Derivatives Markets

984 Pages·2012·6.01 MB·English

by  Robert L. McDonald

Derivatives Markets Page: i
Derivatives Markets Page: iii
Brief Contents Page: vii
Preface Page: xx
What is New in the Third Edition Page: xxi
Plan of the Book Page: xxii
Navigating the Material Page: xxiii
A Note on Examples Page: xxiv
Supplements Page: xxiv
Derivatives Markets Page: xxix
1 Introduction to Derivatives Page: 1
1.1 What is a Derivative? Page: 1
1.2 An Overview of Financial Markets Page: 2
Trading of Financial Assets Page: 2
Measures of Market Size and Activity Page: 4
Stock and Bond Markets Page: 5
Derivatives Markets Page: 6
1.3 The Role Of Financial Markets Page: 9
Financial Markets and the Averages Page: 9
Risk-Sharing Page: 10
1.4 The Uses Of Derivatives Page: 11
Uses of Derivatives Page: 11
Perspectives on Derivatives Page: 13
Financial Engineering and Security Design Page: 14
1.5 Buying And Short-Selling Financial Assets Page: 14
Transaction Costs and the Bid-Ask Spread Page: 14
Example 1.1 Page: 15
Ways to Buy or Sell Page: 15
Short-Selling Page: 16
Example: Short-Selling Wine Page: 17
Example: Short-Selling Stock Page: 18
The Lease Rate of an Asset Page: 18
Risk and Scarcity in Short-Selling Page: 18
Credit Risk Page: 19
Scarcity Page: 19
Chapter Summary Page: 19
Further Reading Page: 20
Problems Page: 20
Part 1 Insurance, Hedging, and Simple Strategies Page: 23
2 An Introduction to Forwards and Options Page: 25
2.1 Forward Contracts Page: 25
The Payoff on a Forward Contract Page: 29
Example 2.1 Page: 30
Graphing the Payoff on a Forward Contract Page: 30
Comparing a Forward and Outright Purchase Page: 30
Zero-Coupon Bonds in Payoff and Profit Diagrams Page: 33
Cash Settlement Versus Delivery Page: 34
Example 2.2 Page: 34
Credit Risk Page: 34
2.2 Call Options Page: 35
Example 2.3 Page: 35
Example 2.4 Page: 35
Option Terminology Page: 35
Payoff and Profit for a Purchased Call Option Page: 36
Example 2.5 Page: 36
Example 2.6 Page: 37
Payoff and Profit for a Written Call Option Page: 38
Example 2.7 Page: 40
2.3 Put Options Page: 40
Example 2.8 Page: 41
Payoff and Profit for a Purchased Put Option Page: 41
Example 2.9 Page: 41
Example 2.10 Page: 42
Payoff and Profit for a Written Put Option Page: 42
Example 2.11 Page: 44
The “Moneyness” of an Option Page: 44
2.4 Summary of Forward and Option Positions Page: 45
Positions Long with Respect to the Index Page: 45
Positions Short with Respect to the Index Page: 46
2.5 Options are Insurance Page: 47
Homeowner’s Insurance Is a Put Option Page: 48
But I Thought Insurance Is Prudent and Put Options Are Risky … Page: 48
Call Options Are Also Insurance Page: 49
2.6 Example: Equity-Linked CDS Page: 50
Graphing the Payoff on the CD Page: 50
Economics of the CD Page: 51
Why Equity-Linked CDs? Page: 52
Chapter Summary Page: 53
Further Reading Page: 54
Problems Page: 54
Appendix 2.A More on Buying a Stock Option Page: 57
Dividends Page: 57
Exercise Page: 57
Margins for Written Options Page: 58
Taxes Page: 58
3 Insurance, Collars, and Other Strategies Page: 61
3.1 Basic Insurance Strategies Page: 61
Insuring a Long Position: Floors Page: 61
Insuring a Short Position: Caps Page: 64
Selling Insurance Page: 65
Covered Call Writing. Page: 66
Covered Puts. Page: 66
3.2 Put-Call Parity Page: 68
Synthetic Forwards Page: 68
The Put-Call Parity Equation Page: 70
Example 3.1 Page: 70
Equivalence of Different Positions. Page: 70
No Arbitrage. Page: 71
3.3 Spreads and Collars Page: 71
Bull and Bear Spreads Page: 71
Example 3.2 Page: 72
Box Spreads Page: 73
Example 3.3 Page: 74
Ratio Spreads Page: 74
Collars Page: 74
Example 3.4 Page: 74
Example 3.5 Page: 76
Zero-Cost Collars. Page: 76
Understanding Collars. Page: 77
The Cost of the Collar and the Forward Price. Page: 78
3.4 Speculating on Volatility Page: 79
Straddles Page: 79
Strangle. Page: 79
Written Straddle. Page: 80
Butterfly Spreads Page: 80
Asymmetric Butterfly Spreads Page: 82
Chapter Summary Page: 84
Further Reading Page: 85
Problems Page: 86
4 Introduction to Risk Management Page: 89
4.1 Basic Risk Management: The Producer’s Perspective Page: 89
Hedging with a Forward Contract Page: 90
Insurance: Guaranteeing a Minimum Price with a Put Option Page: 91
Insuring by Selling a Call Page: 93
Adjusting the Amount of Insurance Page: 95
4.2 Basic Risk Management: The Buyer’s Perspective Page: 96
Hedging with a Forward Contract Page: 97
Insurance: Guaranteeing a Maximum Price with a Call Option Page: 97
4.3 Why Do Firms Manage Risk? Page: 99
An Example Where Hedging Adds Value Page: 100
Reasons to Hedge Page: 102
Taxes. Page: 102
Bankruptcy and Distress Costs. Page: 102
Costly External Financing. Page: 102
