F I F T H E D I T I O N JOHN C.HULL PRENTICE HALL FINANCE SERIES Personal Finance Keown, Personal Finance: Turning Monev into Wealth, Second Edition Trivoli, Personal Portfolio Management: Fundamentals & Strategies Winger/Frasca, Personal Finance: An Integrated Planning Approach, Sixth Edition Undergraduate Investments/Portfolio Management Alexander/Sharpe/Bailey, Fundamentals of Investments, Third Edition Fabozzi, Investment Management, Second Edition Haugen, Modern Investment Theory, Fifth Edition Haugen, The New Finance, Second Edition Haugen, The Beast on Wall Street Haugen, The Inefficient Stock Market, Second Edition Holden, Spreadsheet Modeling: A Book and CD-ROM Series (Available in Graduate and Undergraduate Versions) Nofsinger The Psychology of Investing Taggart, Quantitative Analysis for Investment Management Winger/Frasca, Investments, Third Edition Graduate Investments/Portfolio Management Fischer/Jordan, Security Analysis and Portfolio Management, Sixth Edition 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Series (Available in Graduate and Undergraduate Versions) Keown/Martin/Petty/Scott, Financial Management, Ninth Edition Keown/Martin/Petty/Scott, Financial Management, 9/e activehook M Keown/Martin/Petty/Scott, Foundations of Finance: The Logic and Practice of Financial Management, Third Edition Keown/Martin/Petty/Scott, Foundations of Finance, 3je activebook ' Mathis, Corporate Finance Live: A Web-based Math Tutorial Shapiro/Balbirer, Modern Corporate Finance: A Multidiseiplinary Approach to Value Creation Van Horne/Wachowicz, Fundamentals of Financial Management, Eleventh Edition Mastering Finance CD-ROM Fifth Edition OPTIONS, FUTURES, & OTHER DERIVATIVES John C Hull Maple Financial Group Professor of Derivatives and Risk Management Director, Bonham Center for Finance Joseph L Rotman School of Management University of Toronto Prentice Hall P R E N T I C E H A L L , U P P E R S A D D L E R I V E R , N E W JERSEY CONTENTS Preface Introduction 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Mechanics of 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 xix Exchange-traded markets Over-the-counter markets Forward contracts Futures contracts Options Types of traders Other derivatives Summary Questions and problems Assignment questions 1 2 10 14 15 16 17 futures markets Trading futures contracts Specification of the futures contract Convergence of futures price to spot price Operation of margins Newspaper quotes Keynes and Hicks Delivery Types of traders Regulation Accounting and tax Forward contracts vs futures contracts Summary Suggestions for further reading Questions and problems Assignment questions 19 19 20 23 24 27 31 31 32 33 35 36 37 38 38 40 Determination of forward and futures prices 3.1 Investment assets vs consumption assets 3.2 Short selling 3.3 Measuring interest rates 3.4 Assumptions and notation 3.5 Forward price for an investment asset 3.6 Known income 3.7 Known yield 3.8 Valuing forward contracts 3.9 Are forward prices and futures prices equal? 3.10 Stock index futures 3.11 Forward and futures contracts on currencies 3.12 Futures on commodities ; 41 41 41 42 44 45 47 49 49 51 52 55 58 ix Contents 3.13 3.14 3.15 Cost of carry Delivery options Futures prices and the expected future spot price Summary Suggestions for further reading Questions and problems Assignment questions Appendix 3A: Proof that forward and futures prices are equal when interest rates are constant Hedging strategies using futures 4.1 Basic principles 4.2 Arguments for and against hedging 4.3 Basis risk 4.4 Minimum variance hedge ratio 4.5 Stock index futures 4.6 Rolling the hedge forward Summary Suggestions for further reading Questions and problems Assignment questions Appendix 4A: Proof of the minimum variance hedge ratio formula Interest rate 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 markets Types of rates Zero rates Bond pricing Determining zero rates Forward rates Forward rate agreements Theories of the term structure Day count conventions Quotations Treasury bond futures Eurodollar futures The LIBOR zero curve Duration Duration-based hedging strategies Summary Suggestions for further reading Questions and problems Assignment questions Swaps 6.1 6.2 6.3 6.4 6.5 6.6 6.7 Mechanics of interest rate swaps The comparative-advantage argument Swap quotes and LIBOR zero rates Valuation of interest rate swaps Currency swaps Valuation of currency swaps Credit risk Summary Suggestions for further reading Questions and problems Assignment questions 60 60 61 63 64 65 67 68 70 70 72 75 78 82 86 87 88 88 90 92 93 93 94 94 96 98 100 102 102 103 104 110 Ill 112 116 118 119 120 123 125 125 131 134 136 140 143 145 146 147 147 149 Contents xi Mechanics of 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 options markets Underlying assets Specification of stock options Newspaper quotes Trading Commissions Margins The options clearing corporation Regulation Taxation Warrants, executive stock options, and convertibles Over-the-counter markets Summary Suggestions for further reading Questions and problems Assignment questions 151 151 152 155 157 157 158 160 161 161 162 163 163 164 164 165 Properties of 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 stock options Factors affecting option prices Assumptions and notation Upper and lower bounds for option prices Put-call parity Early exercise: calls on a non-dividend-paying stock Early exercise: puts on a non-dividend-paying stock Effect of dividends Empirical research Summary Suggestions for further reading Questions and problems Assignment questions 167 167 170 171 174 175 177 178 179 180 181 182 183 Trading strategies involving options 9.1 Strategies- involving a single option and a stock 9.2 Spreads 9.3 Combinations 9.4 Other payoffs Summary Suggestions for further reading Questions and problems Assignment questions 