Mathematics of the Financial Markets E-Book

Alain Ruttiens

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

Seitenzahl: 538

Veröffentlichungsjahr: 2013

Beschreibung

Mathematics of the Financial Markets Financial Instruments and Derivatives Modeling, Valuation and Risk Issues "Alain Ruttiens has the ability to turn extremely complex concepts and theories into very easy to understand notions. I wish I had read his book when I started my career!" Marco Dion, Global Head of Equity Quant Strategy, J.P. Morgan "The financial industry is built on a vast collection of financial securities that can be valued and risk profiled using a set of miscellaneous mathematical models. The comprehension of these models is fundamental to the modern portfolio and risk manager in order to achieve a deep understanding of the capabilities and limitations of these methods in the approximation of the market. In his book, Alain Ruttiens exposes these models for a wide range of financial instruments by using a detailed and user friendly approach backed up with real-life data examples. The result is an excellent entry-level and reference book that will help any student and current practitioner up their mathematical modeling skills in the increasingly demanding domain of asset and risk management." Virgile Rostand, Consultant, Toronto ON "Alain Ruttiens not only presents the reader with a synthesis between mathematics and practical market dealing, but, more importantly a synthesis of his thinking and of his life." René Chopard, CEO, Centro di Studi Bancari Lugano, Vezia / Professor, Università dell'Insubria, Varese "Alain Ruttiens has written a book on quantitative finance that covers a wide range of financial instruments, examples and models. Starting from first principles, the book should be accessible to anyone who is comfortable with trading strategies, numbers and formulas." Dr Yuh-Dauh Lyuu, Professor of Finance & Professor of Computer Science & Information Engineering, National Taiwan University

