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Derivatives markets products and participants: an overview
By contrast speculators attempts to profit from anticipating changes in market prices or rates or credit events by entering a derivative contract. According to
Counterparty Risk in the Over-The-Counter Derivatives Market
Here we discuss counterparty risk that may stem from the OTC derivatives markets and attempt to assess the scope of potential cascade effects. This risk is
International Standards for Derivatives Market Intermediary Regulation
12-Jun-2012 into account the distinctions between the OTC derivatives market and ... making a market or intermediating transactions in OTC derivatives.
OTC derivatives market activity in the first half of 2011
Overall interest rate derivative market values fell by around 10%. •. FX derivatives (Table 2): The notional amounts of FX derivatives increased by 12% with
Collateral Netting and Systemic Risk in the OTC Derivatives Market
08-Apr-2010 cognizant that large banks active in the OTC derivatives market do not hold collateral against all the positions in their trading book and ...
Triennial Central Bank Survey 2004 - BIS - March 2005
1.1 Global daily turnover in OTC derivatives markets . foreign exchange and derivatives market data and to which requests for.
OTC derivatives market activity in the second half of 2010
Gross market values provide a more accurate measure of the scale of financial risk transfer taking place in derivatives markets. Gross positive and negative
Monetary and Economic Department OTC derivatives market activity
Gross market values of OTC interest rate derivatives grew by 29% to $9.3 trillion driven primarily by interest rate swaps
Frequently Asked Questions on Derivatives Trading At NSE
Derivatives such as futures or options
Triennial Central Bank Survey Interest rate derivatives market
08-Dec-2019 Interest rate derivatives market turnover in April 2019. By instrument currency
[PDF] Derivatives Markets
Trading of Financial Assets 2 Measures of Market Size and Activity 4 Stock and Bond Markets 5 Derivatives Markets 6 1 3 The Role of Financial Markets 9
[PDF] Derivatives markets - Bank for International Settlements
Derivatives markets The aggregate turnover of exchange-traded financial derivatives contracts monitored by Activity was weaker across the major market risk
[PDF] The Global Derivatives Market A Blueprint for Market Safety and
Derivatives are financial instruments that are traded among market participants over the counter (OTC) or via regulated markets (on-exchange), whereby the
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829_2derivatives_markets_3e_0.pdf Derivatives Markets
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Derivatives MarketsTHIRD EDITIONRobert L. McDonaldNorthwestern University Kellogg School of ManagementBoston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei TokyoEditor in Chief:Donna Battista
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printed in initial caps or all caps. Library of Congress Cataloging-in-Publication DataMcDonald, Robert L. (Robert Lynch)
Derivatives markets / Robert L. McDonald. " 3rd ed. p. cm.Includes bibliographical references and index.
ISBN-13: 978-0-321-54308-0 (hardcover)
ISBN-10: 0-321-54308-4 (hardcover)
1. Derivative securities. I. Title.
HG6024.A3M394 2013
332.6457"dc23 2012029875
10987654321ISBN 10: 0-321-54308-4
ISBN 13: 978-0-321-54308-0
ForIrene,Claire,David, andHenry
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Brief ContentsPreface xxi1 Introduction to Derivatives 1PART ONE Insurance, Hedging, and Simple Strategies 232 An Introduction to Forwards and Options 253 Insurance, Collars, and Other Strategies 61
4 Introduction to Risk Management 89PART TWO
Forwards, Futures, and Swaps 1235 Financial Forwards and Futures 1256 Commodity Forwards and Futures 163
7 Interest Rate Forwards and Futures 195
8 Swaps 233PART THREE
Options 2639 Parity and Other Option Relationships 26510 Binomial Option Pricing: Basic Concepts 293
11 Binomial Option Pricing: Selected Topics 323
12 The Black-Scholes Formula 349
13 Market-Making and Delta-Hedging 381
14 Exotic Options: I 409
vii viiiBrief ContentsPART FOUR Financial Engineering and Applications 43515 Financial Engineering and Security Design 43716 Corporate Applications 469
17 Real Options 509PART FIVE
Advanced Pricing Theory and Applications 54318 The Lognormal Distribution 54519 Monte Carlo Valuation 573
20 Brownian Motion and It
os Lemma 60321 The Black-Scholes-Merton Equation 627
22 Risk-Neutral and Martingale Pricing 649
23 Exotic Options: II 683
24 Volatility 717
25 Interest Rate and Bond Derivatives 751
26 Value at Risk 789
27 Credit Risk 815
Appendix A The Greek Alphabet 851
Appendix B Continuous Compounding 853
Appendix C Jensens Inequality 859
Appendix D An Introduction to Visual Basic for Applications 863Glossary 883References 897
Index 915
ContentsPreface xxiChapter 1