Increase Debt Capacity. Page: 103
Managerial Risk Aversion. Page: 103
Nonfinancial Risk Management. Page: 103
Reasons Not to Hedge Page: 103
Empirical Evidence on Hedging Page: 104
4.4 Golddiggers Revisited Page: 107
Selling the Gain: Collars Page: 107
A 420–440 Collar. Page: 107
A Zero-Cost Collar. Page: 108
The Forward Contract as a Zero-Cost Collar. Page: 109
Synthetic Forwards at Prices Other Than $420. Page: 110
Other Collar Strategies Page: 111
Paylater Strategies Page: 111
4.5 Selecting The Hedge Ratio Page: 112
Cross-Hedging Page: 112
Example 4.1 Page: 113
Quantity Uncertainty Page: 114
Chapter Summary Page: 117
Further Reading Page: 118
Problems Page: 118
Part 2 Forwards, Futures, and Swaps Page: 123
5 Financial Forwards and Futures Page: 125
5.1 Alternative Ways to Buy a Stock Page: 125
5.2 Prepaid Forward Contracts on Stock Page: 126
Pricing the Prepaid Forward by Analogy Page: 127
Pricing the Prepaid Forward by Discounted Present Value Page: 127
Pricing the Prepaid Forward by Arbitrage Page: 127
Pricing Prepaid Forwards with Dividends Page: 128
Discrete Dividends Page: 129
Example 5.1 Page: 129
Continuous Dividends Page: 129
Example 5.2 Page: 130
5.3 Forward Contracts on Stock Page: 131
Does the Forward Price Predict the Future Spot Price? Page: 132
Creating a Synthetic Forward Contract Page: 133
Synthetic Forwards in Market-Making and Arbitrage Page: 135
No-Arbitrage Bounds with Transaction Costs Page: 136
Quasi-Arbitrage Page: 137
An Interpretation of the Forward Pricing Formula Page: 138
5.4 Futures Contracts Page: 138
The S&P 500 Futures Contract Page: 139
Margins and Marking to Market Page: 140
Comparing Futures and Forward Prices Page: 143
Arbitrage in Practice: S&P 500 Index Arbitrage Page: 143
Quanto Index Contracts Page: 145
5.5 Uses of Index Futures Page: 146
Asset Allocation Page: 146
Switching from Stocks to T-bills Page: 146
General Asset Allocation Page: 146
Cross-hedging with Index Futures Page: 147
Cross-hedging with Perfect Correlation Page: 147
Cross-Hedging with Imperfect Correlation Page: 148
Example 5.3 Page: 149
Risk Management for Stock-Pickers Page: 150
5.6 Currency Contracts Page: 150
Currency Prepaid Forward Page: 150
Example 5.4 Page: 151
Currency Forward Page: 151
Example 5.5 Page: 152
Covered Interest Arbitrage Page: 152
Example 5.6 Page: 152
5.7 Eurodollar Futures Page: 153
Chapter Summary Page: 156
Further Reading Page: 158
Problems Page: 158
Appendix 5.A Taxes and the Forward Rate Page: 161
Appendix 5.B Equating Forwards and Futures Page: 162
Appendix 5.C Forward and Futures Prices Page: 162
6 Commodity Forwards and Futures Page: 163
6.1 Introduction to Commodity Forwards Page: 164
Examples of Commodity Futures Prices Page: 164
Differences Between Commodities and Financial Assets Page: 165
Commodity Terminology Page: 166
6.2 Equilibrium Pricing of Commodity Forwards Page: 167
6.3 Pricing Commodity Forwards by Arbitrage Page: 168
An Apparent Arbitrage Page: 168
Short-selling and the Lease Rate Page: 170
No-Arbitrage Pricing Incorporating Storage Costs Page: 171
Cash-and-Carry Arbitrage Page: 172
Example 6.1 Page: 173
Reverse Cash-and-Carry Arbitrage Page: 173
Convenience Yields Page: 174
Summary Page: 175
6.4 Gold Page: 175
Gold Leasing Page: 176
Evaluation of Gold Production Page: 177
Example 6.2 Page: 177
6.5 Corn Page: 178
6.6 Energy Markets Page: 179
Electricity Page: 179
Natural Gas Page: 180
Oil Page: 182
Oil Distillate Spreads Page: 184
Example 6.3 Page: 185
6.7 Hedging Strategies Page: 185
Basis Risk Page: 185
Hedging Jet Fuel with Crude Oil Page: 187
Weather Derivatives Page: 188
6.8 Synthetic Commodities Page: 189
Chapter Summary Page: 191
Further Reading Page: 191
Problems Page: 192
7 Interest Rate Forwards and Futures Page: 195
7.1 Bond Basics Page: 195
Zero-Coupon Bonds Page: 196
Implied Forward Rates Page: 197
Example 7.1 Page: 198
Coupon Bonds Page: 199
Example 7.2 Page: 199
Zeros from Coupons Page: 200
Interpreting the Coupon Rate Page: 201
Continuously Compounded Yields Page: 202
7.2 Forward Rate Agreements, Eurodollar Futures, and Hedging Page: 202
Forward Rate Agreements Page: 203
FRA Settlement in Arrears. Page: 203
FRA Settlement at the Time of Borrowing Page: 203
Synthetic FRAs Page: 204
Example 7.3 Page: 205
Eurodollar Futures Page: 206
Convexity Bias and Tailing Page: 207
LIBOR Versus 3-Month T-Bills. Page: 209
7.3 Duration and Convexity Page: 211
Price Value of a Basis Point and DV01 Page: 211
Example 7.4 Page: 212
Duration Page: 212
Example 7.5 Page: 213
Example 7.6 Page: 213
Example 7.7 Page: 214
Duration Matching Page: 214
Example 7.8 Page: 215
Convexity Page: 215
Example 7.9 Page: 216
7.4 Treasury-Bond and Treasury-Note Futures Page: 217
