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