185 185 187 194 197 197 198 198 199 10 Introduction 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 200 200 203 205 208 209 210 211 212 213 214 214 215 to binomial trees A one-step binomial model Risk-neutral valuation Two-step binomial trees A put example American options Delta Matching volatility with u and d Binomial trees in practice Summary Suggestions for further reading Questions and problems Assignment questions xii Contents 11 A model of the behavior of stock prices 11.1 The Markov property 11.2 Continuous-time stochastic processes 11.3 The process for stock prices 11.4 Review of the model 11.5 The parameters 11.6 Ito's lemma 11.7 The lognormal property Summary Suggestions for further reading Questions and problems Assignment questions Appendix 11 A: Derivation of Ito's lemma 216 216 217 222 223 225 226 227 228 229 229 230 232 12 The Black-Scholes model 234 12.1 Lognormal property of stock prices 234 12.2 The distribution of the rate of return 236 12.3 The expected return 237 12.4 Volatility 238 12.5 Concepts underlying the Black-Scholes-Merton differential equation 241 12.6 Derivation of the Black-Scholes-Merton differential equation 242 12.7 Risk-neutral valuation 244 12.8 Black-Scholes pricing formulas 246 12.9 Cumulative normal distribution function 248 12.10 Warrants issued by a company on its own stock 249 12.11 Implied volatilities 250 12.12 The causes of volatility 251 12.13 Dividends 252 Summary 256 Suggestions for further reading 257 Questions and problems 258 Assignment questions 261 Appendix 12A: Proof of Black-Scholes-Merton formula 262 Appendix 12B: Exact procedure for calculating the values of American calls on dividend-paying stocks 265 Appendix 12C: Calculation of cumulative probability in bivariate normal distribution 266 13 Options on 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 stock indices, currencies, and futures Results for a stock paying a known dividend yield Option pricing formulas Options on stock indices Currency options Futures options Valuation of futures options using binomial trees Futures price analogy Black's model for valuing futures options Futures options vs spot options Summary Suggestions for further reading Questions and problems Assignment questions Appendix 13 A: Derivation of differential equation satisfied by a derivative dependent on a stock providing a dividend yield 267 267 268 270 276 278 284 286 287 288 289 290 291 294 295 Contents xiii Appendix 13B: Derivation of differential equation satisfied by a derivative dependent on a futures price 297 14 The Greek letters 14.1 Illustration 14.2 Naked and covered positions 14.3 A stop-loss strategy 14.4 Delta hedging 14.5 Theta 14.6 Gamma 14.7 Relationship between delta, theta, and gamma 14.8 Vega 14.9 Rho 14.10 Hedging in practice 14.11 Scenario analysis 14.12 Portfolio insurance 14.13 Stock market volatility Summary Suggestions for further reading Questions and problems : Assignment questions Appendix 14A: Taylor series expansions and hedge parameters 299 299 300 300 302 309 312 315 316 318 319 319 320 323 323 324 326 327 329 15 Volatility smiles 15.1 Put-call parity revisited 15.2 Foreign currency options 15.3 Equity options 15.4 The volatility term structure and volatility surfaces 15.5 Greek letters 15.6 When a single large jump is anticipated 15.7 Empirical research Summary Suggestions for further reading Questions and problems Assignment questions Appendix 15A: Determining implied risk-neutral distributions from volatility smiles 330 330 331 334 336 337 338 339 341 341 343 344 16 Value at risk 16.1 The VaR measure 16.2 Historical simulation 16.3 Model-building approach 16.4 Linear model 16.5 Quadratic model 16.6 Monte Carlo simulation 16.7 Comparison of approaches 16.8 Stress testing and back testing 16.9 Principal components analysis Summary Suggestions for further reading Questions and problems Assignment questions Appendix 16A: Cash-flow mapping Appendix 16B: Use of the Cornish-Fisher expansion to estimate VaR 346 346 348 350 352 356 359 359 360 360 364 364 365 366 368 370 345 xiv Contents 17 Estimating volatilities and correlations 17.1 Estimating volatility 17.2 The exponentially weighted moving average model 17.3 The GARCH(1,1) model 17.4 Choosing between the models 17.5 Maximum likelihood methods 17.6 Using GARCHfl, 1) to forecast future volatility 17.7 Correlations Summary Suggestions for further reading Questions and problems Assignment questions 372 372 374 376 377 378 382 385 388 388 389 391 18 Numerical procedures 18.1 Binomial trees 18.2 Using the binomial tree for options on indices, currencies, and futures contracts 18.3 Binomial model for a dividend-paying stock ' 18.4 Extensions to the basic tree approach 18.5 Alternative procedures for constructing trees 18.6 Monte Carlo simulation 18.7 Variance reduction procedures 18.8 Finite difference methods 18.9 Analytic approximation to American option prices Summary Suggestions for further reading Questions and problems Assignment questions Appendix 18A: Analytic approximation to American option prices of MacMillan and of Barone-Adesi and Whaley 392 392 19 Exotic options 19.1 Packages 19.2 Nonstandard American options 19.3 Forward start options 19.4 Compound options 19.5 Chooser options 19.6 Barrier options 19.7 Binary options 19.8 Lookback options 19.9 Shout options 19.10 Asian options 19.11 Options to exchange one asset for another 19.12 Basket options 19.13 Hedging issues 19.14 Static options replication Summary Suggestions for further reading Questions and problems Assignment questions Appendix 19A: Calculation of the first two moments of arithmetic averages and baskets 435 435 436 437 437 438 439 441 441 443 443 445 446 447 447 449 449 451 452 20 More on models and numerical procedures 20.1 The CEV model 20.2 The jump diffusion model 456 456 457 399 402 405 406 410 414 418 427 427 428 430 432 433 454 