Monte Carlo Frameworks

This is one of the first books that describe all the steps that are needed in order to analyze, design and implement Monte Carlo applications. It discusses the financial theory as well as the mathematical and numerical background that is needed to write flexible and efficient C++ code using state-of-the art design and system patterns, object-oriented and generic programming models in combination with standard libraries and tools.

Includes a CD containing the source code for all examples. It is strongly advised that you experiment with the code by compiling it and extending it to suit your needs. Support is offered via a user forum on www.datasimfinancial.com where you can post queries and communicate with other purchasers of the book.

This book is for those professionals who design and develop models in computational finance. This book assumes that you have a working knowledge of C ++.

Autorentext
DANIEL J. DUFFY has been working with numerical methods in finance, industry and engineering since 1979. He has written four books on financial models and numerical methods and C++ for computational finance and he has also developed a number of new schemes for this field. He is the founder of Datasim Education and has a PhD in Numerical Analysis from Trinity College, Dublin. JÖRG KIENITZ is the head of Quantitative Analysis at Deutsche Postbank AG. He is primarily involved in the developing and implementation of models for pricing of complex derivatives structures and for asset allocation. He is also lecturing at university level on advanced financial modelling and gives courses on 'Applications of Monte Carlo Methods in Finance' and on other financial topics including Lévy processes and interest rate models. Joerg holds a Ph.D. in stochastic analysis and probability theory.

Klappentext
The Monte Carlo method is now acknowledged as being one of the most robust tools for a range of applications in finance, from option pricing to risk management and optimization. One of the best languages for the development of Monte Carlo applications and frameworks is C++, an object-oriented and generic programming language which is also an industry standard. This is one of the first books that describe all the steps that are needed in order to analyze, design and implement Monte Carlo applications. It discusses the financial theory as well as the mathematical and numerical background that is needed to write flexible and efficient C++ code using state-of-the-art design and system patterns, object-oriented and generic programming models in combination with standard libraries and tools.

The book is divided into four parts, each one dealing with one major aspect of the current problem domain. The features and topics are:

• Option pricing for a range of one-factor and n-factor models;
• European, Asian, baskets, Heston, jump models;
• Early exercises, calculating option sensitivities;
• The mathematical theory of n-factor Stochastic Differential Equations (SDE);
• An introduction to the numerical analysis of SDE;
• Modelling SDE and the Finite Difference Method (FDM) in C++;
• Applying design and system patterns (GOF, POSA) for improved design;
• Extensive use of the STL and boost libraries;
• Multi-threading and parallel programming (OpenMP) techniques for Monte Carlo;
• Creating Excel-based applications using xlw, Automation and COM;
• Extra discussion of mathematical foundations for Monte Carlo;
• Working source code is provided along with numerous examples, exercises and projects related to the extension of the C++ framework.
  The book is accompanied by a CD which contains the source code for all the examples. It is strongly advised that you experiment with the code by compiling it and extending it to suit your needs. Support is offered via a user forum on www.datasimfinancial.com where you can post queries and communicate with other purchasers of the book.

Inhalt
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 ...