**UFRJ**

# derivatives and internal models

(Parte **1** de 5)

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Derivatives and Internal Models

Fourth Edition Dr Hans-Peter Deutsch

© Hans-Peter Deutsch 2009

All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission.

No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6-10 Kirby Street, London EC1N 8TS.

Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages.

The author has asserted his right to be identiﬁed as the author of this work in accordance with the Copyright, Designs and Patents Act 1988.

First published in 2009 by PALGRAVE MACMILLAN

Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number 785998, of Houndmills, Basingstoke, Hampshire RG21 6XS.

Palgrave Macmillan in the US is a division of St Martin’s Press LLC, 175 Fifth Avenue, New York, NY 10010.

Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world.

Palgrave™ and Macmillan™ are registered trademarks in the United States, the United Kingdom, Europe and other countries.

ISBN-13: 978–0–230–22215–1 ISBN-10: 0–230–22215–3

This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin.

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

Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne

Contents

List of Figures xi List of Tables xv Preface xvii

Part I Fundamentals

2 Fundamental Risk Factors of Financial Markets 7 2.1 Interest rates 7 2.2 Market prices 21 2.3 An intuitive model for ﬁnancial risk factors 2 2.4 Ito processes and stochastic analysis 32

3 Financial Instruments: A System of Derivatives and

Part I Methods

4 Overview of the Assumptions 65

5 PresentValue Methods,Yields, and Traditional Risk Measures 68 5.1 Present value and yield to maturity 68 5.2 Internal rate of return and net present value 70 5.3 Accrued interest, residual debt, and par rates 73 5.4 Traditional sensitivities of interest rate instruments 76

vi CONTENTS

7 The Black-Scholes Differential Equation 94 7.1 The Black-Scholes equation from arbitrage arguments 95 7.2 The Black-Scholes equation and the backward equation 101 7.3 The relationship to the heat equation 105

8 Integral Forms and Analytic Solutions in the Black-ScholesWorld 109 8.1 Option prices as solutions of the heat equation 109 8.2 Option prices and transition probabilities 1 8.3 Compilation of Black-Scholes option prices for different underlyings 114

9 Numerical Solutions Using Finite Differences 121 9.1 Discretizing the Black-Scholes equation 122 9.2 Difference schemes 129 9.3 Convergence criteria 151 9.4 Discrete dividends 156 9.5 Example 157

10 Binomial and Trinomial Trees 161 10.1 General trees 161 10.2 Recombinant trees 165 10.3 The relationship between random walk and binomial parameters 173 10.4 The binomial model with inﬁnitesimal steps 176 10.5 Trinomial trees 178

1 Monte Carlo Simulations 184 1.1 A simple example: The area of a disk 186 1.2 The general approach to Monte Carlo simulations 190 1.3 Monte Carlo simulation of risk factors 191 1.4 Pricing 197

12 Hedging 199 12.1 Replicating portfolios as synthetic derivatives 199 12.2 Hedging derivatives with spot transactions 200 12.3 Hedging derivatives with forward contracts 203 12.4 Hedge-ratios for arbitrary combinations of ﬁnancial instruments 209 12.5 “Greek” risk management with sensitivities 211 12.6 Computation of the greek risk variables 218

13 Martingales and Numeraires 224 13.1 The martingale property 224 13.2 The numeraire 226 13.3 Self-ﬁnancing portfolio strategies 230

CONTENTS vii

13.4 Generalization to continuous time 233 13.5 The drift 243 13.6 The market price of risk 247 13.7 Tradable underlyings 248 13.8 Applications in the Black-Scholes world 249