Contents xv 20.3 20.4 20.5 20.6 20.7 20.8 20.9 Stochastic volatility models The IVF model Path-dependent derivatives Lookback options Barrier options Options on two correlated assets Monte Carlo simulation and American options Summary Suggestions for further reading Questions and problems Assignment questions 458 460 461 465 467 472 474 478 479 480 481 21 Martingales 21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8 21.9 and measures The market price of risk Several state variables • Martingales Alternative choices for the numeraire Extension to multiple independent factors Applications Change of numeraire Quantos Siegel's paradox Summary Suggestions for further reading Questions and problems Assignment questions Appendix 21 A: Generalizations of Ito's lemma Appendix 2IB: Expected excess return when there are multiple sources of uncertainty 483 484 487 488 489 492 493 495 497 499 500 500 501 502 504 22 Interest rate 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9 derivatives: the standard market models Black's model Bond options Interest rate caps European swap options Generalizations Convexity adjustments Timing adjustments Natural time lags Hedging interest rate derivatives Summary Suggestions for further reading Questions and problems Assignment questions Appendix 22A: Proof of the convexity adjustment formula 508 508 511 515 520 524 524 527 529 530 531 531 532 534 536 23 Interest rate 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 derivatives: models of the short rate Equilibrium models One-factor equilibrium models The Rendleman and Bartter model The Vasicek model The Cox, Ingersoll, and Ross model Two-factor equilibrium models No-arbitrage models The Ho and Lee model The Hull and White model 537 537 538 538 539 542 543 543 544 546 506 730 Bearish calendar spread, 193 Bear spread, 189-90, 701 Bermudan option, 436, 701 Bermudan swap option, 586 Beta, 82, 701 changing, 84-85 Bid, 2, 134-35, 157, 701 Bid-ask spread, 701 Bid-offer spread, 135, 157, 158, 701 Binary credit default swap, 642-43 Binary option,' 441, 701 Binomial model, 701 Binomial tree, 701 alternative for constructing, 406-10 American options example, 209-10 control variate technique, 406 denned, 200 delta and, 210-11 dollar dividend-paying stocks, 403-5 European options examples, 200-209 futures option, 284-86 matching volatility, 211-12, 212-13 nondividend-paying stock, 392-97 one-step, 200-5, 269-70 options on index, currency, and futures, 399-402 risk-neutral valuation and, 203-5 stock paying a known dividend yield, 267-70, 402-3 time-dependent interest rates and, 405-6 two-step, 205-10 Bivariate normal distribution, 701 Black Monday, 55, 322, 360, 692 Black's approximation, 701 American call option, 255-56 Black's model, 701 European interest rate derivatives, 508-10 forward risk-neutral valuation and, 514-15, 519-20, 523-24 Black-Scholes bias, equity option and, 334-36 foreign currency options and, 331 -34 single jumps in asset price anticipated, 338-39 Black-Scholes-Merton differential equation, 241-44 Black-Scholes model, 701 cumulative normal distribution function, 248-49 Subject Index delta and, 303 difference between binomial tree model and, 241-42 dividend, 252-56 known dividend yield, 268-69 expected return, 237-38 European option on non-dividend-paying stock, 246-48 forward risk-neutral valuation and, 493-94 implied volatility, 250-51, 331 See also Volatility smile pricing formulas, 246-48, 268-69 risk-neutral valuation and, 244-45, 247 volatility, 238-41, 251-52, 330 Board broker, 701 Board order, 33 Bolsa de Mercadorias y Futuros (BM&F), 19 Bond, pricing, 94-96 Bond option, 511, 701 embedded, 511 European, 511-13 forward risk-neutral valuation and Black's model, 514-15 valuation using Vasicek model, 540-42 on coupon bearing bonds, 549-50 Bond yield, 95, 701 Bond yield volatilities, 513-14 Bootstrap method, 96-98, 701 Boston option, 436, 701 Bottom straddle, 195 Bottom vertical combinations, 196 Boundary conditions, 243 Brady Commission report, 322-23 Break forward, 436, 701 Brent index price, 680 Brownian motion, 218 Bullish calendar spread, 193 Bull spread, 187-89, 701 Business day conventions, 131 Business valuation, 666-67 Butterfly spread, 190-92, 701 Buying on margin, 158 CAC-40 Index, 53 Calendar spread, 192-94, 701 Calibrating instruments, 564 Calibration, 564-65, 584-85, 702 Callable bond, 511, 702 Subject Index Call option, 6-7, 151, 702 Cancelable forward, 436 Cancelable swap, 603-4, 702 Cancelable compounding swaps, 604-5 Cap, interest rate, 515-20, 702 valuation, 517-18 Caplet, interest rate, 516, 702 Cap rate, 515, 702 Capital asset pricing model, 61, 82, 273, 621, 660-61, 702 relation to market price of risk, 485, 487, 665-66 Capital investment appraisal, 660-61 Cash-flow mapping, 354, 702 Cash-or-nothing call, 441, 702 Cash-or-nothing put, 441, 702 Cash price, bond and Treasury bill, 104, 512, 561 See also Dirty price Cash settlement, 32, 702 CAT bond, catastrophic bond, 683, 702 Changing the measure, 212, 495-96 Cheapest-to-deliver bond, 108, 702 Chemical Bank, 687, 690 Chicago Board of Trade (CBOT), 1, 5, 6, 19, 21, 22, 34, 52, 104, 279, 683 Chicago Board Options Exchange (CBOE), 1-2, 11, 151, 152, 153, 154, 155, 157, 270, 273 CBOEdirect, 157 Chicago Mercantile Exchange (CME), 1, 5, 19, 21-22, 32, 34, 52, 53, 55, 56, 110, 111, 279, 497, 679 Cholesky decomposition, 413, 702 Chooser option, 438, 702 Citron, Robert, 686, 688, 689, 693 Claim amount, 615, 618 Clean price, bond, 512, 702 See also Quoted price Clearing margin, 26, 702 Clearinghouse, 26, 702 Collar, interest rate, 517, 702 Collateralization, 625 Collateralized debt obligation (CDO), 646-47, 702 Collateralized mortgage obligation (CMO), 587, 702 Combination, option trading strategy, 194-96, 702 Commission brokers, 32, 702 731 Commission, stock option, 157-58 Commodity Futures Trading Commission (CFTC), 