Derivatives Principle and Practice

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    Derivatives: Principles and Practice

     ARTICLE

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    2 AUTHORS:

    Rangarajan K. Sundaram

    New York University

    67 PUBLICATIONS  8,020 CITATIONS 

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    Sanjiv R. Das

    Santa Clara University

    103 PUBLICATIONS  3,203 CITATIONS 

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    Available from: Sanjiv R. Das

    Retrieved on: 16 January 2016

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    Derivatives:

    Principles and

    Practice

    Rangarajan K. Sundaram

    Stern School of  usiness

    New York University

    New   York NY 10012

    Sanjiv R. Das

    Leavey School of  usiness

    Santa Clara University

    Santa Clara CA 95053

    I McGraw Hill

    I Irwin

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    Contents

    Author B iographies  xv

    Preface  xvi

    Acknowledgments  xxi

    Ch a p t e r 1

    Introduction  1

    1.1  Forward and Futures Contracts  5

    1.2

      Options

      9 -

    1.3  Swaps  10

    1.4

      Using Derivatives: Some Com ments

    1 5

      The Structure of this Book  14

    1.6

      Exercises

      15

    11

    P A R T O N E

    Futures and Forwards

    17

    Chap te r 2

    Futures Markets

    19

    2.1 Introduction  19

    2.2 The Changing Face of Futures Markets

      19

    2.3 The Functioning of Futures Exchang es  21

    2.4 The Standardization of Futures Contracts

      30

    2.5 Closing Out Positions  34

    2.6 Margin Requiremen ts and Default Risk

      36

    2.7 Case Studies in Futures Markets  39

    2.8 Exercises

      53

    A p p e n d i x 2 A   Futures Trading and US Regulation:

    A Brief History

      57

    Chapter

     3

    Pricing Forwards and Futures I: The Basic

    Theory  60

    3.1 Introduction

      60

    3.2 Pricing Forwards by Replication  61

    3.3 Examples

      63

    3.4 Forward Pricing on Currencies and Related

    Assets

      66

    3.5 Forward-Rate Agreem ents  69

    3.6 Concept Check

      69

    3.7 The Marked-to-Market Value of a Forward

    Contract

      70

    3.8 Futures Prices  72

    3.9 Exercises  74

    A p p e n d i x 3A

      Compounding Frequency

      79

    A p p e n d i x 3 B

      Forward and Futures Prices with

    Constant Interest Rates  81

    A p p e n d i x 3 C  Rolling Over Futures Contracts

      83

    Chapter 4

    Pricing Forwards and Futures II: Building

    on the Foundations 85

    4 1

      Introduction

      85

    4. 2 From Theory to Reality  85

    4. 3 The Implied Repo Rate

      89

    4. 4 Transactions Costs

      92

    4. 5 Forward Prices and Future Spot Prices

      92

    4. 6 Index Arbitrage  93

    4.7 Exercises

      97

    A p p e n d i x 4A   Forward Prices with Convenience

    Yields

      100

    Chapter 5

    Hedging with Futures and Forwards

    101

    5 1  Introduction  101

    5.2 A Guide to the Main Results

      103

    5.3 The Cash Flow from a Hedged Position  104

    5.4 The Case of No Basis Risk

      105

    5 5  The Minimum -Variance Hedge Ratio  106

    5.6 Examples

      109

    5.7 Implementation  111

    5.8 Further Issues in Implem entation

      112

    5.9 Index Futures and Changing Equity Risk  114

    5 10  Fixed-Income Futures and Duration-Based

    Hedging  115

    5 11  Exercises

      115

    A p p e n d i x 5A

      Derivation of the Optimal Tailed

    Hedge Ratio

     h

    120

    Chapter

     6

    Interest-Rate Forwards and Futures

    6.1 Introduction  122

    6. 2 Eurodollars and Libor Rates

      122

    6.3 Forward-Rate Agreements  123

    6. 4 Eurodollar Futures

      129

    122

    viii

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    Contents 

    6.5 Treasury Bond Futures 136

    6.6 Treasury Note Futures 139

    6.7 Treasury Bill Futures 139

    6. 8 Duration-Based Hedging 140

    6.9 Exercises 143

    A p p e n d ix 6 A   Deriving the Arbitrage-Free

    FRA Rate 147

    A p p e n d i x 6 B  PVBP-Based Hedging Using

    Eurodollar Futures 148

    A p p e n d i x 6 C  Calculating the Conversion

    Factor 149

    A p p e n d ix 6 D  Duration as a Sensitivity

    Measure 150

    A p p e n d ix 6E  The Duration of

     a

     Futures

    Contract 151

    P AR T T W O

    Options 153

    Chap te r 7

    Options Markets

    155

    7.1 Introduction 155

    7.2 Definitions and Terminology 155

    7 3

      Options as Financial Insurance 156

    7 4  Naked Option Positions 158

    7 5

      Options as Views on Market Direction

    and Volatility 162

    7.6 Exercises 165

    A p p e n d i x 7A   Options Markets 167

    Chapter 8

    Options: Payoffs and Trading

    Strategies 171

    8.1 Introduction 171

    8.2 Trading Strategies I: Covered Calls and

    Protective Puts 171

    8.3 Trading Strategies II: Spreads 174

    8.4 Trading Strategies III: Com binations 182

    8.5 Trading Strategies IV: Other Strategies 185

    8.6 Wh ich Strategies Are the Most Widely

    Used? 189

    8.7 The Barings Case 189

    8.8 Exercises 192

    A p p e n d i x 8 A   Asym metric Butterfly

    Spreads 195

    Chapter 9

    No-Arbitrage Restrictions on Option

    Prices 196

    9 1

      Introduction 196

    9.2 Motivating Exam ples 196

    9 3  Notation and Other Preliminaries 198

    9 4  Maximum and Minimum Prices for

    Options 199

    9 5  The Insurance Value of an Option 204

    9. 6 Option Prices and Contract Parameters 205

    9 7  Numerical Examples 208

    9.8 Exercises 210

    Chapter 1

    Early Exercise and Put-Call Parity

    213

    10 1  Introduction 213

    10 .2 A Decomposition of Option Prices 213

    1 0 3  The Optimality of Early Exercise 216

    1 0 4  Put-Call Parity. 220

    1 0 5  Exercises 226

    Chapter 11

    Option Pricing: An Introduction

    228

    11 1  Overview 228

    11 .2 The Binomial Model 229

    1 1 3  Pricing by Replication in a One-Period

    Binomial Model 231

    1 1 4  Comments 235

    1 1 5  Riskless Hedge Portfolios 237

    1 1 .6 Pricing Using Risk-Neutral

    Probabilities 238

    1 1 7  The One-Period Model in General

    Notation 242

    11 .8 The Delta of an Option 242

    1 1 9

      An Application: Portfolio Insurance 246

