Financial Modelling E-Book

Joerg Kienitz

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Series
• Sprache: Englisch

Seitenzahl: 840

Veröffentlichungsjahr: 2012

Beschreibung

Financial modelling
Theory, Implementation and Practice with MATLAB Source

Jörg Kienitz and Daniel Wetterau

Financial Modelling - Theory, Implementation and Practice with MATLAB Source is a unique combination of quantitative techniques, the application to financial problems and programming using Matlab. The book enables the reader to model, design and implement a wide range of financial models for derivatives pricing and asset allocation, providing practitioners with complete financial modelling workflow, from model choice, deriving prices and Greeks using (semi-) analytic and simulation techniques, and calibration even for exotic options.

The book is split into three parts. The first part considers financial markets in general and looks at the complex models needed to handle observed structures, reviewing models based on diffusions including stochastic-local volatility models and (pure) jump processes. It shows the possible risk-neutral densities, implied volatility surfaces, option pricing and typical paths for a variety of models including SABR, Heston, Bates, Bates-Hull-White, Displaced-Heston, or stochastic volatility versions of Variance Gamma, respectively Normal Inverse Gaussian models and finally, multi-dimensional models. The stochastic-local-volatility Libor market model with time-dependent parameters is considered and as an application how to price and risk-manage CMS spread products is demonstrated.

The second part of the book deals with numerical methods which enables the reader to use the models of the first part for pricing and risk management, covering methods based on direct integration and Fourier transforms, and detailing the implementation of the COS, CONV, Carr-Madan method or Fourier-Space-Time Stepping. This is applied to pricing of European, Bermudan and exotic options as well as the calculation of the Greeks. The Monte Carlo simulation technique is outlined and bridge sampling is discussed in a Gaussian setting and for Lévy processes. Computation of Greeks is covered using likelihood ratio methods and adjoint techniques. A chapter on state-of-the-art optimization algorithms rounds up the toolkit for applying advanced mathematical models to financial problems and the last chapter in this section of the book also serves as an introduction to model risk.

The third part is devoted to the usage of Matlab, introducing the software package by describing the basic functions applied for financial engineering. The programming is approached from an object-oriented perspective with examples to propose a framework for calibration, hedging and the adjoint method for calculating Greeks in a Libor market model.

Source code used for producing the results and analysing the models is provided on the author's dedicated website, http://www.mathworks.de/matlabcentral/fileexchange/authors/246981.