33-34, 161, 702 Commodity price, 667-70 Commodity swap, 605, 702 Comparative-advantage argument interest rate swap, 131-34 currency swap, 141-43 Compound option, 437, 703 Compounding frequency, 42-43, 703 Compounding swap, 595-98, 703 Conditional VaR (C-VaR), 347-48, 703 Confirmation, 130, 595, 596, 602, 703 Constant elasticity of variance (CEV) model, 456-57 Constant maturity swap (CMS), 599-600, 703 Constant maturity Treasury swap (CMT), 600-601, 703 Constructive sale, 162 Consumption asset, 41, 58-59, 703 market price of risk, 485 Contango, 31, 703 Continuous compounding, 43-44, 703 Continuous-time stochastic process, 216, 217-22 Continuous variable, 216 Contract size, 21 Contraction option, 671 Control areas, electricity-producing region, 681 Control variate technique, variance reduction procedure, 406, 415, 703 Convenience yield, 59-60, 703 Convergence arbitrage, 692 Conversion factor, 106-7, 703 Convertible bond, 163, 652-54, 703 conversion ratio, 652 Convexity, 115, 117-18,703 Convexity adjustment, 524-27, 703 constant maturity swap (CMS), 600 forward rates and futures rates, 111, 566 LIBOR-in-arrears swap, 526, 599 swap rates, 527 Cooling degree days (CDD), 679, 702 Copula, 703 Corner the market, 35 Cornish-Fisher expansion, 358, 703 Correlation, 385-88 Correlation swap, 605 Cost of carry, 60, 703 Counterparty, 703 732 Coupon, 94, 703 Covariance, 385-88 consistency condition, 387-88 updating using EWMA, 386 updating using GARCH, 387 Covariance swap, 605 Covered call, 159-60, 703 covered position, 300 Crank-Nicolson scheme, 426 Crashophobia, 335-36 Credit default swap (CDS), 637-44, 703 approximate no-arbitrage arguments, 641 implying default probabilities, 641-42 quotes, 638 recovery rate estimates, 642 valuation, 639-40 Credit default swap spread (CDS spread), 640 Credit derivative, 15, 637, 703 adjusting price for default risk, 647-49 Credit event, 637 Credit rating, 610, 703 Credit rating migration, 626-27 Credit ratings transition matrix, 626-27, 703 Credit risk, 2, 703 interest rate and, 93 of swaps, 145-46 Credit Risk Plus, 630-31 Credit spread option, 645-46, 703 Credit Suisse Financial Products, 630-31 Credit value at risk, 630, 704 CreditMetrics, 632 Cross-currency derivative, 497-99 Cross gamma, 358 Crude oil derivative, 680 Cumulative normal distribution function, 248-49, 704 polynomial approximation, 248 Currency forward and futures, 55-58 quotation, 3, 57 Currency futures option, 279 Currency option, 151 early exercise, 278 implied distribution and lognormal distribution, 331-33 quotation, 276-77 valuation, binomial tree, 399-402 valuation, Black-Scholes, 276-78 volatility smile, 331-34 Subject Index Currency swaps, 140-43, 598, 704 comparative advantage argument, 141-43 impact of default risk, 649-51 to transform loans and assets, 141 valuation of, 143-45 Curvature, 312 Cylinder option, 435 Daiwa, 686, 687 DAX-30 Index, 53 Day count conventions, 102-3, 130, 704 Day order, 33 Day trader, 32-33 Day trade, 26, 704 Default correlation, 627-30, 704 reduced-form models, 629 structural models, 630 Default probability, see Probability of default Default probability density, 613-14, 704 Default risk adjusting derivative prices, 647-49 Deferred payment option, 436, 704 Deferred swap, 521, 704 Delivery, 21-22, 60-61 Delivery price, 4, 704 Delta, 210, 302, 704 estimating, using binomial tree, 398 European stock options, 210-11, 302, 303-5 ' forward contract, 305 futures contract, 305-6 interest rate derivatives, 530 portfolio, 308-9 relationship with theta and gamma, 315-16 Delta hedging, 210, 302-9, 704 dynamic aspects, 306-8 impact of stochastic volatility on delta hedging, 459-60 performance measure, 307, 308 transaction cost, 309 Delta-neutral portfolio, 303, 704 DerivaGem, 715-19, 704 Derivative, 1, 704 nonstandard, 14 plain vanilla, 14 standard, 14 Deutsche Bank, 691 Diagonal spread, 194, 704 Differential swap (diff swap), 601, 704 Subject Index Diffusion process, 704 Dirty price, bond, 512 See also Cash price Discount bond, 704 Discount broker, 158 Discount instrument, 704 Discount rate, 104, 704 Discrete-time stochastic process, 216, 223-25 Discrete variable, 216 Discretionary order, 33 Diversification, 352, 689 Dividend, 402, 704 American Call option valuation, using Black-Scholes model, 254-56 binomial model for stocks paying dollar dividend, 402-3 bounds of option prices, 178-79 European option valuation, using BlackScholes model, 252-53, stock option and, 154-55, 170, 252-56 stock prices and, 154-55, 170 stock splits and, 154-55 Dividend yield, 704 binomial tree and, 269-70, 403-5 DJ Euro Stoxx 50 Index, 53-54 DJ Stoxx 50 Index, 53-54 Dow Jones Industrial Average (DJX), 52, 152, 270, 322 futures, 52, 54 options, 152, 270, 273, Down-and-in call, 439, 704 Down-and-in put, 440, 704 Down-and-out call, 439, 704 Down-and-out put, 440, 704 Downgrade trigger, 625, 704 Drift rate, 219, 704 Duration, 704 bond, 112-14 bond portfolio, 115 modified, 114-15 Duration-based hedge ratio, 117 Duration-based hedging, 116-18 Duration matching, 116, 704 Dynamic hedging scheme, 303, 704 Dynamic options replication, 447 Early exercise, 8, 175-78, 179, 278, 705 Earth Satellite Corporation, 679 Efficient market hypothesis, 217, 705 733 Electricity derivatives, 15, 681 Electronic trading, 2, 705 Embedded options, 511, 705 Empirical research, 705 Energy derivatives, 680-82 Energy prices, modeling, 681-82 Equilibrium model, interest rates, 537-38, 543, 705 Equity swap, 601-2, 705 Equivalent martingale measure result, 483, 488-89 Eurex, 19, 54, 157, 721 Eurocurrency, 705 Eurodollar, 110,705 Eurodollar futures, 110-11, 705 Eurodollar futures option, 282 Eurodollar interest rate, 110, 705 Euronext, 721 European option, 6, 151, 705 binomial trees, 200-209, 269-70, 284-86, 402-5 Black-Scholes model, on a non-dividendpaying stock, 246-48 Black-Scholes model, on a stock paying a known dividend yield, 