Example 7.10 Page: 218
7.5 Repurchase Agreements Page: 220
Example 7.11 Page: 220
Chapter Summary Page: 222
Further Reading Page: 224
Problems Page: 224
Appendix 7.A INTEREST RATE AND BOND PRICE CONVENTIONS Page: 228
Bonds Page: 228
Example 7.12 Page: 229
Example 7.13 Page: 229
Example 7.14 Page: 230
Bills Page: 230
8 Swaps Page: 233
8.1 An Example of A Commodity Swap Page: 233
Physical Versus Financial Settlement Page: 234
Why Is the Swap Price Not $110.50? Page: 236
The Swap Counterparty Page: 237
The Market Value of a Swap Page: 238
8.2 Computing The Swap Rate in General Page: 240
Fixed Quantity Swaps Page: 240
Swaps with Variable Quantity and Price Page: 241
8.3 Interest Rate Swaps Page: 243
A Simple Interest Rate Swap Page: 243
Pricing and the Swap Counterparty Page: 244
Swap Rate and Bond Calculations Page: 246
Example 8.1 Page: 246
The Swap Curve Page: 246
The Swap’s Implicit Loan Balance Page: 248
Deferred Swaps Page: 249
Related Swaps Page: 250
Why Swap Interest Rates? Page: 251
Amortizing and Accreting Swaps Page: 252
8.4 Currency Swaps Page: 252
Example 8.2 Page: 254
Example 8.3 Page: 254
Currency Swap Formulas Page: 255
Other Currency Swaps Page: 256
8.5 Swaptions Page: 256
Example 8.4 Page: 257
8.6 Total Return Swaps Page: 257
Example 8.5 Page: 258
Chapter Summary Page: 259
Further Reading Page: 260
Problems Page: 261
Part 3 Options Page: 263
9 Parity and Other Option Relationships Page: 265
9.1 Put-Call Parity Page: 265
Options on Stocks Page: 266
Example 9.1 Page: 267
Example 9.2 Page: 268
Synthetic stock. Page: 268
Synthetic T-bills. Page: 268
Synthetic options. Page: 269
Options on Currencies Page: 269
Options on Bonds Page: 269
Dividend Forward Contracts Page: 269
9.2 Generalized Parity And Exchange Options Page: 270
Example 9.3 Page: 271
Options to Exchange Stock Page: 272
What Are Calls and Puts? Page: 272
Currency Options Page: 273
9.3 Comparing Options With Respect To Style, Maturity, And Strike Page: 275
European Versus American Options Page: 276
Maximum and Minimum Option Prices Page: 276
Calls. Page: 276
Puts. Page: 277
Early Exercise for American Options Page: 277
Calls on a non-dividend-paying stock. Page: 277
Exercising calls just prior to a dividend. Page: 278
Early exercise for puts. Page: 278
Early exercise in general. Page: 279
Time to Expiration Page: 279
American options. Page: 280
European options. Page: 280
European options when the strike price grows over time. Page: 280
Different Strike Prices Page: 281
Example 9.4 Page: 283
Example 9.5 Page: 284
Example 9.6 Page: 284
Exercise and Moneyness Page: 286
Chapter Summary Page: 286
Further Reading Page: 287
Problems Page: 288
Appendix 9.A Parity Bounds For American Options Page: 291
Appendix 9.B Algebraic Proofs Of Strike-Price Relations Page: 292
10 Binomial Option Pricing: Basic Concepts Page: 293
10.1 A One-Period Binomial Tree Page: 293
Computing the Option Price Page: 294
The Binomial Solution Page: 295
Example 10.1 Page: 297
Arbitraging a Mispriced Option Page: 297
The Option is Overpriced Page: 297
The Option is Underpriced Page: 298
A Graphical Interpretation of the Binomial Formula Page: 298
Risk-Neutral Pricing Page: 299
10.2 Constructing a Binomial Tree Page: 300
Continuously Compounded Returns Page: 300
Example 10.2 Page: 301
Example 10.3 Page: 301
Example 10.4 Page: 301
Volatility Page: 301
Constructing u and d Page: 302
Estimating Historical Volatility Page: 303
One-Period Example with a Forward Tree Page: 305
10.3 Two or More Binomial Periods Page: 306
A Two-Period European Call Page: 306
Constructing the Tree Page: 306
Pricing the Call Option Page: 307
Many Binomial Periods Page: 308
10.4 Put Options Page: 309
10.5 American Options Page: 310
10.6 Options on Other Assets Page: 312
Option on a Stock Index Page: 312
Options on Currencies Page: 312
Options on Futures Contracts Page: 314
Options on Commodities Page: 315
Options on Bonds Page: 316
Summary Page: 317
Chapter Summary Page: 318
Further Reading Page: 318
Problems Page: 319
Appendix 10.A Taxes and Option Prices Page: 322
11 Binomial Option Pricing: Selected Topics Page: 323
11.1 Understanding Early Exercise Page: 323
11.2 Understanding Risk-Neutral Pricing Page: 325
The Risk-Neutral Probability Page: 326
Pricing an Option Using Real Probabilities Page: 327
A One-Period Example Page: 328
A Multi-Period Example Page: 329
11.3 The Binomial Tree and Lognormality Page: 330
The Random Walk Model Page: 330
Modeling Stock Prices as a Random Walk Page: 331
The Binomial Model Page: 332
Lognormality and the Binomial Model Page: 333
Alternative Binomial Trees Page: 335
The Cox-Ross-Rubinstein Binomial Tree Page: 335