14 Interest Rates and Term Structure Models 254 14.1 Instantaneous spot rates and instantaneous forward rates 255 14.2 Important numeraire instruments 257 14.3 The special case of deterministic interest rates 261 14.4 Tradable and nontradable variables 263 14.5 Convexity adjustments 264 14.6 Arbitrage-free interest rate trees 272 14.7 Market rates vs. instantaneous rates 282 14.8 Explicit speciﬁcation of short rate models 288 14.9 The example program TermStructureModels.xls 296 14.10 Monte Carlo on the tree 303 14.1 The drift in term structure models 304 14.12 Short rate models with discrete compounding 310

Part I Instruments

15 Spot Transactions on Interest Rates 315 15.1 Zero bonds 316 15.2 Floaters 317 15.3 Coupon bonds 319 15.4 Swaps 326 15.5 Annuity loans 331

16 Forward Transactions on Interest Rates 343 16.1 Forward rate agreements 343 16.2 Interest rate futures 344 16.3 Forward swaps 348 16.4 Forward bonds 353

17 PlainVanilla Options 357 17.1 Options on spot and forward prices 358 17.2 Index options and index futures 361 17.3 Foreign exchange options and futures 362 17.4 Interest rate options 364

18 Exotic Options 378 18.1 Traditional and general deﬁnition of an option 378 18.2 Payoff proﬁles for selected exotics 378 viii CONTENTS

18.3 Black-Scholes for exotics 383 18.4 Numerical pricing methods for exotics 394

Part IV Risk

19 Fundamental Risk Concepts 407 19.1 Conﬁdence, percentile, and risk 408 19.2 The value at risk of a single risk factor 411 19.3 Approximations in the distribution of risk factors 417 19.4 The covariance matrix 419

20 TheVariance-Covariance Method 430 20.1 Portfolios vs. ﬁnancial instruments 432 20.2 The delta-normal method 434 20.3 The delta-gamma method 439

21 Simulation Methods 462 21.1 Monte Carlo simulation 462 21.2 Historical simulation 465 21.3 Crash and stress testing: Worst case scenarios 467

2 Interest Rate Risk and Cash Flows 470 2.1 Cash ﬂow structures of ﬁnancial instruments 470 2.2 Interpolation and cash ﬂow mapping 485

23 Example of aVaR Computation 491 23.1 The portfolio 491 23.2 Market data 492 23.3 Cash ﬂow mapping 493 23.4 Calculation of risk 493

24 Backtesting: Checking the Applied Methods 496 24.1 Proﬁt-loss computations 496 24.2 The trafﬁc light approach of the supervising authorities 497

Part V Portfolios

25 Classical Portfolio Management 507 25.1 From risk management to portfolio management 508 25.2 Portfolio optimization 517

26 Attributes and their Characteristic Portfolios 535 26.1 General properties of characteristic portfolios 536 26.2 The leverage 539 26.3 The excess return 540

CONTENTS ix

26.4 The optimal portfolio 544 26.5 The efﬁcient frontier revisited 547

27 Active Management and Benchmarking 552 27.1 The capital asset pricing model 552 27.2 Benchmarking against an index 554 27.3 Benchmark and characteristic portfolios 559 27.4 Relations between Sharpe ratio and information ratio 567

PartVI Market Data

30 Market Parameter from Historical Time Series 615 30.1 Historical yields, volatility, and correlation 615 30.2 Error estimates 617 30.3 Return and covariance estimates 627

31 Time Series Modeling 634 31.1 Stationary time series and autoregressive models 637 31.2 Calibration of time series models 648

32 Forecasting with Time Series Models 656 32.1 Forecasting with autoregressive models 657 32.2 Volatility forecasts with GARCH(p,q) processes 660 32.3 Volatility forecasts with GARCH(1,1) processes 665 32.4 Volatility forecasts with moving averages 668

3 Principal Component Analysis 671 3.1 The general procedure 671 3.2 Principal component analysis of the German term structure 678

34 Pre-Treatment of Time Series and Assessment of Models 682 34.1 Pre-treatment of time series 682 34.2 Measuring the goodness of time series models 686