268-69 delta, 210-11 futures option compared to spot option, 288-89 stock paying a known dividend, 267-70 dividend-paying stock, 178-79, 252-53, 267-68 non-dividend-paying stock, 171-74, 246-48 put-call parity, 174-75, 179, 186-87, 268, 330-31 risk-neutral valuation, 269 EWMA, see Exponentially weighted moving average Excess cost layer, reinsurance, 683 Excess-of-loss reinsurance contract, 683 Exchange clearinghouse, 26 Exchange option, 445, 705 Exchange rates, Black-Scholes and, 331-34 Exchange-traded market, 1-2 difference between over-the-counter market and, for options, 152, 154-55 Ex-dividend date, 275, 705 Executive stock option, 163, 705 734 Exercise boundary parametrization approach, Monte Carlo simulation, 477-78 Exercise date, Exercise limit, 155, 705 Exercise price, 6, 705 Exotic option, 14, 163, 435, 705 Exotics, 14 Expansion option, 671 Expectations theory, shape of zero curve, 102, 705 Expected default losses on bonds, 611-12 Expected future price, 31 Expected return, stock's, 237-38 stock option price and, 203 Expected value of a variable, 705 Expiration date, 6, 152, 705 Explicit finite difference method, 422-23, 705 relation to trinomial tree approach, 424-25 Exponential weighting, 705 Exponentially weighted moving average (EWMA), 374-75, 705 compared with GARCH, 377-78 estimating parameters, maximum likelihood methods, 380 Exposure, 705 Extendable bond, 705 Extendable swap, 705 F Factor, 705 Factor analysis, 705 Factor loading, 361 Factor score, 361 FASB Statement No 133, 35 FBI, 34 Federal National Mortgage Association (FNMA), 586 Federal Reserve Board, 34 Fill-or-kill order, 33 Financial Accounting Standards Board (FASB), 35, 706 Statement No 133, 35 Financial intermediary, 129-30, 706 Finite difference method, 418-27, 706 applications of, 427 explicit, 422-23 implicit, 418-21 other, 426-27 relation to trinomial tree approach, 424-26 Subject Index First notice day, 32 First-to-default basket credit default swap, 643 Flat volatility, 518-19, 706 Flex option, 154, 273, 706 Flexi cap, 581 Flexible forwards, 14-15, 435 Flight to quality, 692 Floor, interest rate, 517, 706 valuation, 517-18 Floor-ceiling agreement, see Collar Floorlets, interest rate, 517 Foreign currency option, 706 See also Currency option Forward band, 435 Forward contract, 2-5, 125, 706 delivery, 60 delivery price, delta, 305 difference between futures and, 6, 19, 36-37 difference between options and, 6, 151 foreign exchange quotes, 3, 37 hedging, using, 11 valuing, 49-50, 226-27 Forward difference approximation, 419 Forward exchange rate, 706 Forward induction, 460-61 Forward interest rate, 98-100, 111, 566, 706 volatilities, 579 Forward price, 4, 5, 706 for an investment asset that provides a known cash income, 47-49 for an investment asset that provides a known yield, 49 for an investment asset that provides no income, 45-47 Ito's lemma, applied to, 226-27 relation to futures price, 51-52 risk-neutral valuation, 245-46 Forward-rate agreement (FRA), 100-101, 706 Forward risk-neutral valuation, 489, 524 Black-Scholes formula to price European stock options and, 493-94, bond option, 514-15 interest rate caps, 519-20 option to exchange one asset for another, ' 494-95 swap options, 523-24 Forward risk-neutral world, 489, 706 Forward start option, 437, 706 Subject Index Forward swap, 521, 706 Forward with optional exit, 436 Front office, 690 FTSE 100 Index, 53 Full-service broker, 158 Futures contract, 5-6, 706 asset underlying, 20-21 closing out positions, 20 commodities, 58-61 contract size, 21 currencies, 55-58 delivery, 21-22, 31-32, 60-61 delivery month, 22, 77 delta, 305-6 difference between forward contracts and, 6, 19, 36-37 difference between options and, 6, 151 foreign exchange quotes, 37 long position, 20 margins and, 24-27 marking to market, 24-26 price quotes, 22 quotations, 27-30, 53, 57, 105-6 risk, 61-62 short position, 20 specification of, 20-23 Treasury bond and Treasury note futures, 21, 22, 104-10 Futures interest rate, 111, 566 Futures market, regulation of, 33-34 Futures option, 152, 278-79, 706 interest rate futures option, 279-82 popularity of, 283 put-call parity, 283-84 quotations, 279-82 spot options compared to, 288-89 valuation, using binomial trees, 284-86, 399-402 valuation, using Black's model, 287-88 Futures price, 20, 706 convergence to spot price, 23-24 cost of carry, 60 expected future spot prices and, 31, 61-63 expected growth rate, 286-87 patterns of, 31 relationship to forward prices, 51-52 735 GAP management, 116 Gamma, 312, 706 cross gamma, 358 estimating, using binomial tree, 398 formula, 314 interest rate derivatives, 531 relationship with delta and theta, 315-16 effect on the probability distribution of the value of a portfolio, 356 Gamma-neutral portfolio, 313-14, 706 GARCH, Generalized autoregressive conditional heteroscedasticity, 376-77, 706 compared with EWMA, 377-78 estimating parameters, maximum likelihood methods, 379-82 forcasting future volatility, 382-85 Gaussian copula, 628-29, 632 Generalized Wiener process, 218-21, 706 Geometric average, 238, 706 Geometric Brownian motion, 223, 706 Gibson Greetings, 688, 689, 693, 694 Girsanov's theorem, 212 Globex, 721 Goldman Sachs Commodity Index (GSCI), 53 Goldman Sachs, 53 Good-till-canceled order, 33 Government National Mortgage Association (GNMA), 586 Greek letters, Greeks, 299, 337, 706 See also Delta, Theta, Gamma, Vega, and Rho estimating using binomial tree, 398-99 estimating using finite difference method, 427 estimating using Monte Carlo simulation, 414 interest rate derivatives, 530-31 Gross basis, 26-27 Gross return, stock's, 238 Growth factor, 394 H Hammersmith and Fulham, 688, 613-14 Hazard rate, 613-14, 706 Heating degree days (HDD), 679, 707 Hedge accounting, 35 Hedge-and-forget strategy, 70, 303 Hedger, 10, 11, 31, 61, 693-94, 707 Hedging/hedge, 707 arguments for and against, 72-75 an equity