Introduction to Derivatives 11.1 What Is a Derivative? 21.2 An Overview of Financial Markets 2
Trading of Financial Assets 2
Measures of Market Size and Activity 4
Stock and Bond Markets 5
Derivatives Markets 6
1.3 The Role of Financial Markets 9
Financial Markets and the Averages 9
Risk-Sharing 10
1.4 The Uses of Derivatives 11
Uses of Derivatives 11
Perspectives on Derivatives 13
Financial Engineering and Security
Design 14
1.5 Buying and Short-Selling Financial
Assets 14
Transaction Costs and the Bid-Ask
Spread 14
Ways to Buy or Sell 15
Short-Selling 16
The Lease Rate of an Asset 18
Risk and Scarcity in Short-Selling 18
Chapter Summary 20
Further Reading 20
Problems 20PART ONEInsurance, Hedging, and Simple
Strategies 23Chapter 2
An Introduction to Forwards and
Options 252.1 Forward Contracts 25
The Payoff on a Forward Contract 29
Graphing the Payoff on a Forward
Contract 30
Comparing a Forward and Outright
Purchase 30
Zero-Coupon Bonds in Payoff and Prot
Diagrams 33
Cash Settlement Versus Delivery 34
Credit Risk 34
2.2 Call Options 35
Option Terminology 35
Payoff and Prot for a Purchased Call
Option 36
Payoff and Prot for a Written Call
Option 38
2.3 Put Options 41
Payoff and Prot for a Purchased Put
Option 41
Payoff and Prot for a Written Put
Option 42
The Moneyness of an Option 44ix
xContents2.4 Summary of Forward and Option
Positions 45
Positions Long with Respect to the
Index 45
Positions Short with Respect to the
Index 46
2.5 Options Are Insurance 47
Homeowners Insurance Is a Put
Option 48
But I Thought Insurance Is Prudent and
Put Options Are Risky... 48
Call Options Are Also Insurance 49
2.6 Example: Equity-Linked CDs 50
Graphing the Payoff on the CD 50
Economics of the CD 52
Why Equity-Linked CDs? 52
Chapter Summary 53
Further Reading 54
Problems 55
2.A More on Buying a Stock Option 57
Dividends 57
Exercise 57
Margins for Written Options 58
Taxes 58Chapter 3
Insurance, Collars, and Other
Strategies 613.1 Basic Insurance Strategies 61
Insuring a Long Position: Floors 61
Insuring a Short Position: Caps 64
Selling Insurance 66
3.2 Put-Call Parity 68
Synthetic Forwards 68
The Put-Call Parity Equation 70
3.3 Spreads and Collars 71
Bull and Bear Spreads 71
Box Spreads 73
Ratio Spreads 74
Collars 74
3.4 Speculating on Volatility 79
Straddles 79
Butter"y Spreads 80
Asymmetric Butter"y Spreads 82
Chapter Summary 84
Further Reading 86
Problems 86Chapter 4
Introduction to Risk Management 894.1 Basic Risk Management: The ProducersPerspective 89
Hedging with a Forward Contract 90
Insurance: Guaranteeing a Minimum Price
with a Put Option 91Insuring by Selling a Call 93
Adjusting the Amount of Insurance 95
4.2 Basic Risk Management: The Buyers
Perspective 96
Hedging with a Forward Contract 97
Insurance: Guaranteeing a Maximum Price
with a Call Option 974.3 Why Do Firms Manage Risk? 99
An Example Where Hedging Adds
Value 100
Reasons to Hedge 102
ReasonsNotto Hedge 104
Empirical Evidence on Hedging 104
4.4 Golddiggers Revisited 107
Selling the Gain: Collars 107
Other Collar Strategies 111
Paylater Strategies 111
4.5 Selecting the Hedge Ratio 112
Cross-Hedging 112
Quantity Uncertainty 114
Chapter Summary 117
Further Reading 118
Problems 118PART TWOForwards, Futures, and
Swaps 123Chapter 5
Financial Forwards and Futures 1255.1 Alternative Ways to Buy a Stock 1255.2 Prepaid Forward Contracts on Stock 126
Pricing the Prepaid Forward by
Analogy 127
Pricing the Prepaid Forward by Discounted
Present Value 127
ContentsxiPricing the Prepaid Forward by
Arbitrage 127
Pricing Prepaid Forwards with
Dividends 129
5.3 Forward Contracts on Stock 131
Does the Forward Price Predict the Future
Spot Price? 132
Creating a Synthetic Forward
Contract 133
Synthetic Forwards in Market-Making and
Arbitrage 135
No-Arbitrage Bounds with Transaction
Costs 136
Quasi-Arbitrage 137
An Interpretation of the Forward Pricing
Formula 138
5.4 Futures Contracts 138
The S&P 500 Futures Contract 139
Margins and Marking to Market 140
Comparing Futures and Forward
Prices 143
Arbitrage in Practice: S&P 500 Index
Arbitrage 143
Quanto Index Contracts 145
5.5 Uses of Index Futures 146
Asset Allocation 146
Cross-hedging with Index Futures 147
5.6 Currency Contracts 150
Currency Prepaid Forward 150
Currency Forward 152
Covered Interest Arbitrage 152
5.7 Eurodollar Futures 153
Chapter Summary 157
Further Reading 158
Problems 158
5.A Taxes and the Forward Rate 161
5.B Equating Forwards and Futures 162
5.C Forward and Futures Prices 162Chapter 6
Commodity Forwards and
Futures 1636.1 Introduction to Commodity
Forwards 164
Examples of Commodity Futures
Prices 164Differences Between Commodities and
Financial Assets 166
Commodity Terminology 166
6.2 Equilibrium Pricing of Commodity
Forwards 167
6.3 Pricing Commodity Forwards by
Arbitrage 168
An Apparent Arbitrage 168
Short-selling and the Lease Rate 170
No-Arbitrage Pricing Incorporating
Storage Costs 172
Convenience Yields 174
Summary 175
6.4 Gold 175
Gold Leasing 176