    1 1 1 0  Exercises 248

    A p p e n d i x 11 A

      Riskless Hedge Portfolios

    and Option Pricing 252

    A p p e n d i x 11 B  Risk-Neutral Probabilities

    and Arrow Security Prices 254

    A p p e n d i x 11 C

      The Risk-Neutral Probability,

    No-Arbitrage, and M arket

    Completeness 255

    A p p e n d i x 1 1 D  Equivalent Martingale

    Measures 257

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    x  Contents

    Chapter 12

    Binomial Option Pricing

    259

    12 1  Introduction 259

    1 2 2  The Two-Period Binomial Tree 261

    12 3  Pricing Two-Period European Options 262

    1 2 4  European Option Pricing in General w-Period

    Trees 269

    12 5  Pricing Am erican Options: Preliminary

    Comments 269

    12 6  Am erican Puts on Non-D ividend-Paying

    Stocks 270

    12 7  Cash Dividends in the Binomial Tree 272

    1

    2 8

      An Alternative Approach to Cash

    Dividends 275

    1

    2 9

      Dividend Yields in Binom ial Trees 279

    12 10  Exercises 282

    A ppendix 12A

      A General Representation of

    European Option Prices 286

    Chapter 13

    Implementing the Binomial Model 289

    13 1

      Introduction 289

    1 3 2  The Lognormal Distribution 289

    13 3

      Binomial Approxim ations of the

    Lognormal 294

    1

    3 4

      Computer Implem entation of the Binomial

    Model 298

    13 5

      Exercises 303

    A ppendix 13A   Estimating Historical

    Volatility 306

    Chapter 14

    The Black-Scholes M odel

    308

    14 1  Introduction 308

    1

    4 2

      Option Pricing in the Black-Scholes

    Setting 310

    14 3

      Remarks on the Formula 313

    1 4 4  Working with the Formulae I: Plotting Option

    Prices 314

    14 5  Working with the Formulae II: Algebraic

    Manipulation 315

    14 6  Dividends in the Black-Scholes Model 319

    1 4 7

      Options on Indices, Currencies,

    and Futures 324

    1 4 8

      Testing the Black-Sch oles Model: Implied

    Volatility 327

    1 4 9

      The VIX and Its Derivatives 332

    1 4 1 0  Exercises 335

    A p p e n d i x 14 A   Further Properties of the

    Black-Sch oles Delta 338

    A p p e n d i x 14 B

      Variance and Volatility Swaps

    339

    Chapter 15

    The M athem atics of Black-Scholes 344

    344

    15 1  Introduction 344

    1 5 2  Geom etric Brownian Motion Defined

    1 5 3  The Black-Scholes Formula via

    Replication 348

    15 4  The Black-Scholes Formula via Risk-Neutral

    Pricing 351

    1 5 5  The Black-Scholes Formula via CAPM 353

    1 5 6

      Exercises 354

    Chapte r 16

    Options Modeling:

    Beyond Black-Scholes

    357

    16 1

      Introduction 357

    1 6 2  Jump-Diffusion Mo dels 358

    1 6 3

      Stocha stic Volatility 368

    1 6 4  GARCH Models 374

    1 6 5

      Other Approaches 378

    1 6 6  Implied Binom ial Trees/Local Volatility

    Models 379

    1 6 7  Summary 389

    1 6 8

      Exercises 389

    A p p e n d i x 16 A   Program Code for Jump -

    Diffusions 393

    A p p e n d i x 16 B  Program Code for a Stochastic

    Volatility Mo del 394

    A p p e n d i x 1 6 C  Heuristic Com ments on Option

    Pricing under Stochastic

    Volatility 396

    A p p e n d ix 1 6 D

      Program Code for Simulating

    GARCH Stock Prices

    Distributions 399

    A p p e n d i x 16 E   Local Volatility Models: The Fourth

    Period of

     the

     Example 400

    Chapter 17

    Sensitivity Analysis: The Option

      Gre eks 404

    17 1

      Introduction 404

    17 2  Interpreting the Greeks: A Snapshot

    View 404

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    Contents 

    17 3  The Option Delta 408

    1

    7 4

      The Option Gamma 412

    17 5  The Option Theta 418

    1

    7 6

      The Option Vega 423

    1  The Option Rho 426

    1

    7 8

      Portfolio Greeks 429

    17 9  Exercises 432

    Appendix 17A

      Deriving the Black-Scholes

    Option Greeks 436

    Chapter  8

    Exotic Options I: Path-Independent

    Options  440

    18 1

      Introduction 440

    18 2

      Forward Start Options 442

    18 3

      Binary Options 445

    18 4

      Chooser Options 450

    18 5

      Compound Options 453

    18 6

      Exchange Options 458

    18 7

      Quanta Options 460

    1

    8 8

      Variants on the Exchange

    Option Theme 462

    18 9

      Exercises 465

    Chapter  9

    Exotic Options II: Path-Dependent

    Options

      470

    19 1

      Path-Dependent Exotic

    Options 470

     