Leseprobe

Contents

Cover

Series

Title Page

Copyright

Dedication

Foreword

Main Notations

Introduction

Part I: The Deterministic Environment

Chapter 1: Prior to the yield curve: spot and forward rates

1.1 INTEREST RATES, PRESENT AND FUTURE VALUES, INTEREST COMPOUNDING

1.2 DISCOUNT FACTORS

1.3 CONTINUOUS COMPOUNDING AND CONTINUOUS RATES

1.4 FORWARD RATES

1.5 THE NO ARBITRAGE CONDITION

FURTHER READING

Chapter 2: The term structure or yield curve

2.1 INTRODUCTION TO THE YIELD CURVE

2.2 THE YIELD CURVE COMPONENTS

2.3 BUILDING A YIELD CURVE: METHODOLOGY

2.4 AN EXAMPLE OF YIELD CURVE POINTS DETERMINATION

2.5 INTERPOLATIONS ON A YIELD CURVE

FURTHER READING

Chapter 3: Spot instruments

3.1 SHORT-TERM RATES

3.2 BONDS

3.3 CURRENCIES

FURTHER READING

Chapter 4: Equities and stock indexes

4.1 STOCKS VALUATION

4.2 STOCK INDEXES

4.3 THE PORTFOLIO THEORY

FURTHER READING

Chapter 5: Forward instruments

5.1 THE FORWARD FOREIGN EXCHANGE

5.2 FRAs

5.3 OTHER FORWARD CONTRACTS

5.4 CONTRACTS FOR DIFFERENCE (CFD)

FURTHER READING

Chapter 6: Swaps

6.1 DEFINITIONS AND FIRST EXAMPLES

6.2 PRIOR TO AN IRS SWAP PRICING METHOD

6.3 PRICING OF AN IRS SWAP

6.4 (RE)VALUATION OF AN IRS SWAP

6.5 THE SWAP (RATES) MARKET

6.6 PRICING OF A CRS SWAP

6.7 PRICING OF SECOND-GENERATION SWAPS

FURTHER READING

Chapter 7: Futures

7.1 INTRODUCTION TO FUTURES

7.2 FUTURES PRICING

7.3 FUTURES ON EQUITIES AND STOCK INDEXES

7.4 FUTURES ON SHORT-TERM INTEREST RATES

7.5 FUTURES ON BONDS

7.6 FUTURES ON CURRENCIES

7.7 FUTURES ON (NON-FINANCIAL) COMMODITIES

FURTHER READING

Part II: The Probabilistic Environment

Chapter 8: The basis of stochastic calculus

8.1 STOCHASTIC PROCESSES

8.2 THE STANDARD WIENER PROCESS, OR BROWNIAN MOTION

8.3 THE GENERAL WIENER PROCESS

8.4 THE ITÔ PROCESS

8.5 APPLICATION OF THE GENERAL WIENER PROCESS

8.6 THE ITÔ LEMMA

8.7 APPLICATION OF THE ITô LEMMA

8.8 NOTION OF RISK NEUTRAL PROBABILITY

8.9 NOTION OF MARTINGALE

ANNEX 8.1: PROOFS OF THE PROPERTIES OF dZ(t)

ANNEX 8.2: PROOF OF THE ITÔ LEMMA

FURTHER READING

Chapter 9: Other financial models: from ARMA to the GARCH family

9.1 THE AUTOREGRESSIVE (AR) PROCESS

9.2 THE MOVING AVERAGE (MA) PROCESS

9.3 THE AUTOREGRESSION MOVING AVERAGE (ARMA) PROCESS

9.4 THE AUTOREGRESSIVE INTEGRATED MOVING AVERAGE (ARIMA) PROCESS

9.5 THE ARCH PROCESS

9.6 THE GARCH PROCESS

9.7 VARIANTS OF (G)ARCH PROCESSES

9.8 THE MIDAS PROCESS

FURTHER READING

Chapter 10: Option pricing in general

10.1 INTRODUCTION TO OPTION PRICING

10.2 THE BLACK–SCHOLES FORMULA

10.3 FINITE DIFFERENCE METHODS: THE COX–ROSS–RUBINSTEIN (CRR) OPTION PRICING MODEL

10.4 MONTE CARLO SIMULATIONS

10.5 OPTION PRICING SENSITIVITIES

FURTHER READING

Chapter 11: Options on specific underlyings and exotic options

11.1 CURRENCY OPTIONS

11.2 OPTIONS ON BONDS

11.3 OPTIONS ON INTEREST RATES

11.4 EXCHANGE OPTIONS

11.5 BASKET OPTIONS

11.6 BERMUDAN OPTIONS

11.7 OPTIONS ON NON-FINANCIAL UNDERLYINGS

11.8 SECOND-GENERATION OPTIONS, OR EXOTICS

FURTHER READING

Chapter 12: Volatility and volatility derivatives

12.1 PRACTICAL ISSUES ABOUT THE VOLATILITY

12.2 MODELING THE VOLATILITY

12.3 REALIZED VOLATILITY MODELS

12.4 MODELING THE CORRELATION

12.5 VOLATILITY AND VARIANCE SWAPS

FURTHER READING

Chapter 13: Credit derivatives

13.1 INTRODUCTION TO CREDIT DERIVATIVES

13.2 VALUATION OF CREDIT DERIVATIVES

13.3 CONCLUSION

FURTHER READING

Chapter 14: Market performance and risk measures

14.1 RETURN AND RISK MEASURES

14.2 VaR OR VALUE-AT-RISK

FURTHER READING

Chapter 15: Beyond the Gaussian hypothesis: potential troubles with derivatives valuation

15.1 ALTERNATIVES TO THE GAUSSIAN HYPOTHESIS

15.2 POTENTIAL TROUBLES WITH DERIVATIVES VALUATION

FURTHER READING

Bibliography

Index

For other titles in the Wiley Finance series please see www.wiley.com/finance

This edition first published 2013 Copyright © 2013 Alain Ruttiens

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Library of Congress Cataloging-in-Publication Data to follow

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

ISBN 978-1-118-51345-3 (hardback) ISBN 978-1-118-51347-7 (ebk) ISBN 978-1-118-51348-4 (ebk) ISBN 978-1-118-51349-1 (ebk)