Leseprobe

Contents

Cover

Series

Title Page

Copyright

Dedication

Introduction

1 INTRODUCTION AND MANAGEMENT SUMMARY

2 WHY WE HAVE WRITTEN THIS BOOK

3 WHY YOU SHOULD READ THIS BOOK

4 THE AUDIENCE

5 THE STRUCTURE OF THIS BOOK

6 WHAT THIS BOOK DOES NOT COVER

7 CREDITS

8 CODE

Part I: Financial Markets and Popular Models

Chapter 1: Financial Markets – Data, Basics and Derivatives

1.1 INTRODUCTION AND OBJECTIVES

1.2 FINANCIAL TIME-SERIES, STATISTICAL PROPERTIES OF MARKET DATA AND INVARIANTS

1.3 IMPLIED VOLATILITY SURFACES AND VOLATILITY DYNAMICS

1.4 APPLICATIONS

1.5 GENERAL REMARKS ON NOTATION

1.6 SUMMARY AND CONCLUSIONS

1.7 APPENDIX – QUOTES

Chapter 2: Diffusion Models

2.1 INTRODUCTION AND OBJECTIVES

2.2 LOCAL VOLATILITY MODELS

2.3 STOCHASTIC VOLATILITY MODELS

2.4 STOCHASTIC VOLATILITY AND STOCHASTIC RATES MODELS

2.5 SUMMARY AND CONCLUSIONS

Chapter 3: Models with Jumps

3.1 INTRODUCTION AND OBJECTIVES

3.2 POISSON PROCESSES AND JUMP DIFFUSIONS

3.3 EXPONENTIAL LéVY MODELS

3.4 OTHER MODELS

3.5 MARTINGALE CORRECTION

3.6 SUMMARY AND CONCLUSIONS

Chapter 4: Multi-Dimensional Models

4.1 INTRODUCTION AND OBJECTIVES

4.2 MULTI-DIMENSIONAL DIFFUSIONS

4.3 MULTI-DIMENSIONAL HESTON AND SABR MODELS

4.4 PARAMETER AVERAGING

4.5 MARKOVIAN PROJECTION

4.6 COPULAE

4.7 MULTI-DIMENSIONAL VARIANCE GAMMA PROCESSES

4.8 SUMMARY AND CONCLUSIONS

Part II: Numerical Methods and Recipes

Chapter 5: Option Pricing by Transform Techniques and Direct Integration

5.1 INTRODUCTION AND OBJECTIVES

5.2 FOURIER TRANSFORM

5.3 THE CARR–MADAN METHOD

5.4 THE LEWIS METHOD

5.5 THE ATTARI METHOD

5.6 THE CONVOLUTION METHOD

5.7 THE COSINE METHOD

5.8 COMPARISON, STABILITY AND PERFORMANCE

5.9 EXTENDING THE METHODS TO FORWARD START OPTIONS

5.10 DENSITY RECOVERY

5.11 SUMMARY AND CONCLUSIONS

Chapter 6: Advanced Topics Using Transform Techniques

6.1 INTRODUCTION AND OBJECTIVES

6.2 PRICING NON-STANDARD VANILLA OPTIONS

6.3 BERMUDAN AND AMERICAN OPTIONS

6.4 THE COSINE METHOD AND BARRIER OPTIONS

6.5 GREEKS

6.6 SUMMARY AND CONCLUSIONS

Chapter 7: Monte Carlo Simulation and Applications

7.1 INTRODUCTION AND OBJECTIVES

7.2 SAMPLING DIFFUSION PROCESSES

7.3 SPECIAL PURPOSE SCHEMES

7.4 ADDING JUMPS

7.5 BRIDGE SAMPLING

7.6 LIBOR MARKET MODEL

7.7 MULTI-DIMENSIONAL LéVY MODELS

7.8 COPULAE

7.9 SUMMARY AND CONCLUSIONS

Chapter 8: Monte Carlo Simulation – Advanced Issues

8.1 INTRODUCTION AND OBJECTIVES

8.2 MONTE CARLO AND EARLY EXERCISE

8.3 GREEKS WITH MONTE CARLO

8.4 EULER SCHEMES AND GENERAL GREEKS

8.5 APPLICATION TO TRIGGER SWAP

8.6 SUMMARY AND CONCLUSIONS

8.7 APPENDIX – TREES

Chapter 9: Calibration and Optimization

9.1 INTRODUCTION AND OBJECTIVES

9.2 THE NELDER–MEAD METHOD

9.3 THE LEVENBERG–MARQUARDT METHOD

9.4 THE L-BFGS METHOD

9.5 THE SQP METHOD

9.6 DIFFERENTIAL EVOLUTION

9.7 SIMULATED ANNEALING

9.8 SUMMARY AND CONCLUSIONS

Chapter 10: Model Risk – Calibration, Pricing and Hedging

10.1 INTRODUCTION AND OBJECTIVES

10.2 CALIBRATION

10.3 PRICING EXOTIC OPTIONS

10.4 HEDGING

10.5 SUMMARY AND CONCLUSIONS

Part III: Implementation, Software Design and Mathematics

Chapter 11: Matlab – Basics

11.1 INTRODUCTION AND OBJECTIVES

11.2 GENERAL REMARKS

11.3 MATRICES, VECTORS AND CELL ARRAYS

11.4 FUNCTIONS AND FUNCTION HANDLES

11.5 TOOLBOXES

11.6 USEFUL FUNCTIONS AND METHODS

11.7 PLOTTING

11.8 SUMMARY AND CONCLUSIONS

Chapter 12: Matlab – Object Oriented Development

12.1 INTRODUCTION AND OBJECTIVES

12.2 THE MATLAB OO MODEL

12.3 A MODEL CLASS HIERARCHY

12.4 A PRICER CLASS HIERARCHY

12.5 AN OPTIMIZER CLASS HIERARCHY

12.6 DESIGN PATTERNS

12.7 EXAMPLE – CALIBRATION ENGINE

12.8 EXAMPLE – THE LIBOR MARKET MODEL AND GREEKS

12.9 SUMMARY AND CONCLUSIONS

Chapter 13: Math Fundamentals

13.1 INTRODUCTION AND OBJECTIVES

13.2 PROBABILITY THEORY AND STOCHASTIC PROCESSES

