derivatives principle and practice
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Derivatives: Principles and Practice
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Rangarajan K. Sundaram
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Santa Clara University
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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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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