The Lognormal Tree Page: 336
Is the Binomial Model Realistic? Page: 336
11.4 Stocks Paying Discrete Dividends Page: 336
Modeling Discrete Dividends Page: 337
Problems with the Discrete Dividend Tree Page: 337
A Binomial Tree Using the Prepaid Forward Page: 338
Chapter Summary Page: 340
Further Reading Page: 340
Problems Page: 341
Appendix 11.A Pricing Options with True Probabilities Page: 343
Appendix 11.B Why Does Risk-Neutral Pricing Work? Page: 343
Utility-Based Valuation Page: 344
Standard Discounted Cash Flow Page: 345
Risk-Neutral Pricing Page: 345
Physical vs. Risk-Neutral Probabilities Page: 346
Example Page: 347
State Prices Page: 347
Valuing the Risk-Free Bond Page: 347
Valuing the Risky Stock Using Real Probabilities Page: 347
Risk-Neutral Valuation of the Stock Page: 347
12 The Black-Scholes Formula Page: 349
12.1 Introduction to the Black-Scholes Formula Page: 349
Call Options Page: 349
Example 12.1 Page: 351
Put Options Page: 352
Example 12.2 Page: 352
When Is the Black-Scholes Formula Valid? Page: 352
12.2 Applying the Formula to Other Assets Page: 353
Options on Stocks with Discrete Dividends Page: 354
Example 12.3 Page: 354
Options on Currencies Page: 354
Example 12.4 Page: 355
Options on Futures Page: 355
Example 12.5 Page: 355
12.3 Option Greeks Page: 355
Definition of the Greeks Page: 356
Delta Page: 356
Gamma Page: 358
Vega Page: 359
Theta Page: 359
Rho Page: 360
Psi Page: 360
Greek Measures for Portfolios Page: 361
Example 12.6 Page: 361
Option Elasticity Page: 362
Dollar Risk of the Option Page: 362
Example 12.7 Page: 362
Percentage Risk of the Option Page: 362
Example 12.8 Page: 363
The Volatility of an Option Page: 363
The Risk Premium and Beta of an Option Page: 363
The Sharpe Ratio of an Option Page: 365
The Elasticity and Risk Premium of a Portfolio Page: 365
12.4 Profit Diagrams Before Maturity Page: 366
Purchased Call Option Page: 366
Example 12.9 Page: 366
Calendar Spreads Page: 367
12.5 Implied Volatility Page: 368
Computing Implied Volatility Page: 369
Example 12.10 Page: 369
Using Implied Volatility Page: 370
12.6 Perpetual American Options Page: 372
Valuing Perpetual Options Page: 373
Example 12.11 Page: 374
Barrier Present Values Page: 374
Chapter Summary Page: 374
Further Reading Page: 375
Problems Page: 375
Appendix 12.A THE STANDARD NORMAL DISTRIBUTION Page: 378
Appendix 12.B FORMULAS FOR OPTION GREEKS Page: 378
Delta (Δ) Page: 379
Gamma (Γ) Page: 379
Theta (θ) Page: 379
Vega Page: 379
Rho (ρ) Page: 380
Psi (ψ) Page: 380
13 Market-Making and Delta-Hedging Page: 381
13.1 What do Market-Makers do? Page: 381
13.2 Market-Maker Risk Page: 382
Option Risk in the Absence of Hedging Page: 382
Delta and Gamma as Measures of Exposure Page: 383
13.3 Delta-Hedging Page: 384
An Example of Delta-Hedging for 2 Days Page: 385
Day 0 Page: 385
Day 1: Marking-to-Market Page: 385
Day 1: Rebalancing the Portfolio Page: 385
Day 2: Marking-to-Market Page: 385
Interpreting the Profit Calculation Page: 385
Delta-Hedging for Several Days Page: 387
A Self-Financing Portfolio: The Stock Moves One σ Page: 389
13.4 The Mathematics of Delta-Hedging Page: 389
Using Gamma to Better Approximate the Change in the Option Price Page: 390
Example 13.1 Page: 390
Delta-Gamma Approximations Page: 391
Theta: Accounting for Time Page: 392
Example 13.2 Page: 393
Understanding the Market-Maker’s Profit Page: 393
13.5 The Black-Scholes Analysis Page: 395
The Black-Scholes Argument Page: 395
Delta-Hedging of American Options Page: 396
What Is the Advantage to Frequent Re-Hedging? Page: 397
Example 13.3 Page: 398
Delta-Hedging in Practice Page: 398
Gamma-Neutrality Page: 399
13.6 Market-Making as Insurance Page: 402
Insurance Page: 402
Market-Makers Page: 403
Chapter Summary Page: 403
Further Reading Page: 404
Problems Page: 404
Appendix 13.A TAYLOR SERIES APPROXIMATIONS Page: 406
Appendix 13.B GREEKS IN THE BINOMIAL MODEL Page: 407
14 Exotic Options: I Page: 409
14.1 Introduction Page: 409
14.2 Asian Options Page: 410
XYZ’s Hedging Problem Page: 410
Options on the Average Page: 411
The Definition of the Average Page: 411
Example 14.1 Page: 412
Whether the Average Is Used as the Asset Price or the Strike Page: 412
Comparing Asian Options Page: 412
An Asian Solution for XYZ Page: 413
14.3 Barrier Options Page: 414
Types of Barrier Options Page: 415
Currency Hedging Page: 416
14.4 Compound Options Page: 418
Compound Option Parity Page: 419
Options on Dividend-Paying Stocks Page: 419
Example 14.2 Page: 420
Currency Hedging with Compound Options Page: 421
14.5 Gap Options Page: 421
14.6 Exchange Options Page: 423
European Exchange Options Page: 424
Example 14.3 Page: 425
Chapter Summary Page: 425