    19 2  Barrier Options 470

    19 3

      Asian Options 479

    19 4  Lookback Options 485

    19 5

      Cliquets 488

    19 6  Shout Options 490

    19 7

      Exercises 492

    Appendix 19A  Barrier Option Pricing

    Formulae 496

    Chapter

     20

    Value-at-Risk

    498

    20 1

      Introduction 498

    20 2

      Value-at-Risk 498

    20 3

      Risk Decomposition 505

    20 4

      Coherent Risk Measures 511

    20 5

      Exercises 515

    Chapter 21

    Convertible Bonds

    519

    21 1

      Introduction 519

    21 2

      Convertible Bond Terminology 519

    21

    .3 Main Features of Convertible Bonds 520

    21

    .4 Breakeven Analysis 522

    21

    .5 Pricing Convertibles: A First Pass 523

    21.6 Incorporating Credit Risk 530

    21 7

      Convertible Greeks 534

    21.8 Convertible Arbitrage 542

    21

    .9 Summary 542

    21 10  Exercises 543

    Appendix 21A

      Octave Code for the Blended

    Discount Rate Valuation Tree 545

    Appendix 21B

      Octave Code for the Simplified

    Das-Sundaram Model 546

    Chapter

     22

    Real Options

    548

    22 1

      Introduction 548

    22 2  Preliminary Analysis and Examples

    22 3

      A Real Options Case Study 554

    22 4  Creating the State Space 560

    22 5

      Applications of Real Options 563

    22 6  Summary 564

    22 7

      Exercises 564

    550

    Appendix 22A

    Derivation of Cash-Flow Value

    in the Waiting-to-Invest

    Example 568

      RT THR

    Swaps  569

    Chapter

     23

    Interest Rate Swaps and Floating-Rate

    Products

      571

    23 1  Introduction 571

    23 2

      Floating-Rate Notes 571

    23 3

      Interest Rate Swaps 575

    23 4

      Uses of Swaps 576

    23 5  Swap Payoffs 579

    23 6

      Valuing and Pricing Swaps 582

    23 7  Extending the Pricing Arguments 586

    23 8

      Case Study: The Procter & Gamble-Bankers

    Trust 5/30 Swap 589

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    xii

      Contents

    2 3 9  Case Study: A Long-Term Capital

    Management Convergence Trade 593

    2 3 10 Credit Risk and Credit Exposure 596

    23 11

      Hedging Swaps 597

    2 3 1 2  Caps, Floors, and Swaptions 599

    2 3 1 3  The Black Model for Pricing Caps, Floors ,

    and Swaptions 604

    2 3 1 4  Summary 609

    2 3 1 5

      Exercises 609

    Chapter 24

    Equity Swaps

    613

    24 1  Introduction 613

    24 2  Uses of Equity Swaps 614

    24 3  Payoffs from Equity Swaps 616

    24 4

      Valuation and Pricing of Equity Swaps

    24 5  Summary 628

    24 6  Exercises 628

    622

    Chapter 25

    Currency and Commodity Swaps

    25 1  Introduction 631

    25 2

      Currency Swaps 631

    25 3  Comm odity Swaps 639

    25 4  Summary 643

    25 5  Exercises 644

    631

    P A R T F O U R

    Interest Rate Modeling

    647

    Chapter 26

    The Term Structure of Interest Rates:

    Concepts 649

    26 1  Introduction 649

    2 6 2

      The Yield-to-Maturity 649

    2 6 3  The Term Structure of Interest Rates 651

    2 6 4

      Discount Functions 652

    2 6 5  Zero-Coupon Rates 653

      6 6  Forward Rates 654

    2 6 7

      Yield-to-Maturity, Zero-Coupon Rates,

    and Forward Rates 656

    2 6 8

      Constructing the Yield-to-Maturity Curve:

    An Em pirical Illustration 657

    2 6 9

      Summary 661

    2 6 1 0  Exercises 662

    A p p e n d i x 26 A   The Raw YTM Data 664

    Chapter 27

    Estima ting the Yield C urve

    667

    27 1  Introduction 667

    2 7 2

      Bootstrapping 667

    2 7 3  Splines 669

    2 7 4

      Polynomial Splines 670

    2 7 5  Exponential Splines 673

    2 7 6  Implementation Issues with Splines 674

    2 7 7

      The Nelson-Siegel-Svensson Approach 674

    2 7 8  Summary 676

    2 7 9

      Exercises 676

    A p p e n d i x 27 A   Bootstrapping by Matrix

    Inversion 680

    A p p e n d i x 2 7 B  Implementation with Exponential

    Splines 681

    Chapter 28

    Modeling Term -Structure Movements 684

    28 1

      Introduction 684

    2 8 2  Interest-Rate Mo deling versus Equity

    Modeling 684

    2 8 3  Arbitrage Violations: A Simple

    Example 685

    2 8 4

      A Gentle Introduction to No-Arbitrage

    Modeling 687

    2 8 5

      No-Arbitrage and Equilibrium

    Models 693

    2 8 6  Summary 697

    2 8 7  Exercises 697

    Chapter 29

    Factor Models of the Term Structure 700

    2 9 1  Overview 700

    2 9 2

      The Black-Derman-Toy Model 701

    2 9 3  The Ho-Lee Model 710

    2 9 4  One-Factor Models in Continuous Time 714

    2 9 5  Multifactor Models 720

    2 9 6  Affine Factor Mo dels 722

    2 9 7

      Summary 725

    2 9 8  Exercises 726

    A p p e n d i x 2 9 A

      Deriving the Fundamen tal PDE

    in Factor Models 729

    Chapter 3

    The Heath-Jarrow-Morton and Libor

    M arket Models 731

    30.1 Overview 731

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    Contents

      xiii

    30 2

    30 3

    30 4

    30 5

    30 6

    30 7

    30 8

    30 9

    30 10

    30 11

    30 12

    30 13

    30 14

    30 15

    The HJM Framework: Preliminary

    Comments 731

    A One-Factor HJM Model 733

    A Two-Factor HJM Setting 742

    The HJM Risk-Neutral Drifts: An

    Derivation 746

    Libor Market Models 749

    Mathematical Excursion: Marting;