13.3 NUMERICAL METHODS FOR STOCHASTIC PROCESSES

13.4 BASICS ON COMPLEX ANALYSIS

13.5 THE CHARACTERISTIC FUNCTION AND FOURIER TRANSFORM

13.6 SUMMARY AND CONCLUSIONS

List of Figures

List of Tables

Bibliography

Index

For other titles in the Wiley Finance series

please see www.wiley.com/finance

Copyright © 2012 John Wiley & Sons Ltd

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Library of Congress Cataloging-in-Publication Data

Kienitz, Joerg. Financial modelling : theory, implementation and practice (with Matlab source) / Joerg Kienitz, Daniel Wetterau. p. cm. Includes bibliographical references and index. ISBN 978-0-470-74489-5 (cloth) – ISBN 978-1-118-41331-9 (ebk) – ISBN 978-1-118-41330-2 (ebk) – ISBN 978-1-118-41329-6 (ebk) 1. MATLAB. 2. Finance–Mathematical models. 3. Numerical analysis. 4. Finance–Mathematical models–Computer programs. 5. Numerical analysis–Computer programs. I. Wetterau, Daniel, 1981– II. Title. HG106.K53 2012 332.0285′53–dc23 2012029238

A catalogue record for this book is available from the British Library.

To Amberley, Beatrice and Benoît

Jörg

To Sabine

Daniel

Introduction

1 INTRODUCTION AND MANAGEMENT SUMMARY

The goal of the book is to fill a gap in the literature on financial modelling. Many books and research articles in the field are incomplete – a new model is introduced, an existing model is extended or a certain model topic is analysed but often there is not a hint on implementation or on the practical application of the model. Robustness of implementation, special purpose algorithms, performance analysis of the algorithms or the stability in terms of the model's parameters are all left out. In some cases either the mathematical theory is detailed or a fully flexible software design is suggested where the algorithms for implementing sophisticated models are not presented. In this book we aim to link all the steps of model development. To this end we describe many well-known financial models and provide algorithms as Matlab based software (we use this software as the basis for applying the models in terms of calibration, pricing and simulation). We show the design of an application and provide a source code for all the methods and models discussed. The source code can be used with the software package Matlab, but it is also possible to translate the code into other programming languages such as C#, C++ or Java.

Our aim is a clear separation of concerns. Thus, we separate the analysis of models and their properties from the numerical techniques, and the numerical techniques from the source code of the algorithms written in Matlab. We believe that this is necessary in order to clearly present the material and that, furthermore, this illustrates the life-cycle – model choice, model implementation, model application.

A quantitative analyst, trader or risk manager has to select a model appropriate to the problem under consideration and with respect to its likely progression. Therefore, a qualitative understanding – model properties and applicability – and a quantitative understanding–numerics and implementation – of the model and how given market data are applied to parameterize the model are necessary. Once a model is chosen it has to be implemented, tested and applied to the problem under consideration. To this end fast and reliable pricing methods for simple options have to be implemented. This is necessary since we wish to apply these methods to match market prices with model prices by adjusting the model parameters. This is in fact equivalent to solving a backward problem. Despite the fact that this method is frowned upon by academics and practitioners it is market practice. Finally, exotic payoffs including path-dependency or early exercise features often have to be considered and hedge sensitivities calculated.

