Mathematics of Financial Markets

Mathematics of Financial Markets

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Mathematics of Financial Markets

Financial Instruments and Derivatives Modeling,

Valuation and Risk Issues

Alain Ruttiens

A John Wiley & Sons, Ltd., Publication

This edition first published 2013

Copyright C 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)

Set in 10/12pt Times by Aptara, Inc., New Delhi, India

Printed in Great Britain by CPI Group (UK) Ltd, Croydon, CR0 4YY

To Prof. Didier Marteau,

without whom this book would not exist

Contents

Foreword by A.G. MALLIARIS, Loyola University, Chicago xi

Main Notations xiii

Introduction xv

PART I THE DETERMINISTIC ENVIRONMENT

1 Prior to the Yield Curve: Spot and Forward Rates 3

1.1 Interest Rates, Present and Future Values, Interest Compounding 3

1.2 Discount Factors 5

1.3 Continuous Compounding and Continuous Rates 6

1.4 Forward Rates 8

1.5 The No Arbitrage Condition 11

Further Reading 12

2 The Term Structure or Yield Curve 13

2.1 Introduction to the Yield Curve 13

2.2 The Yield Curve Components 15

2.3 Building a Yield Curve: Methodology 17

2.4 An Example of Yield Curve Points Determination 21

2.5 Interpolations on a Yield Curve 21

Further Reading 22

3 Spot Instruments 23

3.1 Short-Term Rates 23

3.2 Bonds 24

3.3 Currencies 43

Further Reading 45

4 Equities and Stock Indexes 47

4.1 Stocks Valuation 47

4.2 Stock Indexes 51

viii Contents

4.3 The Portfolio Theory 52

Further Reading 73

5 Forward Instruments 75

5.1 The Forward Foreign Exchange 75

5.2 FRAs 84

5.3 Other Forward Contracts 86

5.4 Contracts for Difference (CFD) 88

Further Reading 89

6 Swaps 91

6.1 Definitions and First Examples 91

6.2 Prior to an IRS Swap Pricing Method 94

6.3 Pricing of an IRS Swap 99

6.4 (Re)Valuation of an IRS Swap 102

6.5 The Swap (Rates) Market 103

6.6 Pricing of a CRS Swap 105

6.7 Pricing of Second-Generation Swaps 108

Further Reading 118

7 Futures 119

7.1 Introduction to Futures 119

7.2 Futures Pricing 123

7.3 Futures on Equities and Stock Indexes 127

7.4 Futures on Short-Term Interest Rates 130

7.5 Futures on Bonds 132

7.6 Futures on Currencies 138

7.7 Futures on (Non-Financial) Commodities 139

Further Reading 144

PART II THE PROBABILISTIC ENVIRONMENT

8 The Basis of Stochastic Calculus 147

8.1 Stochastic Processes 147

8.2 The Standard Wiener Process, or Brownian Motion 150

8.3 The General Wiener Process 152

8.4 The Ito Process ˆ 152

8.5 Application of the General Wiener Process 153

8.6 The Ito Lemma ˆ 155

8.7 Application of the Ito Lemma ˆ 156

8.8 Notion of Risk Neutral Probability 158

8.9 Notion of Martingale 159

Annex 8.1: Proofs of the Properties of dZ(t) 161

Annex 8.2: Proof of the Ito Lemma ˆ 163

Further Reading 164

Contents ix

9 Other Financial Models: From ARMA to the GARCH Family 165

9.1 The Autoregressive (AR) Process 165

9.2 The Moving Average (MA) Process 166

9.3 The Autoregression Moving Average (ARMA) Process 168

9.4 The Autoregressive Integrated Moving Average (ARIMA) Process 168

9.5 The ARCH Process 171

9.6 The GARCH Process 172

9.7 Variants of (G)ARCH Processes 173

9.8 The MIDAS Process 174

Further Reading 174

10 Option Pricing in General 175

10.1 Introduction to Option Pricing 175

10.2 The Black–Scholes Formula 179

10.3 Finite Difference Methods: The Cox–Ross–Rubinstein (CRR)

Option Pricing Model 186

10.4 Monte Carlo Simulations 191

10.5 Option Pricing Sensitivities 195

Further Reading 207

11 Options on Specific Underlyings and Exotic Options 209

11.1 Currency Options 209

11.2 Options on Bonds 211

11.3 Options on Interest Rates 219

11.4 Exchange Options 227

11.5 Basket Options 228

11.6 Bermudan Options 230

11.7 Options on Non-Financial Underlyings 230

11.8 Second-Generation Options, or Exotics 231

Further Reading 235

12 Volatility and Volatility Derivatives 237

12.1 Practical Issues About the Volatility 238

12.2 Modeling the Volatility 247

12.3 Realized Volatility Models 251

12.4 Modeling the Correlation 252

12.5 Volatility and Variance Swaps 254

Further Reading 256

13 Credit Derivatives 257

13.1 Introduction to Credit Derivatives 257

13.2 Valuation of Credit Derivatives 263

13.3 Conclusion 273

Further Reading 274

14 Market Performance and Risk Measures 275

14.1 Return and Risk Measures 275

x Contents

14.2 VaR or Value-at-Risk 292

Further Reading 302

15 Beyond the Gaussian Hypothesis: Potential Troubles with

Derivatives Valuation 303

15.1 Alternatives to the Gaussian Hypothesis 303

15.2 Potential Troubles with Derivatives Valuation 312

Further Reading 318

Bibliography 319

Index 323

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 quan￾titative 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 statis￾tical 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

B bond price

c coupon rate of a bond

C convexity, or call price, in function of the context

cov(.) covariance of (.)

d dividend paid by a stock

D duration

Dt discount factor relative to time t

E(.) expected value of (.)

F forward price, or future price (depends on the context)

FV future value

-ibor generic for LIBOR, EURIBOR, or any other inter-bank market rate

K strike price of an option

κ kurtosis

M month or million, depending on context

MD modified duration

MtM “Marked to Market” (= valued to the observed current market price)

μ drift of a stochastic process

N total number of a series (integer number), or nominal (notional) amount (depends

on the context)

N (.) Gaussian (normal) density distribution function

N(.) Gaussian (normal) cumulative distribution function

P put price

P{.} probability of {.}

PV present value

Q(.) Poisson density distribution function

r generic symbol for a rate of return

rf risk-free return

ρ(.) correlation of (.)

skew skewness

S spot price of an asset (equity, currency, etc.), as specified by the context

STD(.) standard deviation of (.)

σ volatility of a stochastic process

t current time, or time in general (depends on the context)