Further Reading Page: 426
Problems Page: 426
Appendix 14.A Pricing Formulas for Exotic Options Page: 429
Asian Options Based on the Geometric Average Page: 430
Average Price Options Page: 430
Average Strike Options Page: 430
Compound Options Page: 431
Infinitely Lived Exchange Option Page: 432
Part 4 Financial Engineering and Applications Page: 435
15 Financial Engineering and Security Design Page: 437
15.1 The Modigliani-Miller Theorem Page: 437
15.2 Structured Notes without Options Page: 438
Single Payment Bonds Page: 438
Zero-coupon equity-linked bond Page: 440
Example 15.1 Page: 440
Example 15.2 Page: 440
Zero-coupon commodity-linked bond Page: 440
Example 15.3 Page: 440
Zero-Coupon Currency-Linked Bond Page: 441
Multiple Payment Bonds Page: 441
Equity-linked bonds Page: 442
Example 15.4 Page: 443
Commodity-linked bonds Page: 443
Example 15.5 Page: 443
Perpetuities Page: 444
Currency-linked bonds Page: 444
15.3 Structured Notes with Options Page: 445
Convertible Bonds Page: 446
Valuing and Structuring an Equity-Linked CD Page: 447
Structuring the Product Page: 448
Alternative Structures Page: 448
Example 15.6 Page: 449
Reverse Convertible Bonds Page: 449
Tranched Payoffs Page: 451
Variable Prepaid Forwards Page: 452
Example 15.7 Page: 453
15.4 Strategies Motivated by Tax and Regulatory Considerations Page: 453
Capital Gains Deferral Page: 454
Hedging by Corporate Insiders Page: 455
Tax Deferral for Corporations Page: 456
Marshall & Ilsley SPACES Page: 458
The M&I Issue Page: 458
Design Considerations Page: 459
15.5 Engineered Solutions for Golddiggers Page: 460
Gold-Linked Notes Page: 460
Notes with Embedded Options Page: 462
Chapter Summary Page: 463
Further Reading Page: 464
Problems Page: 464
16 Corporate Applications Page: 469
16.1 Equity, Debt, And Warrants Page: 469
Debt and Equity as Options Page: 469
Example 16.1 Page: 470
Example 16.2 Page: 472
Leverage and the Expected Return on Debt and Equity Page: 472
Example 16.3 Page: 473
Conflicts Between Debt and Equity Page: 475
Multiple Debt Issues Page: 477
Warrants Page: 478
Convertible Bonds Page: 479
Example 16.4 Page: 480
Example 16.5 Page: 480
Callable Bonds Page: 481
Callable Nonconvertible Bonds Page: 482
Callable Convertible Bonds Page: 483
Bond Valuation Based on the Stock Price Page: 485
Other Bond Features Page: 485
Put Warrants Page: 486
16.2 Compensation Options Page: 487
The Use of Compensation Options Page: 487
Valuation of Compensation Options Page: 489
Whose Valuation Page: 489
Valuation Inputs Page: 490
Repricing of Compensation Options Page: 492
Example 16.6 Page: 492
Reload Options Page: 493
Level 3 Communications Page: 495
Example 16.7 Page: 495
Valuing the Outperformance Feature Page: 496
Accounting for the Multiplier Page: 497
16.3 The Use Of Collars In Acquisitions Page: 498
The Northrop Grumman—TRW merger Page: 499
Chapter Summary Page: 502
Further Reading Page: 503
Problems Page: 503
Appendix 16.A An Alternative Approach to Expensing Option Grants Page: 507
17 Real Options Page: 509
17.1 Investment And The Npv Rule Page: 509
Static NPV Page: 510
The Correct Use of NPV Page: 511
The Project as an Option Page: 511
17.2 Investment Under Uncertainty Page: 512
A Simple DCF Problem Page: 513
Example 17.1 Page: 513
Valuing Derivatives on the Cash Flow Page: 514
Example 17.2 Page: 514
Evaluating a Project with a 2-Year Investment Horizon Page: 515
A Tree for Project Value Page: 516
Solving for the Optimal Investment Decision Page: 517
Evaluating the Project with an Infinite Investment Horizon Page: 518
17.3 Real Options In Practice Page: 519
Peak-Load Electricity Generation8 Page: 519
Research and Development Page: 523
17.4 Commodity Extraction As An Option Page: 525
Single-Barrel Extraction under Certainty Page: 525
Optimal Extraction Page: 526
Value and Appreciation of the Land Page: 527
Using the Option Pricing Formula Page: 527
Changing Extraction Costs Page: 527
Gold Extraction Revisited Page: 528
Single-Barrel Extraction under Uncertainty Page: 528
Valuing an Infinite Oil Reserve Page: 530
Valuing the Producing Firm Page: 530
Valuing the Option to Invest Page: 530
Example 17.3 Page: 530
Example 17.4 Page: 531
17.5 Commodity Extraction With Shutdown And Restart Options Page: 531
Permanent Shutting Down Page: 533
Example 17.5 Page: 533
The value of the producing well Page: 534
Investing When Shutdown Is Possible Page: 535
Example 17.6 Page: 536
Restarting Production Page: 536
Example 17.7 Page: 536
Additional Options Page: 537
Chapter Summary Page: 538
Further Reading Page: 538
Problems Page: 538
Appendix 17.A Calculation of Optimal Time to Drill an Oil Well Page: 541