A Probability and Statistics 699

A.1 Probability, expectation, and variance 699 A.2 Multivariate distributions, covariance, correlation, and beta 701 x CONTENTS

A.3 Moments and characteristic functions 705 A.4 A Collection of important distributions 710 A.5 Transformations between distributions 734

Bibliography 739 Index 745

Figures

2.1 Determining the length of a time period using different day count conventions 10

2.2 The general discount factor for the time span from t to T 12

2.3 Interest rates for the same discount factor based on different day count and compounding conventions 18

2.5 The general forward discount factor for the time between T and T′ as seen at time t 19

2.6 Forward rates for periods starting in T = 1, 2, | , 15 years |

for terms T′ − T = 1,2, | ,15 years 21 |

2.7 A random walk in two dimensions 23

2.8 End-of-day values of a stock price over a period of 500 trading days 27

2.9 The distribution from Table 2.5 with μdt = 0 and σ√dt = 13 2 2.10 The trinomial tree used to derive the forward equation 42 2.1 The trinomial tree used to derive the backward equation 45 3.1 Examples of common money market instruments 50 3.2 Examples of common capital market securities 54 5.1 Cash ﬂow table of a portfolio 72 5.2 Summary of the different yields 75

9.1 Part of the solution surface of an American plain vanilla call on an underlying with a discrete dividend payment 155

10.1 The ﬁrst steps in a recombining binomial tree 165 10.2 A simple trinomial tree 179 1.1 A disk with diameter 2m in a square with sides of length 2m 187

xii FIGURES

1.2 The same random walk as in Figure 1.2 but with a drift (mean return) of 6% per year 193

14.1 The binomial tree with the indexing showing the number of up- and down-moves required to get to the nodes starting from node (0,0) 276

14.2 A monetary unit (black dot) at node (2,2) generates ADPs at nodes (2, 1) and (1, 2) and also at all the earlier “striped” nodes 284

14.3 Flow of information when constructing the short rate tree 291

15.1 Valuation of bonds with annual coupon payments using spot rates and par rates with annual compounding 322

15.2 The sensitivities of the coupon bonds from Figure 15.1 326

15.3 Reducing the interest rate risk of a bond by using a plain vanilla interest rate swap 326

15.4 Cash ﬂow table of a swap 328

15.5 The equivalent coupon bond for an annuity loan with ﬁxed rate 8.40% per year and semi-annual payments 335

15.6 Cash ﬂow table of the annuity loan from Figure 15.5 339

16.1 Forward swap ﬁx vs. 12-month ﬂoating, starting in three years for ten years. 353

16.2 The forward par swap corresponding to Figure 16.1 354 17.1 The input parameters for all of the following examples 357

17.2 Valuation of European plain vanilla spot and futures options using the Black-Scholes model 358

17.3 Valuation of European plain vanilla options using the binomial distribution. 359

17.4 Valuation of an American call c on the spot price S, and of an American call c′ on the forward price S′ using a binomial tree 360

17.5 Interest rate options on the 12-month ﬂoating rate 372 17.6 Pricing swaptions 375

18.1 The range of possible straight lines through to points where the points are known only up to a certain error represented by the bars 397

20.1 Black-Scholes price of a straddle 432 23.1 The portfolio 492

23.2 Market data of the portfolio’s risk factors given in their original currency 492

FIGURES xiii

23.3 Volatilities and correlations of the risk factors 493

23.4 Calculation of the weights needed for the cash ﬂow mapping 494

23.5 The portfolio after the cash ﬂow mapping 494 23.6 Three ways for calculating the value at risk 494

24.1 Value at risk backtesting by means of a binomial test for 250 backtesting periods and 9% VaR conﬁdence 503

24.2 The table of add-ons for 250 backtesting periods as shown in the text of the German Principle I (corresponding to CAD I) 503

25.1 A risk return diagram showing the efﬁcient frontier, the capital market line, and some important characteristic portfolios 523