Evaluation of Gold Production 177
6.5 Corn 178
6.6 Energy Markets 179
Electricity 180
Natural Gas 180
Oil 182
Oil Distillate Spreads 184
6.7 Hedging Strategies 185
Basis Risk 186
Hedging Jet Fuel with Crude Oil 187
Weather Derivatives 188
6.8 Synthetic Commodities 189
Chapter Summary 191
Further Reading 192
Problems 192Chapter 7
Interest Rate Forwards and
Futures 1957.1 Bond Basics 195
Zero-Coupon Bonds 196
Implied Forward Rates 197
Coupon Bonds 199
Zeros from Coupons 200
Interpreting the Coupon Rate 201
Continuously Compounded Yields 202
7.2 Forward Rate Agreements, Eurodollar
Futures, and Hedging 202
Forward Rate Agreements 203
Synthetic FRAs 204
Eurodollar Futures 206
xiiContents7.3 Duration and Convexity 211
Price Value of a Basis Point and DV01 211
Duration 213
Duration Matching 214
Convexity 215
7.4 Treasury-Bond and Treasury-Note
Futures 217
7.5 Repurchase Agreements 220
Chapter Summary 222
Further Reading 224
Problems 225
7.A Interest Rate and Bond Price
Conventions 228
Bonds 228
Bills 230Chapter 8
Swaps 2338.1 An Example of a Commodity Swap 233
Physical Versus Financial Settlement 234
Why Is the Swap Price Not $110.50? 236
The Swap Counterparty 237
The Market Value of a Swap 238
8.2 ComputingtheSwapRateinGeneral 240
Fixed Quantity Swaps 240
Swaps with Variable Quantity and
Price 241
8.3 Interest Rate Swaps 243
A Simple Interest Rate Swap 243
Pricing and the Swap Counterparty 244
Swap Rate and Bond Calculations 246
The Swap Curve 247
The Swaps Implicit Loan Balance 248
Deferred Swaps 249
Related Swaps 250
Why Swap Interest Rates? 251
Amortizing and Accreting Swaps 252
8.4 Currency Swaps 252
Currency Swap Formulas 255
Other Currency Swaps 256
8.5 Swaptions 256
8.6 Total Return Swaps 257
Chapter Summary 259
Further Reading 260
Problems 261PART THREEOptions 263Chapter 9
Parity and Other Option
Relationships 2659.1 Put-Call Parity 265
Options on Stocks 266
Options on Currencies 269
Options on Bonds 269
Dividend Forward Contracts 269
9.2 Generalized Parity and Exchange
Options 270
Options to Exchange Stock 272
What Are Calls and Puts? 272
Currency Options 273
9.3 Comparing Options with Respect to Style,
Maturity, and Strike 275
European Versus American Options 276
Maximum and Minimum Option
Prices 276
Early Exercise for American Options 277
Time to Expiration 280
Different Strike Prices 281
Exercise and Moneyness 286
Chapter Summary 286
Further Reading 287
Problems 288
9.A Parity Bounds for American Options 291
9.B Algebraic Proofs of Strike-Price
Relations 292Chapter 10
Binomial Option Pricing: Basic
Concepts 29310.1 A One-Period Binomial Tree 293
Computing the Option Price 294
The Binomial Solution 295
Arbitraging a Mispriced Option 297
A Graphical Interpretation of the Binomial
Formula 298
Risk-Neutral Pricing 299
10.2 Constructing a Binomial Tree 300
ContentsxiiiContinuously Compounded Returns 301
Volatility 302
Constructinguandd303
Estimating Historical Volatility 303
One-Period Example with a Forward
Tree 305
10.3 Two or More Binomial Periods 306
A Two-Period European Call 306
Many Binomial Periods 308
10.4 Put Options 309
10.5 American Options 310
10.6 Options on Other Assets 312
Option on a Stock Index 312
Options on Currencies 312
Options on Futures Contracts 314
Options on Commodities 315
Options on Bonds 316
Summary 317
Chapter Summary 318
Further Reading 319
Problems 319
10.A Taxes and Option Prices 322Chapter 11
Binomial Option Pricing: Selected
Topics 32311.1 Understanding Early Exercise 323
11.2 Understanding Risk-Neutral Pricing 326
The Risk-Neutral Probability 326
Pricing an Option Using Real
Probabilities 327
11.3 The Binomial Tree and Lognormality 330
The Random Walk Model 330
Modeling Stock Prices as a Random
Walk 331
The Binomial Model 332
Lognormality and the Binomial Model 333
Alternative Binomial Trees 335
Is the Binomial Model Realistic? 336
11.4 Stocks Paying Discrete Dividends 336
Modeling Discrete Dividends 337
Problems with the Discrete Dividend
Tree 337
A Binomial Tree Using the Prepaid
Forward 339Chapter Summary 340
Further Reading 341
Problems 341
11.A Pricing Options with True
Probabilities 343
11.B Why Does Risk-Neutral Pricing
Work? 344
Utility-Based Valuation 344
Standard Discounted Cash Flow 345
Risk-Neutral Pricing 345
Physical vs. Risk-Neutral Probabilities 346
Example 347Chapter 12
The Black-Scholes Formula 34912.1 Introduction to the Black-ScholesFormula 349
Call Options 349
Put Options 352
When Is the Black-Scholes Formula
Valid? 352
12.2 Applying the Formula to Other
Assets 353
Options on Stocks with Discrete
Dividends 354
Options on Currencies 354
Options on Futures 355
12.3 Option Greeks 356
Denition of the Greeks 356
Greek Measures for Portfolios 361
Option Elasticity 362
12.4 Profit Diagrams Before Maturity 366
Purchased Call Option 366
Calendar Spreads 367
12.5 Implied Volatility 369
Computing Implied Volatility 369
Using Implied Volatility 370
12.6 Perpetual American Options 372
Valuing Perpetual Options 373