To Prof. Didier Marteau,without whom this book would not exist

Foreword

The valuation and risk dimensions of financial instruments, and, to some extent, the way they behave, rest on a vast, complex set of mathematical models grouped into what is called quantitative finance. Today more than ever, it should be required that each and every one involved in financial markets or products has good command of quantitative finance. The problem is that the many books in this field are devoted either to a specific type of financial instruments, combining product description and quantitative aspects, or to a specific mathematical or statistical theory, or otherwise, with an impressive degree of mathematical formalism, which needs a high degree of competence in mathematics and quantitative methods. Alain Ruttiens' text is aiming to offer in a single book what should be needed to be known by a wide readership to master the quantitative finance at large. It covers, on the one hand, all the financial products, from the traditional spot instruments in forex, stocks, interest rates, and so on, to the most complex derivatives, and, on the other hand, the major quantitative tools designed to value them, and to assess their risk potentials. This book should therefore provide the best entry-level reference for anyone concerned in some way with financial markets and products to master their quantitative aspects, or to fill the gaps in areas with which they are less familiar.

At first sight, this ambitious objective seems hard to achieve, given the variety and the complexity of the materials it aims to cover. As a matter of fact, Alain recognizes that fulfilling such an objective implies sorting among a vast array of topics in a rather subjective way. Fortunately, the author had the chance to at least induce a positive bias in such a subjective selection by relying upon his experience as a market practitioner for more than 20 years. He furthermore treats this material in a clear, pedagogical way, requiring no prerequisites in the reader, except the basics of algebra and statistics.

Finally, the reader should appreciate the overall aim of Alain's book, allowing for useful comparisons – some valuation methods appearing to be more robust and trustworthy than others – and often warning against the lack of reliability of some quantitative models, due to the hypotheses on which they are built. This last point is all the more crucial after the recent financial crises, which were at least partially due to some inappropriate uses of quantitative models.

For all of these reasons, my expectation is that Alain's book should be a great success.

A.G. Malliaris Loyola University, Chicago

Main Notations

Bbond priceccoupon rate of a bondCconvexity, or call price, in function of the contextcov(.)covariance of (.)ddividend paid by a stockDdurationDtdiscount factor relative to time tE(.)expected value of (.)Fforward price, or future price (depends on the context)FVfuture value-iborgeneric for LIBOR, EURIBOR, or any other inter-bank market rateKstrike price of an optionκkurtosisMmonth or million, depending on contextMDmodified durationMtM“Marked to Market” (= valued to the observed current market price)μdrift of a stochastic processNtotal number of a series (integer number), or nominal (notional) amount (depends on the context)(.)Gaussian (normal) density distribution functionN(.)Gaussian (normal) cumulative distribution functionPput priceP{.}probability of {.}PVpresent value(.)Poisson density distribution functionrgeneric symbol for a rate of returnrfrisk-free returnρ(.)correlation of (.)skewskewnessSspot price of an asset (equity, currency, etc.), as specified by the contextSTD(.)standard deviation of (.)σvolatility of a stochastic processtcurrent time, or time in general (depends on the context)t0initial timeTmaturity timeτtenor, that is, time interval between current time t and maturity TV(.)variance of (.)(.)stochastic process of (.)stochastic variablezt“zero” or 0-coupon rate of maturity tZstandard Wiener process (Brownian motion, white noise)

Introduction

The world is the excess of possible.1

The aim of this book is to present the quantitative aspects of financial markets instruments and their derivatives. With such a broad scope, it goes without saying that it remains a “general” book, which is why, at the end of each of the chapters, there is a list of further reading for those who want to expand the topic (this also applies at a global level, cf. the end of this Introduction). Ideally, everyone concerned with financial markets – whether a trader, a risk manager, a sales person, an accountant, or managing a fund, an institutional or a bank, and so on, or else a student in finance, of course – should have to be aware of what is happening, quantitatively speaking, behind the financial instruments' behaviors.