26.1 The risk return diagram for the same investment universe as in Figure 25.1, but for a different time period 542

28.1 Bootstrapping for coupon periods of equal length according to the methods represented by Equations 28.2 and 28.3 581

28.2 Determining the yield curve using classical bootstrapping 585 29.1 The original price volatilities and correlations 613 29.2 The volatilities and correlations with respect to EUR 613 29.3 The volatilities and correlations with respect to USD 614

30.1 Measuring the yield, the volatility, and their errors from a (simulated) data series of 250 “measurements” 622

31.1 Daily returns of the FTSE index as an example of a stationary data series 636

31.2 Simulated GARCH(1,1) process 647

31.3 Simulated annealing using m Markov chains with n steps in each chain 654

3.1 The components of the ﬁrst four eigenvectors α1, | ,α4 679 |

34.1 Conditional standard deviation of the GARCH(1,1) process ﬁtted to the FTSE time series 693

34.2 Number of outliers outside a 9% conﬁdence interval as a function of λ for a 10-day EWMA volatility forecast 696

34.3 F(α) as deﬁned in Equation 34.5 for GARCH (solid line), MA (dotted line), and EWMA (dash-dotted line) 696 xiv FIGURES

34.5 Daily standard deviations of the S&P500 returns from

GARCH (solid line), MA (dotted line), and EWMA (dash-dotted line) 697

A.1 2000 iid random numbers uniformly distributed between −1/2 and +1/2 735

A.2 The random numbers from Figure A.1 after Transformation A.97 with n=12 735

Tables

2.1 The commonly used day count conventions 8 2.2 Business day conventions 1 2.3 Effects of business day conventions 1

2.4 Interest rate factors in general notation and their speciﬁc form for the four most commonly used compounding methods 13

2.5 Statistical properties of the logarithm of a risk factor and of the risk factor itself 31

3.1 Some examples of international money market instruments 53

3.2 Interest payment modes in international bond markets 56 6.1 Boundaries for the values of plain vanilla options 89

9.1 Properties of the three most commonly used ﬁnite difference methods 152

10.1 Interpretation of the various components in the

Black-Scholes option pricing formulae for plain vanilla calls and puts 178

12.1 Sensitivities for options and forward contracts with respect to the spot price S(t) 210

12.2 Deﬁnitions of the “Greeks” 212 12.3 Examples of Omega and Delta 215

15.1 Present value and sensitivities of a zero bond in the four most commonly used compounding methods 317

15.2 Long/short conventions for swaps and swaptions 327 17.1 Caplets and ﬂoorlets as options on zero bonds 370 17.2 Swaptions as options on coupon bonds 376

18.1 Parameters for the valuation of knock-out options using Equation 18.6 389 xvi TABLES

18.2 Classiﬁcation and “brute force” valuation methods for exotic options 394

21.1 The pros and cons of the most common value at risk methods 469 34.1 Conﬁdence level α = 9% 694 34.2 Conﬁdence level α = 95% 695

Preface

The philosophy of this book is to provide an introduction to the valuation and risk management of modern ﬁnancial instruments formulated in precise (and mathematically correct) expressions, covering all pertinent topics with a consistent and exact notation and with a depth of detail sufﬁcient to give the reader a truly sound understanding of the material – an understanding which even places the reader in a position to independently develop pricing and risk management algorithms (including actually writing computer programs), should this be necessary. Such tasks will greatly be facilitated by the CDROM accompanying the book. This CD-ROM contains Microsoft ExcelTM workbooks presenting concrete realizations of the concepts discussed in the book in the form of executable algorithms. Of course, the reader has full access to all source codes of the Visual BasicTM modules as well as to all calculations done in the spread sheet cells. The CD-ROM thus contains a collection of literally thousands of examples providing the reader with valuable assistance in understanding the complex material and serving as the potential basis for the further development of the reader’s own particular pricing and risk management procedures.

(Parte **1** de 5)