For each step of the process a practitioner can, of course, consult different books or research articles that outline every single step in detail. We, however, want to show the whole life-cycle of financial modelling and provide an overview of widely applied models in finance in a single volume. In what follows we wish to outline their features and point to the applications of a particular model. Then, we show how to implement the pricing of simple products which are then used to calibrate model parameters. The fitted parameters of the models can then be applied to the pricing of exotic options. The numerics for each model are separated from the ‘recipe’ for the implementation using Matlab. As far as we know, ours is the first book to deal with financial modelling in this way.

We think that the book is special because we link the mathematical theory with numerical examples and source code which can be readily used. The results obtained in this book can be checked immediately by the reader applying the source code. The reader is of course free to extend and modify the code to further explore the numerical methods and gain more insights into advanced financial modelling and the underlying mathematical theory. We provide source code for

Advanced models including stochastic volatility or jumps.Fourier transform based modern pricing methods.Direct integration with efficient numerical schemes.Monte Carlo simulation including special purpose schemes.Optimization and calibration using local and global optimizers.Calculating Greeks and applying early exercise rights.Hedge strategies and model risk.

For example, we price European options using several versions of the fast Fourier transform (FFT). We use this exceptionally fast method for calibrating a wide range of complex models to given market data. Using the derived model parameters we price exotic payoffs applying Monte Carlo simulation techniques with innovative methods such as bridge sampling or quasi random numbers.

Furthermore, we show how to optimize the code in terms of robustness and efficiency and how to extend it to suit your own needs. To this end we show how to incorporate methods from object oriented programming and design patterns in software engineering.

2 WHY WE HAVE WRITTEN THIS BOOK

We wrote this book in order to show how to apply advanced probabilistic and analytic models to financial problems. We also show that it is possible to design and implement flexible and efficient software for pricing and hedging a range of financial instruments. Furthermore, it is possible to obtain model parameters from market data for advanced models. Finally, we have attempted to create an understandable and seamless process, starting with a given financial model, designing its implementation using Matlab and then applying it to real-world situations. To this end we cover the modelling process by describing the characteristics of well-known modelling approaches. We implement each model and finally apply it efficiently to solve financial modelling problems. The mathematical theory is covered but we have placed it in the final chapter since there are many textbooks that review the mathematical fundamentals as well as special issues of advanced numerics and mathematical techniques more rigorously. The interested reader can therefore work through the theoretical aspects but a more practically oriented reader can start right away studying financial models and use the code to work with the models. We believe that this is a feature which makes the book unique; dealing with financial modelling in this way makes it unnecessary to reinvent the wheel since the stony path of implementation has been transformed into a well-paved road by providing all the source code.

3 WHY YOU SHOULD READ THIS BOOK

Advanced probabilistic models such as Lévy processes, stochastic volatility models and numerical procedures to solve financial problems using such models are becoming increasingly popular. They will soon become vital for risk management purposes and pricing applications. To be prepared for this era finance professionals should have studied the most popular models and should have dealt with the numerical methods appropriate to master such models. The numerical methods in this case are Fourier analysis, direct integration and Monte Carlo simulation. The models are able to capture many market phenomena, such as smiles or fat tails, that are often observed in financial markets data and further highlight the risks involved.

4 THE AUDIENCE

For whom is this book intended? We have written this book for those finance professionals who design, develop and apply advanced financial models. Especially we have in mind those people who wish to go beyond the Gaussian model paradigm. We apply modern mathematical tools and discuss their implementation to price and risk manage a wide range of financial instruments. We assume that the reader has a working knowledge of financial modelling, for example as discussed in Wilmott, P. (2007) or Hull, J. (2011), as well as some hands-on experience with Matlab or some other object oriented programming language, for instance C++, as discussed in Duffy, D. (2004 and 2006). Using this book, you can define financial models and integrate them into the Matlab software framework.