    Libor Rates: Notation 751

    Risk-Neutral Pricing in the LMM

    Simulation of the Market Model

    Calibration 757

    Swap Market Models 758

    Swaptions 760

    Summary 761

    Exercises 761

    Appendix 30A

      Risk-Neutral Drifts

    P A R T

    Credit

    and Volatilities in HJM

      IV

    Risk 769

    Algebraic

    ales 750

    753

    757

    765

    Chapter 33

    Reduced-Form Models of Default Risk

    Chapter 3

    Credit Derivative Products

    771

    779

    31 1  Introduction 771

    31 2

      Total Return Swaps 775

    31 3

      Credit Spread Options/Forwards

    31 .4 Credit Default Swaps / 779

    31.5 Credit-Linked Notes

     

    788

    31 6

      Correlation Products 790

    31 7

      Summary 797

    31 8

      Exercises 797

    Appendix 31A

      The CDS Big Bang 800

    Chapter 32

    Structural Models of Default Risk 802

    32 1

    32 2

    32 3

    32 4

    32 5

    32 6

    32 7

    32 8

    Introduction 802

    The Merton 1974) Model

    Issues in Implementation

    A Practitioner Model 817

    803

    812

    Extensions of the Merton Model 819

    Evaluation of the Structural

    Approach 820

    Summary 823

    Exercises 824

    Model

    33 1

    33 2

    33 3

    33 4

    33 5

    33 6

    33 7

    33 8

    33 9

    33 10

    Introduction 829

    Modeling Default I: Intensity Processes

      \

    Modeling Default II: Recovery Rate

    Conventions 834

    The Litterman-Iben Model 836

    The Duffie-Singleton Result 841

    Defaultable HJM Models 843

    Ratings-Based Modeling: The JLT

    Model 845

    An Application of Reduced-Form Models:

    Pricing CDS 853

    Summary 855

    Exercises 855

    Appendix 33A

      Duffle-Singleton

    in Discrete Time 859

    Appendix 33B

      Derivation of the Drift-Volatility

    Relationship 860

    Chapter 34

    Modeling Correlated Default 863

    34 1

    34 2

    34 3

    34 4

    34 5

    34 6

    34 7

    34 8

    34 9

    34 10

    34 11

    Introduction 863

    Examples of Correlated Default

    Products 863

    Simple Correlated Default Math 865

    Structural Models Based on

    Asset Values 868

    Reduced-Form Models 874

    Multiperiod Correlated Default 875

    Fast Computation of Credit Portfolio Loss

    Distributions without Simulation 878

    Copula Functions 881

    Top-Down Modeling of Credit

    Portfolio Loss 893

    Summary 897

    Exercises 898

    Bibliography B-l

    Index 1-1

    829

    830

    Appendix 32A  The Delianedis-Geske

    Model 826

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    9/9

    xiv   Contents

    The following Web chapters are

    available at w ww .mhhe.com /sdle:

    PART SIX

    Computation 901

    Chapter 35

    Derivative Pricing with Finite

    Differencing 903

    35 1  Introduction 903

    35 2  Solving Differential Equations 904

    35 3  A First Approach to Pricing Equity

    Options 907

    35 4  Imp licit Finite Differencing 913

    35 5  The Crank-Nicholson Scheme 917

    35 6

      Finite Differencing for Term-S tructure

    Models 919

    35 7

      Summary 921

    35 8  Exercises 922

    Chapter 36

    Derivative Pricing with Monte Carlo

    Simulation 923

    36 1  Introduction 923

    36 2

      Simulating No rmal Random Variables 924

    36 3  Bivariate Rando m Variables 925

    36 4

      Cholesky Decom position 925

    36 5  Stochastic Processes for Equity Prices 927

    36 6

      ARCH Models 929

    36 7  Interest-Rate Processes 930

    36 8

      Estim ating Histo rical Volatility for

    Equities 932

    36 9

      Estim ating Histo rical Volatility for Interest

    Rates 932

    36 10

      Path-De pendent Options 933

    36 11  Variance Redu ction 935

    36 12

      Mo nte Carlo for Am erican Options 938

    36 13  Summ ary 942

    36 14

      Exercises 943

    Chapter 37

    Using Octave 945

    37 1  Some Simple Commands 945

    37 2

      Reg ression and Integration 948

    37 3  Rea ding in Data, Sorting, and Finding 950

    37 4

      Equ ation Solving 955

    37 5  Screenshots 955