portfolio, 84 736 basic principles, 70-72 basis risk, 75-78 competitors and, 73-74 delta hedging, 302-9 duration-based hedging strategies, 116-18 effectiveness, 79 exotic options, 447 gamma, 312-15 in practice, 319 interest rate derivatives, 530-31, 565-66 long hedge, 71-72 naked and covered position, 300 perfect hedge, 70 performance measure, 302, 307, 308 ratio, 78-79, 707 rho, 318-19 rolling forward, 86-87 shareholders and, 73 short, 71 stop-loss strategy, 300-302 theta, 309-11 using index future, 82-85 vega, 316-18 Hicks, John, 31, 61 Historic volatility, 239, 707 Historical simulation, value at risk, 348-50, 707 compared with model building approach, 359-60 History-dependent derivative, 461-65 Holiday calendar, 707 Hong Kong and Shanghai bank, 687 Hopscotch method, 426 Hunt brothers, 34 Hurricane Andrew, 683 I Implicit finite difference method, 418-21, 707 relation to explicit finite difference method, 422-23 Implied volatility, 250-51, 707 Implied volatility function (IVF) model, 337, 460-61, 707 Implied distribution, 707 currency options, 331-32 stock options, 334-35 Implied repo rate, 707 Implied tree model, 337, 460-61, 707 Importance sampling, variance reduction procedure, 415 Subject Index Inception profit, 691, 707 Index amortizing rate swap, 605, 707 Index arbitrage, 54, 707 Index currency option note (ICON), 14 Index futures, 707 hedging, using index future, 82-85 portfolio insurance, using, 321-22 pricing, 54 quotations, 53 Index options, 152, 270-76, 707 portfolio insurance, 273-75 quotations, 270-73 valuation, binomial tree, 399-402 valuation, Black-Scholes, 275-76 Indexed principal swap, 605, 707 Initial margin, 24-26, 158, 159, 707 Inner barrier, 467 Instantaneous forward rate, 707 Instantaneous short rate, 537 See also Short rate Insurance derivatives, 15, 682-83 Interest only (IO), 587, 708 Interest rate, 42-44 continuous compounding, 43 day count conventions, 102-3, 130 forward, 98-100 forward-rate agreements (FRA), 100-101 term structure theories, 102 types of, 93-94 zero-coupon yield curve, 97-98 Interest rate cap, 515-20, 707 forward risk-neutral valuation and Black's model, 519-20 Interest rate collar, 517, 707 Interest rate derivatives, 508, 707 Black's model, 508-10 bond options, 511-15 convexity adjustment, 524-26 embedded bond option, 511 equilibrium models, 537-43 European bond option, 511-15 European swap option, 520-24 Heath, Jarrow, Morton model, 574-77 hedging, 530-31 interest rate cap, 515-20 LIBOR market model, 577-86 natural time lag, 529-30 no-arbitrage models, 543-44 short rate models, 537-67 Subject Index timing adjustment, 527-29 yield volatility, 513-14 Interest rate floor, 517-18, 707 Interest rate futures Eurodollar futures, 110-11 Relation to forward interest rate, 111 Treasury bond futures, 104-10 Interest rate futures option, 279-82 Eurodollar futures option, 279, 282 Treasury bond futures option, 279, 282, Treasury note futures option, 279 Interest rate models BGM model, 577 equilibrium models, 537-43 Heath, Jarrow, Morton model, 574-77 LIBOR market model, 577-86 no-arbitrage models, 543-44 short rate models, 537-67 standard market models, 508-31 two-factor models, 543, 571-73 Interest rate option, 707 Interest rate parity, 56 Interest rate swap, 707 comparative-advantage argument, 131-34 confirmation, 130-31 day count conventions, 130 financial intermediary, role of, 129-30 impact of default risk, 651 mechanics of, 125-31 nonstandard swaps, 594-606 plain vanilla interest rate swap, 125 to transform an asset, 128-29 to transform a liability, 127-28 valuation, 136-40 Interest rate trees, 550-52 general tree-building procedure, 552-63 trinomial trees, 551-52 nonstandard branching, 552 International Petroleum Exchange (IPE), 680, 681 International Swaps and Derivatives Association (ISDA), 130 Master Agreement, 130-31 In-the-money option, 153, 304, 311, 315, 707 Intrinsic value, 154, 707 Inverted market, 31, 708 Investment asset, 41, 58, 708 market price of risk, 485 Investment grade, 610 737 Ito's lemma, 216, 226-27, 487, 708 Ito's process, 222, 708 Jett, Joseph, 686, 687, 690 J.P Morgan, 375, 632 Jump diffusion model, 457-58, 708 K Kappa, 708 Keynes, John Maynard, 31, 61 Kidder Peabody, 686, 687, 690 KMV, 623 Knock-in and knock-out options, 439 Kurtosis, 332, 708 Lambda, 708 Last notice day, 32 Last trading day, 32 LEAPS, see Long-term equity anticipation securities Least-squares approach, Monte Carlo simulation for American options, 474-77 Leeson, Nick, 686, 687, 690 Levenberg-Marquardt procedure, 565 LIBOR-in-arrears swap, 526-27, 599, 708 Limit move, 22, 708 limit up, 22 limit down, 22 Limit order, 33, 157, 708 Linear model, value at risk, 352-55 Liquidity preference theory, shape of zero curve, 102, 708 Liquidity premium, 708 Liquidity risk, 691-92, 708 Lloyds syndicate, 683 Locals, 32, 708 Lock out period, callable bond, 511 Lognormal property, 227-28, 234-36, 708 London Interbank Bid Rate (LIBID), 93, 708 London Interbank Offer Rate (LIBOR), 45, 93-94, 110,708 zero curve, 111-12, 135-36, 708 See also Swap zero curve London International Financial Futures and Options Exchange (LIFFE), 19, 111, London Stock Exchange, 13, 53 Long hedge, 71-72, 708 Subject Index 738 Long position, 2, 8, 20, 708 Long-Term Capital Management (LTCM), 625, 687, 692 Long Term Credit Bank of Japan, 14 Long-term equity anticipation securities (LEAPS), 153, 273, 708 Lookback option, 441-42, 461-63, 465-67, 708 Loss given default, 623-25 Low-discrepancy sequence, see Quasi-random sequence M Maintenance margin, 24, 159, 708 Margin account, 24, 25, 158 Margin, 24-27, 158-60, 708 clearing margin, 26 gross margining, 27 initial margin, 24-26, 158-59 maintenance margin, 24, 159 margin call, 24, 709 net margining, 27 for stocks, 158-59 for stock options, 159-60 variation margin, 24 Market-if-touched (MIT) order, 33 Market maker, 130, 157, 158, 709 Market model, 709 