Barrier Present Values 374
Chapter Summary 374
Further Reading 375
Problems 375
12.A The Standard Normal Distribution 378
xivContents12.B Formulas for Option Greeks 379
Delta () 379
Gamma () 379
Theta () 379
Vega 380
Rho () 380
Psi () 380Chapter 13
Market-Making and Delta-
Hedging 38113.1 What Do Market-Makers Do? 381
13.2 Market-Maker Risk 382
Option Risk in the Absence of
Hedging 382
Delta and Gamma as Measures of
Exposure 383
13.3 Delta-Hedging 384
An Example of Delta-Hedging for 2
Days 385
Interpreting the Prot Calculation 385
Delta-Hedging for Several Days 387
A Self-Financing Portfolio: The Stock
Moves One389
13.4 The Mathematics of Delta-Hedging 389
Using Gamma to Better Approximate the
Change in the Option Price 390
Delta-Gamma Approximations 391
Theta: Accounting for Time 392
Understanding the Market-Makers
Prot 394
13.5 The Black-Scholes Analysis 395
The Black-Scholes Argument 396
Delta-Hedging of American Options 396
What Is the Advantage to Frequent
Re-Hedging? 397
Delta-Hedging in Practice 398
Gamma-Neutrality 399
13.6 Market-Making as Insurance 402
Insurance 402
Market-Makers 403
Chapter Summary 403
Further Reading 404
Problems 404
13.A Taylor Series Approximations 406
13.B Greeks in the Binomial Model 407Chapter 14
Exotic Options: I 40914.1 Introduction 409
14.2 Asian Options 410
XYZs Hedging Problem 411
Options on the Average 411
Comparing Asian Options 412
An Asian Solution for XYZ 413
14.3 Barrier Options 414
Types of Barrier Options 415
Currency Hedging 416
14.4 Compound Options 418
Compound Option Parity 419
Options on Dividend-Paying Stocks 419
Currency Hedging with Compound
Options 421
14.5 Gap Options 421
14.6 Exchange Options 424
European Exchange Options 424
Chapter Summary 425
Further Reading 426
Problems 426
14.A Pricing Formulas for Exotic Options 430
Asian Options Based on the Geometric
Average 430
Compound Options 431
Innitely Lived Exchange Option 432PART FOURFinancial Engineering andApplications 435Chapter 15
Financial Engineering and Security
Design 43715.1 The Modigliani-Miller Theorem 437
15.2 Structured Notes without Options 438
Single Payment Bonds 438
Multiple Payment Bonds 441
15.3 Structured Notes with Options 445
Convertible Bonds 446
Reverse Convertible Bonds 449
Tranched Payoffs 451
ContentsxvVariable Prepaid Forwards 452
15.4 Strategies Motivated by Tax and
Regulatory Considerations 453
Capital Gains Deferral 454
Marshall & Ilsley SPACES 458
15.5 Engineered Solutions for
Golddiggers 460
Gold-Linked Notes 460
Notes with Embedded Options 462
Chapter Summary 463
Further Reading 464
Problems 464Chapter 16
Corporate Applications 46916.1 Equity, Debt, and Warrants 469Debt and Equity as Options 469
Leverage and the Expected Return on Debt
and Equity 472Multiple Debt Issues 477
Warrants 478
Convertible Bonds 479
Callable Bonds 482
Bond Valuation Based on the Stock
Price 485
Other Bond Features 485
Put Warrants 486
16.2 Compensation Options 487
The Use of Compensation Options 487
Valuation of Compensation Options 489
Repricing of Compensation Options 492
Reload Options 493
Level 3 Communications 495
16.3 The Use of Collars in Acquisitions 499
The Northrop Grumman"TRW
merger 499Chapter Summary 502
Further Reading 503
Problems 503
16.A An Alternative Approach to Expensing
Option Grants 507Chapter 17
Real Options 50917.1 Investment and the NPV Rule 509Static NPV 510The Correct Use of NPV 511
The Project as an Option 511
17.2 Investment under Uncertainty 513
A Simple DCF Problem 513
Valuing Derivatives on the Cash Flow 514
Evaluating a Project with a 2-Year
Investment Horizon 515
Evaluating the Project with an Innite
Investment Horizon 519
17.3 Real Options in Practice 519
Peak-Load Electricity Generation 519
Research and Development 523
17.4 Commodity Extraction as an Option 525
Single-Barrel Extraction under
Certainty 525
Single-Barrel Extraction under
Uncertainty 528
Valuing an Innite Oil Reserve 530
17.5 Commodity Extraction with Shutdown
and Restart Options 531Permanent Shutting Down 533
Investing When Shutdown Is Possible 535
Restarting Production 536
Additional Options 537
Chapter Summary 538
Further Reading 538
Problems 538
17.A Calculation of Optimal Time to Drill an
Oil Well 541
17.B The Solution with Shutting Down and
Restarting 541PART FIVEAdvanced Pricing Theory andApplications 543Chapter 18
The Lognormal Distribution 54518.1 The Normal Distribution 545Converting a Normal Random Variable to
Standard Normal 548
Sums of Normal Random Variables 549
18.2 The Lognormal Distribution 550
18.3 A Lognormal Model of Stock Prices 552
xviContents18.4 Lognormal Probability Calculations 556
Probabilities 556
Lognormal Prediction Intervals 557
The Conditional Expected Price 559
The Black-Scholes Formula 561
18.5 Estimating the Parameters of a Lognormal
Distribution 562
18.6 How Are Asset Prices Distributed? 564
Histograms 564
Normal Probability Plots 566
Chapter Summary 569
Further Reading 569
Problems 570
18.A The Expectation of a Lognormal
Variable 571