Using another programming language such as C++, C# or Java it is possible to translate the code into this language. At the time of writing there are many specialized libraries available that can replace Matlab functions, such as optimizers or fast Fourier transform algorithms.

This book is also aimed at students in financial engineering and related disciplines. It can serve as a basis to extend financial models or it can be used directly to apply models to financial market data for quantitative studies in economics or risk management courses. Furthermore, the material can be used for preparing reports and thesis projects in all quantitative finance disciplines. Finally, the book can function as a bridge between IT and quantitative finance because it demonstrates how Matlab code is produced from financial and mathematical models. Working with other programming languages such as C++ or VBA the methods are also applicable using available mathematical libraries or the Matlab runtime kernel via packaging functions using dll, xll or COM components.

Therefore, the primary target audience are people working in the financial industry and those studying for a degree in finance, or doing research or lecturing in that area. The book provides a wealth of examples and starting points for developing customized solutions. It can act as a guide for those starting out in the world of financial modelling and it can also serve as the basis for setting up complex and advanced models for finance professionals. Hence we recommend this book to financial engineers, quantitative researches, traders, risk managers, risk controllers as well as to students and academics in this field and those engaged in research towards a masters or doctoral thesis. Lecturers will find plenty of illustrated examples in the book, which are very useful for the application of theoretical topics in finance. Furthermore, it will be of use to students on programming courses on quantitative finance and numerics for financial applications.

We believe our text forms the ideal complement to earlier books by Duffy, D. (2004 and 2006) and Duffy, D. and Kienitz, J. (2009), where the focus is on developing software architectures in C++.

This book should be of interest to anyone who would benefit from understanding and using (advanced) financial models and to those who learn how to implement such models.

Since we provide ready to use code for the whole work-flow (model choice, implementation and calibration to market data), the reader can apply each model when studying this book. Tables and figures can be reproduced and certain modelling aspects can be further explored immediately.

Finally, we must alert our readers to the fact that we do not give any warranty for completeness, nor do we guarantee that the code is error free. Any damage or loss incurred in the application of the software and the concepts suggested in the book are entirely the reader's responsibility.

5 THE STRUCTURE OF THIS BOOK

The book has three parts. Each part consists of a number of chapters which, taken together, deal with a specific aspect of the problem of modelling financial problems. The contents of the parts can be summarized as follows:

Part I – Financial Markets and Popular Models In the first part of the book we summarize financial markets facts and argue that fat tails, volatility smiles and other observed phenomena are not in line with the usual modelling approach using Geometric Brownian Motion or VaR models based on the Normal Distribution. To this end we give an overview of well-known financial models and their properties. For instance, we analyse the impact of parameter changes on implied volatility surfaces, risk neutral densities or typical sample paths and returns.Part II – Numerical Methods and Recipes In this part of the book we consider the numerical methods necessary for implementing and using the models introduced in Part I. We describe fundamental techniques such as fast Fourier transform, Quadrature methods, optimization or Monte Carlo simulation. To this end we provide the source code for all techniques discussed. This covers advanced numerical schemes for applying the Fourier transform such as the COS- or the CONV method or special purpose simulation schemes for stochastic volatility models. The code is not restricted to simple applications. We deal with the application of Fourier transforms and Monte Carlo methods to handle early exercises and the stable calculation of Greeks for discontinuous payoffs. In particular we apply the Adjoint Method and Proxy Simulation techniques. Moreover, we give an overview of different approaches to optimization and guide the reader through several techniques including Differential Evolution or Sequential Quadratic Programming. Of course, we provide the implementation for the methods discussed. This is the first time that such methods are both considered and their implementation described in a single book.Part III – Implementation, Software Design and Mathematics The basic and efficient application of Matlab is discussed in the last part of the book. We start with an introduction to the Matlab package and show the basics that a financial engineer should know. Then, we discuss the object oriented approach and illustrate it by developing a calibration engine. In doing so, we discuss design patterns and class hierarchies to achieve our goal.