Market-not-held order, 33 Market order, 33, 157 Market price of risk, 483, 484-86, 709 multiple independent factors, 492-93 Market segmentation theory, shape of zero curve, 102, 709 Marking to market, 24-26, 709 Marking to model, 691 Markov process, 216-17, 709 Markowitz, Harry, 352 Martingale, 483, 488-89, 709 equivalent martingale measure result, 488-89 Maturity date, 6, 709 Maximum likelihood method, 378-82, 709 Mean reversion, 377, 518, 539, 542, 709 Measure, 483, 709 Metallgesellschaft (MG), 87, 688 Mid-curve Eurodollar futures option, 279 Middle office, 690 Midland Bank, 687 Minimum variance hedge ratio, 78-82 Min-max, 435 Model building approach, value at risk, 350-52 compared with historical simulation, 359-60 Modified duration, 114-15, 709 Moment matching, variance reduction procedure, 416 Money market account, 489, 709 Monte Carlo simulation, 224, 301, 410-14, 446, 459, 462, 709 American options and, 474-78 exercise boundary parametrization approach, 477-78 generating random samples, 412-13 Greek letters and, 414 least-squares approach, 474-77 LIBOR market model of short rates, 579-81 number of trials, 413 quantifying seller default risk, 643-44 value at risk measure, 359 valuing derivatives on more than one market variable, 412-13 valuing first-to-default basket credit default swaps, 643 valuing mortgage-backed securities, 588 valuing new business, 666-67, 671-75 Moody's, 610, 638 Risk Management Services, 623 Mortgage-backed security (MBS), 586-88, 709 N Naked option, 159 Naked position, 300, 709 Nasdaq 100 index (NDX), 53 futures, 21, 53 Mini Nasdaq 100 index futures, 21, 53 option, 152, 270 National Association of Securities Dealers Automatic Quotations Service, 53 National Futures Association (NFA), 34 National Westminster Bank, 687, 690 Natural gas derivatives, 680-81 Natural time lags, 529-30 Net basis, 26-27 Net present value (NPV) approach, 660 Netting, 624-25, 709 Neutral calendar spread, 193 New York Cotton Exchange, 21 New York Federal Reserve, 687 New York Mercantile Exchange (NYMEX), 22, 680, 681 Subject Index New York Stock Exchange (NYSE), 13, 52, 55, 323 Newton-Raphson method, 95, 251, 541, 559, 709 Nikkei 225 Stock Average, 52 futures, 52, 55, 497 No-arbitrage assumption, 709 No-arbitrage interest rate model, 543-44, 709 Nonstandard American options, 436 Nonstationary model, 563-64, 709 Nonsystematic risk, 61, 488, 709 Normal backwardation, 31, 709 Normal distribution, 234, 709, 722-23 Normal market, 31, 709 Notice of intention to deliver, 20, 31-32 Notional principal, 126, 637, 709 Numeraire, 488, 709 annuity factor as the numeraire, 491-92 impact of a change in numeraire, 495-97 interest rates when a bond price is the numeraire, 491 money market account as the numeraire, 489-90 numeraire ratio, 496 zero-coupon bond price as the numeraire, 490-91 Numerical procedure, 392, 710 Off-the-run bonds, 691 Offer, 2, 135, 157, 710 Offsetting orders, 157 On-the-run bond, 691 Open interest, 30, 157, 710 Open order, 33 Open outcry trading system, 2, 710 Option-adjusted spread (OAS), 588, 710 Option fence, 435 Options, 6-10, 710 class, 153,702,710 difference between futures (or forward) contracts and, 6, 151 exercise limits, 155 exercising, 160-61 exotic, 163, 435-49 hedging, using, 11 intrinsic value, 154 position limits, 155 positions, 739 regulation of, 161 series, 153, 710 taxation, 161-62 time value, 154 trading, 157 types of, 6, 151-52 See also: Currency options, Futures options, Index options, Stock options, Swaptions Options Clearing Corporation (OCC), 160-61, 710 Options in an investment opportunity, 670-71 Options on two correlated assets, 472-74 Options to exchange one asset for another, 445-46 forward risk-neutral valuation, 494-95 Options to defer, 671 Options to extend, 671 Orange County, 686, 688, 693, 694 Order book official, see Board broker Order, types of, 33 Out-of-the-money option, 153, 304, 311, 315, 710 Outer barrier, 467 Outside model hedging, 565 Overnight repo, 94 Over-the-counter market, 1, 2, 710 difference between exchange-traded market and, for options, 154, 163 Pacific Exchange, 151 Package, 435, 710 Parallel shift, 361, 710 Par bond yield, 95-96, 710 Par value, 710 Partial simulation approach, Monte Carlo simulation, 359 Pass-throughs, 587 Path-dependent derivative, 461-65, 710 Payoff, 3, 710 Perfect hedge, 70 Philadelphia Stock Exchange (PHLX), 151, 276 Plain vanilla product, 435, 710 Poisson process, 630, 710 Portfolio immunization, 116, 710 Portfolio insurance, 273-75, 320-23, 692, 710 stock market volatility and, 323 Position limit, 23, 155, 710 740 Position traders, 33 Premium, 710 Prepayment function, 587, 710 Prices lifetime highs, 30 lifetime lows, 30 opening price, 27 settlement price, 27 Price sensitivity hedge ratio, 117 Prime rates, 133-34 Principal, 710 Principal components analysis, 360-63, 530-31, 584-85, 710 Principal only (PO), 587, 710 Probability measure, 486 Probability of default, 623 assuming no recovery, 612-13 bond prices and, 610-19 bond prices and historical probability of default, 619-20 estimating, using equity prices, 621-23 implied from bond data, 616-17 implied from CDS swaps, 641-42 risk-neutral vs real-world, 620-21 Proctor and Gamble, 605, 688, 689, 693, 694 Program trading, 54, 710 Protective put, 185, 710 Pull-to-par, 711 Put-call parity, 174-75, 179, 186-87, 268, 283-84, 330-31, 438, 711 Put option, 6, 7-8, 151, 711 Puttable bond, 511,711 Puttable swap, 711 Quadratic model, value at risk, 356-59 Quadratic resampling, variance reduction procedure, 416 Quanto, 497-99, 711 adjustment, 601 Quasi-random sequence, variance reduction procedure, 416-17, 711 Quotations, commodity futures, 27-30 currency futures, 57 currency options, 276, 277 futures options, 279-82 stock index futures, 53 stock index options, 270-73 Subject Index stock options, 155-57 Treasury bills, 104 Treasury bonds, 103-4 Treasury bond and