18.B Constructing a Normal Probability
Plot 572Chapter 19
Monte Carlo Valuation 57319.1 Computing the Option Price as aDiscounted Expected Value 573
Valuation with Risk-Neutral
Probabilities 574
Valuation with True Probabilities 575
19.2 Computing Random Numbers 577
19.3 Simulating Lognormal Stock Prices 578
Simulating a Sequence of Stock Prices 578
19.4 Monte Carlo Valuation 580
Monte Carlo Valuation of a European
Call 580
Accuracy of Monte Carlo 581
Arithmetic Asian Option 582
19.5 Efficient Monte Carlo Valuation 584
Control Variate Method 584
Other Monte Carlo Methods 587
19.6 Valuation of American Options 588
19.7 The Poisson Distribution 591
19.8 Simulating Jumps with the Poisson
Distribution 593
Simulating the Stock Price with
Jumps 593
Multiple Jumps 596
19.9 Simulating Correlated Stock Prices 597
GeneratingnCorrelated Lognormal
Random Variables 597Chapter Summary 599
Further Reading 599
Problems 599
19.A Formulas for Geometric Average
Options 602Chapter 20
Brownian Motion and It
osLemma 60320.1 The Black-Scholes Assumption about
Stock Prices 603
20.2 Brownian Motion 604
Denition of Brownian Motion 604
Properties of Brownian Motion 606
Arithmetic Brownian Motion 607
The Ornstein-Uhlenbeck Process 608
20.3 Geometric Brownian Motion 609
Lognormality 609
Relative Importance of the Drift and Noise
Terms 610
Multiplication Rules 610
Modeling Correlated Asset Prices 612
20.4 It
os Lemma 613Functions of an It
o Process 614Multivariate It
os Lemma 61620.5 The Sharpe Ratio 617
20.6 Risk-Neutral Valuation 618
A Claim That PaysS(T)a619
Specic Examples 620
Valuing a Claim onSaQb621
20.7 Jumps in the Stock Price 623
Chapter Summary 624
Further Reading 624
Problems 624
20.A Valuation Using Discounted Cash
Flow 626Chapter 21
The Black-Scholes-Merton
Equation 62721.1 Differential Equations and Valuation under Certainty 627The Valuation Equation 628
Bonds 628
Dividend-Paying Stocks 629
ContentsxviiThe General Structure 629
21.2 The Black-Scholes Equation 629
Verifying the Formula for a Derivative 631
The Black-Scholes Equation and
Equilibrium Returns 634
What If the Underlying Asset Is Not an
Investment Asset? 635
21.3 Risk-Neutral Pricing 637
Interpreting the Black-Scholes
Equation 637
The Backward Equation 637
Derivative Prices as Discounted Expected
Cash Flows 638
21.4 Changing the Numeraire 639
21.5 Option Pricing When the Stock Price Can
Jump 642
Mertons Solution for Diversiable
Jumps 643
Chapter Summary 644
Further Reading 644
Problems 645
21.A Multivariate Black-Scholes Analysis 646
21.B Proof of Proposition 21.1 646
21.C SolutionsforPricesandProbabilities 647Chapter 22
Risk-Neutral and Martingale
Pricing 64922.1 Risk Aversion and Marginal Utility 65022.2 The First-Order Condition for Portfolio
Selection 652
22.3 Change of Measure and Change of
Numeraire 654
Change of Measure 655
The Martingale Property 655
Girsanovs Theorem 657
22.4 Examples of Numeraire and Measure
Change 658
The Money-Market Account as Numeraire
(Risk-Neutral Measure) 659Risky Asset as Numeraire 662
Zero Coupon Bond as Numeraire (Forward
Measure) 662
22.5 Examples of Martingale Pricing 663
Cash-or-Nothing Call 663Asset-or-Nothing Call 665
The Black-Scholes Formula 666
European Outperformance Option 667
Option on a Zero-Coupon Bond 667
22.6 Example:Long-MaturityPutOptions 667
The Black-Scholes Put Price
Calculation 668
Is the Put Price Reasonable? 669
Discussion 671
Chapter Summary 671
Further Reading 673
Problems 673
22.A The Portfolio Selection Problem 676
The One-Period Portfolio Selection
Problem 676
The Risk Premium of an Asset 678
Multiple Consumption and Investment
Periods 679
22.B Girsanovs Theorem 679
The Theorem 679
Constructing Multi-Asset Processes from
Independent Brownian Motions 680
22.C Risk-Neutral Pricing and Marginal Utility
in the Binomial Model 681Chapter 23 Exotic Options: II 68323.1 All-or-Nothing Options 683Terminology 683
Cash-or-Nothing Options 684
Asset-or-Nothing Options 685
Ordinary Options and Gap Options 686
Delta-Hedging All-or-Nothing
Options 687
23.2 All-or-Nothing Barrier Options 688
Cash-or-Nothing Barrier Options 690
Asset-or-Nothing Barrier Options 694
Rebate Options 694
Perpetual American Options 695
23.3 Barrier Options 696
23.4 Quantos 697
The Yen Perspective 698
The Dollar Perspective 699
A Binomial Model for the Dollar-
Denominated Investor 701
xviiiContents23.5 Currency-Linked Options 704
Foreign Equity Call Struck in Foreign
Currency 705
Foreign Equity Call Struck in Domestic
Currency 706
Fixed Exchange Rate Foreign Equity
Call 707
Equity-Linked Foreign Exchange Call 707
23.6 Other Multivariate Options 708
Options on the Best of Two Assets 709
Basket Options 710
Chapter Summary 711
Further Reading 711
Problems 712
23.A The Re"ection Principle 715Chapter 24
Volatility 71724.1 Implied Volatility 718
24.2 Measurement and Behavior of
Volatility 720
Historical Volatility 720
Exponentially Weighted Moving
Average 721
Time-Varying Volatility: ARCH 723
The GARCH Model 727