Appendix 17.B The Solution with Shutting Down and Restarting Page: 541
Part 5 Advanced Pricing Theory and Applications Page: 543
18 The Lognormal Distribution Page: 545
18.1 The Normal Distribution Page: 545
Example 18.1 Page: 548
Converting a Normal Random Variable to Standard Normal Page: 548
Example 18.2 Page: 549
Sums of Normal Random Variables Page: 549
The Central Limit Theorem Page: 550
18.2 The Lognormal Distribution Page: 550
18.3 A Lognormal Model of Stock Prices Page: 552
Example 18.3 Page: 553
Example 18.4 Page: 555
Example 18.5 Page: 555
18.4 Lognormal Probability Calculations Page: 555
Probabilities Page: 556
Lognormal Prediction Intervals Page: 557
Example 18.6 Page: 557
Example 18.7 Page: 558
The Conditional Expected Price Page: 559
The Black-Scholes Formula Page: 561
18.5 Estimating the Parameters of a Lognormal Distribution Page: 562
Example 18.8 Page: 562
18.6 How are Asset Prices Distributed? Page: 564
Histograms Page: 564
Normal Probability Plots Page: 566
Example 18.9 Page: 567
Example 18.10 Page: 567
Chapter Summary Page: 569
Further Reading Page: 569
Problems Page: 570
Appendix 18.A The Expectation of a Lognormal Variable Page: 571
Appendix 18.B Constructing a Normal Probability Plot Page: 572
19 Monte Carlo Valuation Page: 573
19.1 Computing the Option Price as a Discounted Expected Value Page: 573
Valuation with Risk-Neutral Probabilities Page: 574
Valuation with True Probabilities Page: 575
19.2 Computing Random Numbers Page: 577
19.3 Simulating Lognormal Stock Prices Page: 578
Simulating a Sequence of Stock Prices Page: 578
19.4 Monte Carlo Valuation Page: 579
Monte Carlo Valuation of a European Call Page: 580
Example 19.1 Page: 580
Accuracy of Monte Carlo Page: 581
Arithmetic Asian Option Page: 582
Example 19.2 Page: 583
19.5 Efficient Monte Carlo Valuation Page: 584
Control Variate Method Page: 584
Other Monte Carlo Methods Page: 587
19.6 Valuation of American Options Page: 588
19.7 The Poisson Distribution Page: 591
Example 19.3 Page: 592
19.8 Simulating Jumps with the Poisson Distribution Page: 593
Simulating the Stock Price with Jumps Page: 593
Multiple Jumps Page: 596
19.9 Simulating Correlated Stock Prices Page: 597
Generating n Correlated Lognormal Random Variables Page: 597
Chapter Summary Page: 599
Further Reading Page: 599
Problems Page: 599
Appendix 19.A Formulas for Geometric Average Options Page: 602
20 Brownian Motion and Itô’s Lemma Page: 603
20.1 The Black-Scholes Assumption About Stock Prices Page: 603
20.2 Brownian Motion Page: 604
Definition of Brownian Motion Page: 604
Properties of Brownian Motion Page: 606
Arithmetic Brownian Motion Page: 607
The Ornstein-Uhlenbeck Process Page: 608
20.3 Geometric Brownian Motion Page: 608
Lognormality Page: 609
Relative Importance of the Drift and Noise Terms Page: 610
Multiplication Rules Page: 610
Modeling Correlated Asset Prices Page: 612
Example 20.1 Page: 613
20.4 ItÔ’s Lemma Page: 613
Functions of an Itô Process Page: 614
Proposition 20.1 Page: 615
Example 20.2 Page: 615
Example 20.3 Page: 616
Multivariate Itô’s Lemma Page: 616
Proposition 20.2 Page: 616
Example 20.4 Page: 616
Example 20.5 Page: 617
20.5 The Sharpe Ratio Page: 617
20.6 Risk-Neutral Valuation Page: 618
A Claim That Pays S(T)a Page: 619
Proposition 20.3 Page: 619
Specific Examples Page: 620
Valuing a Claim on SaQb Page: 621
Proposition 20.4 Page: 621
20.7 Jumps In The Stock Price Page: 622
Proposition 20.5 Page: 623
Chapter Summary Page: 624
Further Reading Page: 624
Problems Page: 624
Appendix 20.A Valuation Using Discounted Cash Flow Page: 626
b21 The Black-Scholes-Merton Equation Page: 627
21.1 Differential Equations and Valuation Under Certainty Page: 627
The Valuation Equation Page: 627
Bonds Page: 628
Dividend-Paying Stocks Page: 629
The General Structure Page: 629
21.2 The Black-Scholes Equation Page: 629
Verifying the Formula for a Derivative Page: 631
Simple Present Value Calculations. Page: 631
All-Or-Nothing Options Page: 633
The Black-Scholes Equation and Equilibrium Returns Page: 634
What If the Underlying Asset Is Not an Investment Asset? Page: 635
Example 21.1 Page: 636
21.3 Risk-Neutral Pricing Page: 637
Interpreting the Black-Scholes Equation Page: 637
The Backward Equation Page: 637
Derivative Prices as Discounted Expected Cash Flows Page: 638
21.4 Changing the Numeraire Page: 639
Example 21.2 Page: 639
Proposition 21.1 Page: 640
Example 21.3 Page: 641
21.5 Option Pricing When the Stock Price Can Jump Page: 642
Merton’s Solution for Diversifiable Jumps Page: 642
Chapter Summary Page: 644
Further Reading Page: 644
Problems Page: 645
Appendix 21.A Multivariate Black-Scholes Analysis Page: 646
Appendix 21.B Proof of Proposition 21.1 Page: 646