The parts are divided into 13 chapters. The structure of the book is as follows:

Part I Financial Markets and Popular Models

Introduction

Chapter 1 – Financial Markets – Data, Basics, Derivatives

Chapter 2 – Diffusion Models

Chapter 3 – Models with Jumps

Chapter 4 – Multi-Dimensional Models

Part II – Numerical Methods and Recipes

Chapter 5 – Option Pricing by Transform Techniques and Direct Integration

Chapter 6 – Advanced Topics using Transform Techniques

Chapter 7 – Monte Carlo Simulation and Applications

Chapter 8 – Monte Carlo Simulation – Advanced Issues

Chapter 9 – Calibration and Optimization

Chapter 10 – Model Risk – Calibration, Pricing and Hedging

Part III – Implementation, Software Design and Mathematics

Chapter 11 – Matlab–Basics

Chapter 12 – Matlab–Object Oriented Development

Chapter 13 – Math Fundamentals

6 WHAT THIS BOOK DOES NOT COVER

This book assumes some knowledge of finance and we do not aim to discuss the whole theory in this book. Thus, for instance, we do not cover in detail such important techniques as change of numeraire, arbitrage or the fundamental theorem of option pricing. We assume that the reader knows what options and financial instruments are and wishes to apply numerical techniques to such problems and, in doing so, to gain some practical understanding of the mathematics involved. We also assume that the reader has a working knowledge of Matlab. In particular, you should be familiar with the following topics:

Handling matrices and vectors.Fundamental Matlab syntax.Functions and function handles.

We do think, however, that it is possible to learn the Matlab prerequisites in a very short time. For instance, the reader can start with Chapter 11 and work through the given material and experiment with the source code.

Finally, we hope that you enjoy this book as much as we have enjoyed working on it. For feedback, updates and questions concerning the book and its content please feel free to contact [email protected].

Good luck with applying the numerical methods to your problem under consideration!

Jörg Kienitz and Daniel Wetterau

7 CREDITS

At this stage it is time to thank many collaborators and especially students prepraing their thesis during internships within the Quantitative Analytics group of Deutsche Postbank AG. We thank Philipp Beyer for analysing Forward Start options in Lévy process based models, Beyer, P. and Kienitz, J. (2009), Holger Kammeyer for working on the Heston–Hull–White model, Kammeyer, H. and Kienitz, J. (2012a), Kammeyer, H. and Kienitz, J. (2012b), Kammeyer, H. and Kienitz, J. (2012c), Ines Weber for exploring Monte Carlo methods for options with early exercise opportunities, Sven Glaser for working on CMS Spread option pricing and finally Nikolai Nowaczyk for researching into the application of adjoint methods for calculating Greeks, Nowakcyk, N. and Kienitz, J. (2011).

Furthermore, we thank our colleague Manuel Wittke for fruitful discussions and research, Kienitz, J., Wetterau, D. and Wittke, M. (2011), Kienitz, J. and Wittke, M. (2010) and many suggestions on early stages of the manuscript as well as collaborating on the issues presented in Chapter 10. Further joint work is in preparation, Kienitz, J., Wetterau, D. and Wittke, M. (2013).

Christian Fries has to be mentioned for stimulating discussions on mathematical and financial topics.

Daniel J. Duffy is greatly acknowledged for suggestions on early versions of the manuscript.

Special thanks to Dr. Ingo Schneider and Wojciech Slusarski for carefully reading versions of the manuscript and for their many helpful remarks and comments.

Finally, Graeme West has to be mentioned. He started to read the manuscript but sadly died. Joerg Kienitz wishes to thank Graeme for stimulating discussions and for the invitation to lecture at AIMS Summer School in Cape Town, 2011.

8 CODE

We have to stress that the code used for implementing the models and the numerical methods presented in this book is not static. We work on the code by improving it in terms of speed, accuracy or by just fixing bugs. To this end the code available from the download section or directly from the authors' web-site is updated, improved or extended from time to time. We keep the reader informed on the web-site, www.jkienitz.de.