Realized Quadratic Variation 729
24.3 Hedging and Pricing Volatility 731
Variance and Volatility Swaps 731
Pricing Volatility 733
24.4 Extending the Black-Scholes Model 736
Jump Risk and Implied Volatility 737
Constant Elasticity of Variance 737
The Heston Model 740
Evidence 742
Chapter Summary 745
Further Reading 745
Problems 746Chapter 25
Interest Rate and Bond
Derivatives 75125.1 An Introduction to Interest RateDerivatives 752
Bond and Interest Rate Forwards 752Options on Bonds and Rates 753Equivalence of a Bond Put and an Interest
Rate Call 754
Taxonomy of Interest Rate Models 754
25.2 Interest Rate Derivatives and the
Black-Scholes-Merton Approach 756
An Equilibrium Equation for Bonds 757
25.3 Continuous-Time Short-Rate Models 760
The Rendelman-Bartter Model 760
The Vasicek Model 761
The Cox-Ingersoll-Ross Model 762
Comparing Vasicek and CIR 763
Duration and Convexity Revisited 764
25.4 Short-Rate Models and Interest Rate
Trees 765
An Illustrative Tree 765
The Black-Derman-Toy Model 769
Hull-White Model 773
25.5 Market Models 780
The Black Model 780
LIBOR Market Model 781
Chapter Summary 783
Further Reading 784
Problems 784
25.A Constructing the BDT Tree 787Chapter 26
Value at Risk 78926.1 Value at Risk 789
Value at Risk for One Stock 793
VaR for Two or More Stocks 795
VaR for Nonlinear Portfolios 796
VaR for Bonds 801
Estimating Volatility 805
Bootstrapping Return Distributions 806
26.2 Issues with VaR 807
Alternative Risk Measures 807
VaR and the Risk-Neutral Distribution 810
Subadditive Risk Measures 811
Chapter Summary 812
Further Reading 813
Problems 813Chapter 27
Credit Risk 81527.1 Default Concepts and Terminology 815Contentsxix27.2 The Merton Default Model 817
Default at Maturity 817
Related Models 819
27.3 Bond Ratings and Default
Experience 821
Rating Transitions 822
Recovery Rates 824
Reduced Form Bankruptcy Models 824
27.4 Credit Default Swaps 826
Single-Name Credit Default Swaps 826
Pricing a Default Swap 828
CDS Indices 832
Other Credit-Linked Structures 834
27.5 Tranched Structures 834
Collateralized Debt Obligations 836
CDO-Squareds 840
Nth to default baskets 842
Chapter Summary 844
Further Reading 846
Problems 846Appendix A
The Greek Alphabet 851
Appendix B
Continuous Compounding 853B.1 The Language of Interest Rates 853B.2 The Logarithmic and Exponential
Functions 854
Changing Interest Rates 855
SymmetryforIncreasesandDecreases 855
Problems 856Appendix C
Jensens Inequality 859C.1 Example: The Exponential Function 859C.2 Example: The Price of a Call 860
C.3 Proof of Jensens Inequality 861
Problems 862Appendix D
An Introduction to Visual Basic for
Applications 863D.1 Calculations without VBA 863D.2 How to Learn VBA 864D.3 Calculations with VBA 864
Creating a Simple Function 864
A Simple Example of a Subroutine 865
Creating a Button to Invoke a
Subroutine 866
Functions Can Call Functions 867
Illegal Function Names 867
Differences between Functions and
Subroutines 867
D.4 Storing and Retrieving Variables in a
Worksheet 868
Using a Named Range to Read and Write
Numbers from the Spreadsheet 868
Reading and Writing to Cells That Are Not
Named 869
Using the Cells Function to Read and
Write to Cells 870
Reading from within a Function 870
D.5 Using Excel Functions from within
VBA 871
Using VBA to Compute the Black-Scholes
Formula 871
The Object Browser 872
D.6 Checking for Conditions 873
D.7 Arrays 874
Dening Arrays 874
D.8 Iteration 875
A SimpleforLoop 876
Creating a Binomial Tree 876
Other Kinds of Loops 877
D.9 Reading and Writing Arrays 878
Arrays as Output 878
Arrays as Inputs 879
D.10 Miscellany 880
Getting Excel to Generate Macros for
You 880
Using Multiple Modules 881
Recalculation Speed 881
Debugging 882
Creating an Add-In 882
Glossary 883
References 897
Index 915
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PrefaceYou cannot understand modern nance and nancial markets without understanding deriva- tives. This book will help you to understand the derivative instruments that exist, how they are used, how they are priced, and how the tools and concepts underlying derivatives are useful more broadly in nance. Derivativesarenecessarilyananalyticalsubject,butIhavetriedthroughouttoempha- size intuition and to provide a common sense way to think about the formulas. I do assume thatareaderofthisbookalreadyunderstandsbasicnancialconceptssuchaspresentvalue, and elementary statistical concepts such as mean and standard deviation. In order to make the book accessible to readers with widely varying backgrounds and experiences, I use a tiered approach to the mathematics. Chapters 1...9 emphasize present value calculations, and there is almost no calculus until Chapter 18. The last part of the book develops the Black-Scholes-Mertonapproachto pricing derivatives and presents some of the standard mathematical tools used in option pricing, suchasIt osLemma.Therearealsochaptersdealingwithapplicationstocorporatenance,nancial engineering, and real options.