Appendix 21.C Solutions For Prices and Probabilities Page: 647
22 Risk-Neutral and Martingale Pricing Page: 649
22.1 Risk Aversion and Marginal Utility Page: 650
22.2 The First-Order Condition for Portfolio Selection Page: 652
22.3 Change of Measure and Change of Numeraire Page: 654
Change of Measure Page: 655
The Martingale Property Page: 655
Girsanov’s Theorem Page: 657
22.4 Examples of Numeraire and Measure Change Page: 658
The Money-Market Account as Numeraire (Risk-Neutral Measure) Page: 659
The Money-Market Account Page: 659
The Money-Market Account as Numeraire Page: 660
Constructing a Process for Si(t) Page: 660
Interpretation Page: 661
Risky Asset as Numeraire Page: 662
Zero Coupon Bond as Numeraire (Forward Measure) Page: 662
22.5 Examples of Martingale Pricing Page: 663
Cash-or-Nothing Call Page: 663
Interpretation of Volatility Page: 664
Dividends Page: 665
Asset-or-Nothing Call Page: 665
The Black-Scholes Formula Page: 666
European Outperformance Option Page: 667
Option on a Zero-Coupon Bond Page: 667
22.6 Example: Long-Maturity Put Options Page: 667
The Black-Scholes Put Price Calculation Page: 668
Is the Put Price Reasonable? Page: 669
The Likelihood of Exercise and Expected Payoff Page: 669
Understanding the Option Price Page: 669
Discussion Page: 671
Chapter Summary Page: 671
Further Reading Page: 673
Problems Page: 673
Appendix 22.A The Portfolio Selection Problem Page: 676
The One-Period Portfolio Selection Problem Page: 676
The Risk Premium of an Asset Page: 678
Multiple Consumption and Investment Periods Page: 678
Appendix 22.B Girsanov’s Theorem Page: 679
The Theorem Page: 679
Constructing Multi-Asset Processes from Independent Brownian Motions Page: 680
Risk-Neutral Measure Page: 680
Risky Asset as Numeraire Page: 681
Appendix 22.C Risk-Neutral Pricing and Marginal Utility in the Binomial Model Page: 681
23 Exotic Options: II Page: 683
23.1 All-Or-Nothing Options Page: 683
Terminology Page: 683
Cash-or-Nothing Options Page: 684
Example 23.1 Page: 685
Asset-or-Nothing Options Page: 685
Example 23.2 Page: 686
Ordinary Options and Gap Options Page: 686
Example 23.3 Page: 687
Delta-Hedging All-or-Nothing Options Page: 687
23.2 All-Or-Nothing Barrier Options Page: 688
Cash-or-Nothing Barrier Options Page: 690
Down-And-In Cash Call. Page: 691
Deferred Down Rebate Option Page: 691
Down-And-Out Cash Call. Page: 691
Down-And-In Cash Put. Page: 692
Down-And-Out Cash Put. Page: 692
Example 23.4 Page: 692
Example 23.5 Page: 693
Up-And-In Cash Put. Page: 693
Deferred Up Rebate Page: 693
Up-And-Out Cash Put. Page: 694
Up-And-In Cash Call. Page: 694
Up-And-Out Cash Call. Page: 694
Asset-or-Nothing Barrier Options Page: 694
Rebate Options Page: 694
Perpetual American Options Page: 695
23.3 Barrier Options Page: 695
Example 23.6 Page: 696
23.4 Quantos Page: 697
The Yen Perspective Page: 698
Example 23.7 Page: 699
The Dollar Perspective Page: 699
Example 23.8 Page: 701
A Binomial Model for the Dollar-Denominated Investor Page: 701
Example 23.9 Page: 703
23.5 Currency-Linked Options Page: 704
Foreign Equity Call Struck in Foreign Currency Page: 705
Example 23.10 Page: 706
Foreign Equity Call Struck in Domestic Currency Page: 706
Example 23.11 Page: 706
Fixed Exchange Rate Foreign Equity Call Page: 706
Example 23.12 Page: 707
Equity-Linked Foreign Exchange Call Page: 707
Example 23.13 Page: 708
23.6 Other Multivariate Options Page: 708
Options on the Best of Two Assets Page: 708
Basket Options Page: 710
Chapter Summary Page: 711
Further Reading Page: 711
Problems Page: 712
Appendix 23.A The Reflection Principle Page: 715
24 Volatility Page: 717
24.1 Implied Volatility Page: 717
24.2 Measurement and Behavior of Volatility1 Page: 720
Historical Volatility Page: 720
Exponentially Weighted Moving Average Page: 721
Example 24.1 Page: 722
Time-Varying Volatility: ARCH Page: 723
The ARCH Model Page: 724
ARCH Volatility Forecasts Page: 726
The GARCH Model Page: 727
Maximum Likelihood Estimation of a GARCH Model Page: 727
Volatility Forecasts Page: 728
Example 24.2 Page: 729
Realized Quadratic Variation Page: 729
24.3 Hedging and Pricing Volatility Page: 731
Variance and Volatility Swaps Page: 731
Example 24.3 Page: 731
Example 24.4 Page: 732
Pricing Volatility Page: 732
The Log Contract Page: 733
Valuing the Log Contract Page: 734
Computing the VIX Page: 735
24.4 Extending the Black-Scholes Model Page: 736
Jump Risk and Implied Volatility Page: 737
Constant Elasticity of Variance Page: 737
The CEV Pricing Formula Page: 739
Implied Volatility in the CEV Model Page: 740
The Heston Model Page: 740
Evidence Page: 742
Chapter Summary Page: 745
Further Reading Page: 745
