrent Containers 977 30.6 An Introduction to the Asynchronous Agents Library and Event-Based Systems 978 30.7 A Design Plan to Implement a Framework Using Message Passing and Other Approaches 986 30.8 Summary and Conclusions 989 30.9 Exercises and Projects 990 CHAPTER 31 Monte Carlo Simulation, Part I 993 31.1 Introduction and Objectives 993 31.2 The Boost Parameters Library for the Impatient 995 31.3 Monte Carlo Version 1: The Monolith Program (‘Ball of Mud’) 1000 31.4 Policy-Based Design: Dynamic Polymorphism 1003 31.5 Policy-Based Design Approach: CRTP and Static Polymorphism 1011 31.6 Builders and their Subcontractors (Factory Method Pattern) 1013 31.7 Practical Issue: Structuring the Project Directory and File Contents 1014 31.8 Summary and Conclusions 1016 31.9 Exercises and Projects 1017 CHAPTER 32 Monte Carlo Simulation, Part II 1023 32.1 Introduction and Objectives 1023 32.2 Parallel Processing and Monte Carlo Simulation 1023 32.3 A Family of Predictor–Corrector Schemes 1033 32.4 An Example (CEV Model) 1038 32.5 Implementing the Monte Carlo Method Using the Asynchronous Agents Library 1041 32.6 Summary and Conclusions 1047 32.7 Exercises and Projects 1050 Appendix 1: Multiple-Precision Arithmetic 1053 Appendix 2: Computing Implied Volatility 1075 References 1109 Index 1117