Most of the calculations in this book can be replicated using Excel spreadsheets on the CD-ROM that comes with the book.1These allow you to experiment with the pricing models and build your own spreadsheets. The spreadsheets on the CD-ROM contain option pricing functions written in Visual Basic for Applications, the macro language in Excel. You can incorporate these functions into your own spreadsheets. You can also examine and modify the Visual Basic code for the functions. Appendix D explains how to write such functions in Excel, and documentation on the CD-ROM lists the option pricing functionsthat come with the book. Relevant Excel functions are also mentioned throughout the book.WHAT IS NEW IN THE THIRD EDITIONThe reader familiar with the previous editions will nd the same overall plan, but will
discover many changes. Some are small, some are major. In general:1. Some of the advanced calculations are not easy in Excel, for example the Heston option pricing
calculation. As an alternative to Excel I usedR(http://r-project.org) to prepare many of the new graphs
and calculations. In the near future I hope to provide anRtutorial for the interested reader.xxixxiiPreface.Many examples have been updated..There are numerous changes to streamline and clarify exposition..ThereareconnectionsthroughouttoeventsduringthenancialcrisisandtotheDodd-
Frank nancial reform act..New boxes cover Bernie Madoff, Mexicos oil hedge, oil arbitrage, LIBOR during
the nancial crisis, Islamic nance, Bank capital, Google and compensation options,Abacus and Magnetar, and other topics.
Several chapters have also been extensively revised:.Chapter 1 has a new discussion of clearing and the organization and measurement of
markets..The chapter on commodities, Chapter 6, has been reorganized. There is a new intro- ductory discussion and overview of differences between commodities and nancial assets, a discussion of commodity arbitrage using copper, a discussion of commodityindices, and boxes on tanker-based oil-market arbitrage and illegal futures contracts..Chapter 15 has a revamped discussion of structures, a new discussion of reverse
convertibles, and a new discussion of tranching..Chapter 25 has been heavily revised. There is a discussion of the taxonomy of xed
income models, distinguishing short-rate models and market models. New sectionson the Hull-White and LIBOR market models have been added..Chapter 27 also has been heavily revised. One of the most important structuring
issues highlighted by the nancial crisis is the behavior of tranched claims that are themselves based on tranched claims. Many collateralized debt obligations satisfy this description, as do so-called CDO-squared contracts. There is a section on CDO- squareds and a box on Goldman Sachs Abacus transaction and the hedge fund Magnetar. The 2009 standardization of CDS contracts is discussed. Finally, Chapter 22 is new in this edition, focusing on the martingale approach to pricing derivatives. The chapter explains the important connection between investor portfolio decisions and derivatives pricing models. In this context, it provides the rationale forrisk-neutralpricingandfordifferentclassesofxedincomepricingmodels.Thechapter discusses Warren Buffetts critique of the Black-Scholes put pricing formula. You can skip this chapter and still understand the rest of the book, but the material in even the rst fewsections will deepen your understanding of the economic underpinnings of the models.PLAN OF THE BOOKThis book grew from my teaching notes for two MBA derivatives courses at Northwestern
Universitys Kellogg School of Management. The two courses roughly correspond to the rsttwo-thirdsandlastthirdofthebook.Therstcourseisageneralintroductiontoderiva- tive products (principally futures, options, swaps, and structured products), the markets in which they trade, and applications. The second course is for those wanting a deeper under- standing of the pricing models and the ability to perform their own analysis. The advanced course assumes that students know basic statistics and have seen calculus, and from that point develops the Black-Scholes option-pricing framework. A 10-week MBA-level coursePrefacexxiiiwill not produce rocket scientists, but mathematics is the language of derivatives and it
would be cheating students to pretend otherwise. I wrote chapters to allow "exible use of the material, with suggested possible paths through the material below. In many cases it is possible to cover chapters out of order. For example, I wrote the book anticipating that the chapters on lognormality and Monte Carlo simulation might be used in a rst derivatives course. Thebookhasvepartsplusappendixes.Part1introducesthebasicbuildingblocksof derivatives:forwardcontractsandcallandputoptions.Chapters2and3examinethesebasic instruments and some common hedging and investment strategies. Chapter 4 illustrates the use of derivatives as risk management tools and discusses why rms might care about risk management. These chapters focus on understanding the contracts and strategies, but not on pricing. Part 2considers the pricing of forward, futures, and swaps contracts. In these con- tracts, you are obligated to buy an asset at a pre-specied price, at a future date. What is the pre-specied price, and how is it determined? Chapter 5 examines forwards and futures on nancial assets, Chapter 6 discusses commodities, and Chapter 7 looks at bond and inter- est rate forward contracts. Chapter 8 shows how swap prices can be deduced from forward prices. Part 3studies option pricing. Chapter 9 develops intuition about options prior to delving into the mechanics of option pricing. Chapters 10 and 11 cover binomial option pricing and Chapter 12, the Black-Scholes formula and option Greeks. Chapter 13 explains delta-hedging, which is the technique used by market-makers when managing the risk of an option position, and how hedging relates to pricing. Chapter 14 looks at a few important exotic options, including Asian options, barrier options, compound options, and exchange options. The techniques and formulas in earlier chapters are applied inPart 4. Chapter 15 covers nancial engineering, which is the creation of new nancial products from the derivatives building blocks in earlier chapters. Debt and equity pricing, compensation options, and mergers are covered in Chapter 16. Chapter 17 studies real options"the applicationofderivativesmodelstothevaluationandmanagementofphysicalinvestments. Finally,Part5explorespricingandhedgingindepth.Thematerialinthispartexplains in more detail the structure and assumptions underlying the standard derivatives models. Chapter 18 covers the lognormal model and shows how the Black-Scholes formula is a discounted expected value. Chapter 19 discusses Monte Carlo valuation, a powerful and commonly used pricing technique. Chapter 20 explains what it means to say that stock prices follow a diffusion process, and also covers It os Lemma, which is a key result in the study of derivatives. (At this point you will discover that It os Lemma has already been developed intuitively in Chapter 13, using a simple numerical example.) Chapter 21 derives the Black-Scholes-Merton partial differential equation (PDE). Although the Black-Scholesformulais famous, the Black-Scholes-Mertonequation, dis- cussed in this chapter, is the more profound result. The martingale approach to pricing is covered in Chapter 22. We obtain the same pricing formulas as with the PDE, of course, but the perspective is different and helps to lay groundwork for later xed income discussions. Chapter23coversexoticoptionsinmoredetailthanChapter14,includingdigitalbarrierop- tionsandquantos.Chapter24discussesvolatilityestimationandstochasticvolatilitypricing models.Chapter25showshowtheBlack-Scholesandbinomialanalysisapplytobondsand interest rate derivatives. Chapter 26 covers value-at-risk, and Chapter 27 discusses credit products.xxivPrefaceNAVIGATING THE MATERIALThe material is generally presented in order of increasing mathematical and conceptual
difculty, which means that related material is sometimes split across distant chapters. For example, xed income is covered in Chapters 7 and 25, and exotic options in Chapters 14 and 23. As an illustration of one way to use the book, here is a rough outline of material I cover in the courses I teach (within the chapters, I skip specic topics due to timeconstraints):.Introductory course: 1...6, 7.1, 8...10, 12, 13.1...13.3, 14, 16, 17.1, 17.3..Advanced course: 13, 18...22, 7, 8, 15, 23...27.