Furthermore, if you spot any errors or you wish to submit some new numerical method as a Matlab function, a new model or some improvement of the method illustrated in this book you are very welcome. Please send any feedback to [email protected].

For the current book the code can be obtained by download. Thus, there is no CD included in this book. A dedicated website for the code is http://www.mathworks.de/matlabcentral/fileexchange/authors/246981.

Part I

FINANCIAL MARKETS AND POPULAR MODELS

1

Financial Markets – Data, Basics and Derivatives

1.1 INTRODUCTION AND OBJECTIVES

The first chapter is to introduce the models that appear in subsequent chapters and, in so doing, to highlight the necessity of applying advanced numerical techniques. Since we wish to apply mathematical models to financial problems, we first have to analyse the markets under consideration. We have to check the available data upon which we build our models. Then, we have to investigate which models are appropriate and, finally, we need to decide on numerical methods to solve the modelling problem.

We motivate using market data; we highlight the nature of risk and the problems which arise with inappropriate modelling. The final conclusion is that the observed market structures need sophisticated models, numerically challenging implementation and deeply involved special purpose algorithms. Furthermore, we provide answers and suggestions to the following questions:

What kind of objects do we have to model?What kind of distributions are necessary? Do we need anything other than the Gaussian distribution?What kind of patterns do we observe and which model is capable of reproducing such patterns?How complex should a model be?Which mathematical methods do we need? PDE? SDE? Numerical Mathematics?

We do not rate the models, but we do give advice on the numerical methods which can be applied to implement the different models and on what kind of market observation is covered by a certain model. We work out several methods which can be applied. The reader can try the different solutions and – very important – check the implementation, the stability and the robustness. Furthermore, the code provided can be modified to fit the special modelling issues.

Since financial models have to be implemented as computer programs, or they have to be integrated into a pricing library, numerical methods are required. The most fundamental risk of a model is, of course, its inapplicability in a certain setting. To this end we have to analyse which risk factors can be modelled using a certain class of models and we have to be aware of the risk factors that have not been taken into account. But once we have decided to apply a particular model, and we think that we are applying it appropriately, we face the following challenges:

Appropriate numerical techniques.Approximations used should be robust, efficient and accurate.Black box solutions should be avoided.The implementation should be stable and reliable.

1.2 FINANCIAL TIME-SERIES, STATISTICAL PROPERTIES OF MARKET DATA AND INVARIANTS

To use a mathematical model for gaining insights and applying it to financial market data we need to choose some quantities or risk factors which we model. To this end we consider the notion of a market invariant. Fix a starting point in time and an estimation interval τ. The interval τ could be one day or one month, for instance. Suppose from a market data provider we can get the data for an index , with

We regard X(t) as a random variable. A random variable X is called a market invariant for and estimation interval τ if the realizations

are independentare identically distributed.

A simple but effective method to test if a random variable qualifies as an invariant is the following:

Take a time series Xs, of the possible invariant.Split the time series into two partsPlot histograms corresponding to X1 and X2.Plot lagged time series against Xt.

Let us illustrate this test on time series for index and swap data. Before we actually start let us illustrate the dependence structure corresponding to independent, positively and negatively dependent random variables. To this end we take as an example the normal distribution with zero mean and a given covariance matrix, Σ. For our examples we choose three different covariance matrices, namely,

The dependence structure is displayed in Figure 1.1.

Figure 1.1 Time series generated from a normal distribution with covariance given by Σ0 (top left), Σ1 (top right) and Σ2 (bottom) reflecting independence, positive and negative dependence

We call a market invariant Xtime homogeneous if the distribution of X does not depend on the chosen time point tstart. In the sequel we consider Equity, Index, Interest Rate and Option markets. First, we consider index time series for the , the , the and the . We argue that the prices of the indices do not obey the properties necessary to be an invariant.