Problems Page: 746
25 Interest Rate and Bond Derivatives Page: 751
25.1 An Introduction To Interest Rate Derivatives Page: 751
Bond and Interest Rate Forwards Page: 752
Example 25.1 Page: 753
Options on Bonds and Rates Page: 753
Bond Options. Page: 753
Interest Rate Options. Page: 753
Equivalence of a Bond Put and an Interest Rate Call Page: 754
Example 25.2 Page: 754
Taxonomy of Interest Rate Models Page: 754
Short-Rate Models. Page: 754
Market Models. Page: 755
25.2 Interest Rate Derivatives And The Black-Scholes-Merton Approach Page: 756
An Equilibrium Equation for Bonds Page: 757
25.3 Continuous-Time Short-Rate Models Page: 760
The Rendelman-Bartter Model Page: 760
The Vasicek Model Page: 761
The Cox-Ingersoll-Ross Model Page: 762
Comparing Vasicek and CIR Page: 763
Duration and Convexity Revisited Page: 764
Example 25.3 Page: 765
25.4 Short-Rate Models And Interest Rate Trees Page: 765
An Illustrative Tree Page: 765
Zero-Coupon Bond Prices. Page: 766
Example 25.4 Page: 767
Yields and Expected Interest Rates. Page: 768
Option Pricing. Page: 768
Example 25.5 Page: 769
The Black-Derman-Toy Model Page: 769
Example 25.6 Page: 772
Hull-White Model Page: 773
Example 25.7 Page: 774
Constructing the Initial Interest Rate Grid. Page: 774
Probabilities. Page: 774
Matching Zero-Coupon Bond Prices. Page: 776
Valuation. Page: 778
Example 25.8 Page: 779
Example 25.9 Page: 779
25.5 Market Models Page: 779
The Black Model Page: 780
Example 25.10 Page: 781
LIBOR Market Model Page: 781
Chapter Summary Page: 783
Further Reading Page: 784
Problems Page: 784
Appendix 25.A Constructing The Bdt Tree Page: 787
26 Value at Risk Page: 789
26.1 Value at Risk Page: 789
Value at Risk for One Stock Page: 792
Example 26.1 Page: 793
Example 26.2 Page: 794
VaR for Two or More Stocks Page: 795
Example 26.3 Page: 795
VaR for Nonlinear Portfolios Page: 796
Delta Approximation Page: 797
Example 26.4 Page: 797
Example 26.5 Page: 798
Monte Carlo Simulation Page: 799
Example 26.6 Page: 799
Example 26.7 Page: 801
VaR for Bonds Page: 801
Example 26.8 Page: 802
Example 26.9 Page: 803
Example 26.10 Page: 804
Estimating Volatility Page: 805
Bootstrapping Return Distributions Page: 806
26.2 Issues With VaR Page: 806
Alternative Risk Measures Page: 807
Tail VaR Page: 807
Example 26.11 Page: 807
The Cost of Insurance Page: 809
Example 26.12 Page: 810
VaR and the Risk-Neutral Distribution Page: 810
Subadditive Risk Measures Page: 811
Chapter Summary Page: 812
Further Reading Page: 813
Problems Page: 813
27 Credit Risk Page: 815
27.1 Default Concepts and Terminology Page: 815
27.2 The Merton Default Model Page: 817
Default at Maturity Page: 817
Example 27.1 Page: 818
Related Models Page: 819
Example 27.2 Page: 820
27.3 Bond Ratings and Default Experience Page: 820
Rating Transitions Page: 822
Recovery Rates Page: 824
Reduced Form Bankruptcy Models Page: 824
27.4 Credit Default Swaps Page: 826
Single-Name Credit Default Swaps Page: 826
Pricing a Default Swap Page: 828
CDS Indices Page: 832
Other Credit-Linked Structures Page: 833
Total Rate of Return Swaps Page: 834
Credit-Linked Notes Page: 834
Credit Guarantees Page: 834
27.5 Tranched Structures Page: 834
Collateralized Debt Obligations Page: 836
A CDO with Independent Defaults Page: 837
A CDO with Correlated Defaults Page: 839
Synthetic CDOs Page: 839
CDO-Squareds Page: 840
Nth to default baskets Page: 842
Chapter Summary Page: 844
Further Reading Page: 846
Problems Page: 846
Appendixes Page: 849
Appendix A The Greek Alphabet Page: 851
Appendix B Continuous Compounding Page: 853
B.1 The Language of Interest Rates Page: 853
B.2 The Logarithmic and Exponential Functions Page: 853
Problems Page: 856
Appendix C Jensen’s Inequality Page: 858
C.1 Example: The Exponential Function Page: 859
C.2 Example: The Price of a Call Page: 860
C.3 Proof Of Jensen’s Inequality2 Page: 861
Problems Page: 862
Appendix D An Introduction to Visual Basic for Applications Page: 863
D.1 Calculations Without Vba Page: 863
D.2 How To Learn Vba Page: 864
D.3 Calculations With Vba Page: 864
D.4 Storing and Retrieving Variables In a Worksheet Page: 868
D.5 Using Excel Functions From Within Vba Page: 870
D.6 Checking For Conditions Page: 873
D.7 Arrays Page: 874
D.8 Iteration Page: 875
D.9 Reading And Writing Arrays Page: 877
D.10 Miscellany Page: 880
Glossary Page: 883
References Page: 897
Index Page: 915
A Page: 915
B Page: 916
C Page: 919
D Page: 923
E Page: 925
F Page: 926
G Page: 928
H Page: 929
I Page: 930
J Page: 931
K Page: 932
L Page: 932
M Page: 933
N Page: 935
O Page: 936
P Page: 937
Q Page: 940
R Page: 940
S Page: 942
T Page: 945
U Page: 946
V Page: 946
W Page: 947
X Page: 947
Y Page: 947
Z Page: 947