Table P.1 outlines some possible sets of chapters to use in courses that have different emphases. There are a few sections of the book that provide background on topics every readershouldunderstand.Theseincludeshort-sales(Section1.4),continuouscompounding (AppendixB),prepaidforwardcontracts(Sections5.1and5.2),andzero-couponbondsandimplied forward rates (Section 7.1).A NOTE ON EXAMPLESMany of the numerical examples in this book display intermediate steps to assist you in
following the logic and steps of a calculation. Numbers displayed in the text necessarily are rounded to three or four decimal points, while spreadsheet calculations have many more signicant digits. This creates a dilemma: Should results in the book match those you would obtain using a spreadsheet, or those you would obtain by computing the displayed equations? As a general rule,the numerical examples in the book will provide the results you would obtain by entering the equations directly in a spreadsheet.Due to rounding, thedisplayed equations will not necessarily produce the correct result.SUPPLEMENTSA robust package of ancillary materials for both instructors and students accompanies the
text.Instructors ResourcesFor instructors, an extensive set of online tools is available for download from the catalog
page forDerivatives Marketsat www.pearsonhighered.com/mcdonald. An onlineInstructors Solutions Manualby R¨udiger Fahlenbrach,´Ecole Polytech- nique F ´ed´erale de Lausanne, contains complete solutions to all end-of-chapter problems in the text and spreadsheet solutions to selected problems. The onlineTest Bankby Matthew W. Will, University of Indianapolis, features approximately ten to fteen multiple-choice questions, ve short-answer questions, and one longer essay question for each chapter of the book. The Test Bank is available in several electronic formats, including Windows and MacintoshTestGenlesandMicrosoftWordles.TheTestGenandTestBankareavailable online at www.pearsonhighered.com/irc. PrefacexxvTABLE P.1Possible chapters for different courses. Chapters marked with a Y are strongly recommended, those marked with a * are recommended, and those with a t with the track but are optional. The advanced course assumes students have already taken a basic course. Sections 1.4, 5.1, 5.2, 7.1, and Appendix B are recommended background for all introductory courses.IntroductoryRisk Chapter General Futures Options Management Advanced1. Introduction Y Y Y Y
2. Intro. to Forwards and Options Y Y Y Y
3. Insurance, Collars, and Other Strategies Y Y Y Y
4. Intro. to Risk Management * * Y Y
5. Financial Forwards and Futures Y Y Y Y
6. Commodity Forwards and Futures * Y *
7. Interest Rate Forwards and Futures * Y * Y
8. Swaps Y Y Y Y
9. Parity and Other Option Relationships * Y
10. Binomial Option Pricing: I Y * Y Y
11. Binomial Option Pricing: II * *
12. The Black-Scholes Formula Y * Y Y
13. Market-Making and Delta-Hedging Y * Y
14. Exotic Options: I Y *
15. Financial Engineering * * * Y *
16. Corporate Applications * *
17. Real Options * *
18. The Lognormal Distribution * * Y
19. Monte Carlo Valuation * * Y
20. Brownian Motion and It
os Lemma Y21. The Black-Scholes Equation Y
22. Risk-neutral and Martingale Pricing
23. Exotic Options: II Y
24. VolatilityY
25. Interest Rate Models Y
26. Value at Risk Y Y
27. Credit Risk * Y
xxviPreface OnlinePowerPoint slides, developed by Peter Childs, University of Kentucky, pro- videlectureoutlinesandselectedartfromthebook.Copiesoftheslidescanbedownloadedand distributed to students to facilitate note taking during class.Student ResourcesAprintedStudentSolutionsManualbyR¨udigerFahlenbrach,´EcolePolytechniqueF´ed´erale
de Lausanne, provides answers to all the even-numbered problems in the textbook. A printedStudent Problems Manual,byR¨udiger Fahlenbrach, contains additional problems and worked-out solutions for each chapter of the textbook. Spreadsheetswith user-dened option pricing functions in Excel are included on a CD-ROMpackagedwiththebook.TheseExcelfunctionsarewritteninVBA,withthecode accessible and modiable via the Visual Basic editor built into Excel. These spreadsheetsand any updates are also posted on the books website.ACKNOWLEDGMENTSKellogg student Tejinder Singh catalyzed the book in 1994 by asking that the Kellogg
Finance Department offer an advanced derivatives course. Kathleen Hagerty and I initially co-taught that course, and my part of the course notes (developed with Kathleens help and feedback) evolved into the last third of this book. In preparing this revision, I once again received invaluable assistance from R