Statistics of Financial Markets

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An Introduction

Third Edition

•

Christian Matthias Hafner

Jürgen Franke Wolfgang Karl Härdle

Statistics of Financial Markets

Editorial board:

Endre Süli, University of Oxford

Wojbor Woyczynski, Case Western Reserve University ´

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ISBN 978-3-642-16520-7

DOI 10.1007/978-3-642-16521-4

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Germany

Prof. Dr. Jürgen Franke

FB Mathematik

67663 Kaiserslautern

Gebäude 48

Prof. Dr. Wolfgang Karl Härdle

Ladislaus von Bortkiewicz Chair of Stati

Claude Sabbah, CNRS, École Polytechnique

Unter den Linden 6

Carles Casacuberta, Universitat de Barcelona

[email protected] 10099 Berlin

Statistics a

Humboldt-Universität zu Berlin

School of Business and Economics

C.A.S.E. Centre for Applied

Germany

[email protected]

Prof. Dr. Christian Matthias Hafner

Université Catholique de Louvain

Sheldon Axler, San Francisco State University

Inst. Statistique

Vincenzo Capasso, Università degli Studi di Milano

Voie du Roman Pays 20

Angus MacIntyre, Queen Mary, University of London

Belgium

Kenneth Ribet, University of California, Berkeley

[email protected]

TU Kaiserslautern

Erwin-Schrödinger-Str.

1348 Leuven-la-Neuve

and

Graduate Institute of Statistics

National Central University

Taiwan

© Springer-Verlag Berlin Heidelberg 2011

Figure 0.1: Notes of a student for the exam of a course based on this book.

Figure 0.2: Notes of a student for the exam of a course based on this book.

Figure 0.3: Notes of a student for the exam of a course based on this book.

Figure 0.4: Notes of a student for the exam of a course based on this book.

Figure 0.5: Notes of a student for the exam of a course based on this book.

Preface to the Third Edition

The meltdown of financial assets in Autumn of 2008 has made the conse￾quences of the financial crisis clear to the broader public. The rapid loss

of value of asset backed securities, collateralized debt obligations and other

structured products was caused by the devaluation of real estate related prod￾ucts. We have therefore found it important to revise our book further and

present actualized research in financial statistics and econometrics. We have

extended the copulae chapter by introducing hiearchical copulae. In the chap￾ters 11-13 we have updated the data analysis with more recent data. The

majority of chapters now have an exercise section, the solutions which are

provided in the book by S. Borak, W. H¨ardle and B. Lopez Cabrera. In

addition to these changes we have translated all quantlet codes into Matlab

and R. These quantlets may be downloaded from the Springer.com page. Fi￾nally, we would like to thank Richard Song, Julius Mungo, Ostap Okhrin,

Andrija Mihoci, and Gernot Muller ¨ for their help in the management of the

text. Special thanks go to Barbara Choro´s-Tomczyk for the very skillful and

energetic preparation and editing work of this edition.

Kaiserslautern, Berlin and Rotterdam, November 2010

Preface to the Second Edition

After the success of the first edition we felt obliged to catch up with the

rapidly growing literature in financial statistics and econometrics. This sec￾ond edition expands on material that was only briefly covered in the previous

edition. As an example, Chapter 17 on copula is an extensive update of the

literature and describes some of our own research in this area. In the chapter

on time series with stochastic volatility (Chapter 13), we present a critique of

standard stationary GARCH modelling and describe an alternative nonpara￾metric way of modelling based on the idea of a time-varying unconditional

variance, and hence a non-stationary process. This new view of volatility

modelling seems to provide promising results in prediction when compared

with standard GARCH models. We have substantially augmented the section

on risk management (Section 6.3), including the Volga and Vanna coefficients

and the recent work on realised volatility. Another very active part of research

is on multivariate GARCH models, where we provide an updated review in

Section 13.4. We have included a new section on simulation techniques and

an entire chapter on Credit Risk Management. In addition to these changes,

we have eliminated a small number of errors in the first edition. Finally, we

would like to thank Ying Chen, Ekaterina Ignatieva and Kai Detlefsen for

the text management.

Berlin, Kaiserslautern and Louvain-la-Neuve, August 2007

Contents

Preface to the Third Edition

Preface to the Second Edition

I Option Pricing 1

1 Derivatives 3

1.1 Recommended Literature . . . . . . . . . . . . . . . . . . . . 10

1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Introduction to Option Management 13

2.1 Arbitrage Relations . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Portfolio Insurance . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Binary One-Period Model . . . . . . . . . . . . . . . . . . . . 32

2.4 Recommended Literature . . . . . . . . . . . . . . . . . . . . 37

2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Basic Concepts of Probability Theory 43

3.1 Real Valued Random Variables . . . . . . . . . . . . . . . . . 43

3.2 Expectation and Variance . . . . . . . . . . . . . . . . . . . . 46

3.3 Skewness and Kurtosis . . . . . . . . . . . . . . . . . . . . . . 47

3.4 Random Vectors, Dependence, Correlation . . . . . . . . . . . 48

3.5 Conditional Probabilities and Expectations . . . . . . . . . . 49

3.6 Recommended Literature . . . . . . . . . . . . . . . . . . . .

3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Stochastic Processes in Discrete Time 55

4.1 Binomial Processes . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Trinomial Processes . . . . . . . . . . . . . . . . . . . . . . . 59

4.3 General Random Walks . . . . . . . . . . . . . . . . . . . . . 61

4.4 Geometric Random Walks . . . . . . . . . . . . . . . . . . . . 62

4.5 Binomial Models with State Dependent Increments . . . . . . 63

4.6 Recommended Literature . . . . . . . . . . . . . . . . . . . . 64

4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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