In writing this book, my concern was twofold: to sort out what really needs to be mastered, and to write up the text in the most pedagogical way. I hope that with both my 25-year professional experience in financial markets and my teaching activities, this objective will have been reached in a satisfactory way.

As regards the mathematical formulae, they are not proved, except when the proof brings some useful insight. Rather, I have tried to justify as much as possible their importance, and to translate them from algebra into plain English. After all, the vast majority of people involved with financial markets do not compute prices, sensitivities, and so on since they have access to data providers such as Bloomberg, where almost everything is valued. Therefore, it is not a question of replacing the computer but of having some command of these calculations, both for a safety reason – it is better to understand what is behind the data we manipulate – and to be able to appreciate the order of magnitude of the prices we are confronted with. And even sometimes to be capable of drafting some rough calculation aside from the market data.

Also, I have tried as much as possible to avoid excessive formalism – formalism is securing the outputs of research, but may, in other circumstances, burden the understanding by non-mathematicians. This is the case, for example, in Chapter 8, The Basis of Stochastic Calculus. Besides the basics of algebra and probabilities and statistics, there is no prerequisite for using this book.

I warmly thank Renaud Beaupain, Christian Berbé, 2 Frédéric Botteman, Marc Buckens, Simon Dablemont, François Delclaux, Jean-Charles Devin, Andrés Feal, Florena Gaillard, Michel Godefroid, Christian Jaumain, Mahnoosh Mirghaemi and Angelo Pessaris, who each agreed to proofread one or two of the 15 chapters – they helped me significantly with their remarks and comments. As the saying goes, any remaining errors or deficiencies are my own. In the same way, I welcome readers' comments and remarks at [email protected].

Two final, practical remarks:

In the many real market examples, dates are expressed as dd/mm/yy.I am using the “-ibor” notation to globally denote any kind of LIBOR as well as EURIBOR interbank interest rate. By the way, to make the reading of formulae easier, I have tried to choose symbols (see Main Notations) which are as close as possible to what they represent.

Alain Ruttiens

FURTHER READING

As general references:

John Y. CAMPBELL, Andrew W. LO, A. Craigh MACKINLAY, The Econometrics of Fnancial Markets, Princeton University Press, 1996, 632 p.

Sergio M. FOCARDI, Frank J. FABOZZI, The Mathematics of Financial Modeling and Investment Management, John Wiley & Sons, Inc., Hoboken, 2004, 800 p.

Lawrence GALITZ, Financial Times Handbook of Financial Engineering, FT Press, 3rd ed. Scheduled on November 2011, 480 p.

Philippe JORION, Financial Risk Manager Handbook, John Wiley & Sons, Inc., Hoboken, 5th ed., 2009, 752 p.

Tze Leung LAI, Haipeng XING, Statistical Models and Methods for Financial Markets, Springer, 2008, 374 p.

David RUPPERT, Statistics and Finance, An Introduction, Springer, 2004, 482 p.

Dan STEFANICA, A Primer for the Mathematics of Financial Engineering, FE Press, 2011, 352 p.

Robert STEINER, Mastering Financial Calculations, FT Prentice Hall, 1997, 400 p.

John L. TEALL, Financial Market Analytics, Quorum Books, 1999, 328 p. Presents the maths needed to understand quantitative finance, with examples and applications focusing on financial markets.

1. (Translated from French) Thomas RIEN, Cette mémoire du cœur, 1985.

2. Who sadly passed away recently.

Part I

The Deterministic Environment

2

The term structure or yield curve

2.1 INTRODUCTION TO THE YIELD CURVE

A term structure or yield curve can be defined as the graph of spot rates or zeroes1 in function of their maturity. Since most of the time interest rates are higher with longer maturities, one talks of a “normal” yield curve if it is going upwards, and of an “inverse” yield curve if and when longer rates are lower than shorter rates.