note futures, 105-6 Quoted price, bond and Treasury bill, 103-4, 512, 561 See also Clean price Ratchet cap, 581 Rainbow option, 446, 472, 711 Random walk, 232-33 Range forward contract, 14-15, 435, 711 Real option, 1, 660, 711 Rebalancing, 242, 303, 711 Recovery rate, 614-15, 642, 711 Reference entity, 637 Reference obligation, 637 Reinsurance, against catastrophic risks (CAT reinsurance), 682-83 Repo, 94, 711 Repo rate, 94, 711 overnight repo, 94 term repo, 94 Representative sampling through a tree, variance reduction procedure, 417-18 Repurchase agreement, 94 Reset date, 711 Retractable bond, 511 Reverse calendar spreads, 194 Reversion level, 711 Rho, 318-19, 711 estimating, using binomial tree, 398 Rights issue, 155, 711 Risk, see Value at risk back testing, 360 hedging and basis risk, 75-78 market risk, 146 nonsystematic, 61, 488, 713 stress testing, 360, 689 swaps and credit risk, 145-46 systematic, 61, 62, 488, 713 Risk and return, relationship between, 61 Risk-free interest rate, 45, 93, 94, 168-70, 711 Risk-free zero curve, 111 Risk limits, 686-89 Risk-neutral valuation, 203-5, 244-45, 247, 269, 678, 483, 661-65, 711 Risk-neutral world, 204, 205, 711 forward risk-neutral, 489 Subject Index rolling forward risk-neutral, 578 traditional, 483, 486 RiskMetrics, 375 Roll back, 711 Roll-over risk, 136 Rusnak, John, 686, 687, Russell 2000 index (RUT), 53, 270 options, 270 Scalper, 32, 711 Scenario analysis, 319-20, 689, 692, 711 Securities and Exchange Commission (SEC), 34, 161,711 Segmentation theory, shape of zero curve, 102 Seller default risk, 643-44 Settlement price, 27, 711 Share Price Index, 53 Shell, 686, 688 Shorting, 41-42 Short hedge, 71, 711 Short position, 2, 8, 20, 711 Short rate, 537, 711 Cox, Ingersoll, Ross model, 542-43 calibration, 564-65 equilibrium models, 537-38 general tree-building procedure, 552-63 Ho-Lee model, 544-46 Hull-White model, 546-49 interest rate trees, 550-52 no-arbitrage models, 543-44 nonstationary models, 563-64 one-factor equilibrium models, 538 Rendleman and Bartter model, 538-39 two-factor equilibrium models, 543 Vasicek model, 539-42 Short selling, 41-42, 711 Short squeezed investor, 42 Short-term risk-free rate, see Short rate Shout option, 443, 712 Siegel's paradox, 499-500 Sigma, see Vega Simulation, 712 Singapore International Monetary Exchange (SIMEX), 19 Specialist, 712 Speculator, 10, 11-12, 32-33, 693-94, 712 Speculation, using futures, 11-12 741 using options, 12-13 Spot contract, Spot rate, 94, 712 Spot option, 288 futures option compared to, 288-89 Spot price, 4-5, 712 convergence of futures price to, 23-24 futures prices and expected future spot prices, 31, 61-63 Spot volatility, 518-19, 712 Spread option, 712 Spread trading strategy, 187-94 Spread transaction, 26, 712 Standard and Poor's (S&P) Index, 52, 610, 638 100 Index (OEX), 152, 270, 273 500 Index (SPX), 32, 52, 55, 152, 270, 279, 360 500 Index futures, 32, 52, 54 futures option, 279 MidCap 400 Index, 52 Mini S&P 500 futures, 52 options, 152, 270, 273 Standard Oil, 14 Static hedging scheme, 303, 712 Static options replication, 447-49, 712 Step-up swap, 594, 712 Sticky cap, 581-82 Sticky delta rule, 337 Sticky strike rule, 337 Stochastic process, 216, 712 Stochastic variable, 712 Stochastic volatility models, 458-60 Stock dividend, 712 Stock's expected return, 237-38 stock option price and, 203 Stock index, 52-54, 712 Stock index futures, see Index futures Stock index options, see Index options Stock option, 152-55, 712 See also BlackScholes model commissions, 157-58 dividend and stock split, 154-55 executive, 163 expiration dates, 152-53 flex option, 154, 273 long-term equity anticipation securities (LEAPS), 153, 273 margins, 159-60 naked, 159 742 position and exercise limits, 155 quotations, 155-57 regulations of, 161 specification of, 152-55 strike prices, 153, 167-68 taxation, 161-62 terminology, 153-54 trading, 157 Stock option value, American options on dividend-paying stock, 179, 254-56 American options on non-dividend-paying stock, 247 analytic approximation to American option prices, 427 assumptions, 170-71 binomial tree, 200-210, 212-13 binomial tree pricing formulas, 202-3, 207, 212-13,269-70, Black-Scholes model, 268-69 bounds, for dividend-paying stocks, 178-79, 267-68 bounds, for non-dividend-paying stocks, 171-74 dividends, 154-55, 178-79, 186-87, 252-56 dividend yield, 267-70 European options on a dividend-paying stock, 178-79, 252-53 European options on a non-dividend-paying stock, 171-74, 246-48 European options on a stock paying a known dividend yield, 267-70 factors affecting prices, 167-70 implied distribution and lognormal distribution, 334-35 put-call parity, 174-75, 179, 186-87, 268, 330-31 risk-neutral valuation, 269 single large jump in asset anticipated, 338-39 stock's expected return and, 203 volatility smile (skew), 334-36 Stock prices, expected return, 237-38 lognormal property, 227-28, 234-36 rate of return, distribution of, 236-37 the process for, 222-25 volatility, 238-41 Stock split, 154-55, 712 stock dividend and, 154-55 Subject Index stock options and, 155 stock prices and, 154 Stop-and-limit order, 33 Stop-limit order, 33 Stop-loss order, 33 Stop-loss strategy, 300-302 Stop order, 33 Storage cost, 58, 712 Straddle, 194-95, 712 straddle purchase, 195 straddle write, 195 Strangle, 196, 712 Strap, 195-96, 712 Stratified sampling, variance reduction procedure, 415-16 Strengthening of the basis, 75 Stress testing, 360, 689, 692, 712 Strike price, 6, 712 Stripped mortgage-backed securities, 587 Strip, 195-96, 687, 712 Sumitomo, 686, 687, 690 Swap rate, 134-35, 527, 712 bid, 135 offer, 135 Swap zero curve, 111, 136 See also LIBOR zero curve Swaps, 125, 712 accrual, 603 amortizing, 594 basis, 595 cancelable, 603-4 cancelable compounding, 604-5 commodity, 605 comparative-advantage argument, 131-34, 141-43 compounding, 595-98 confirmations, 130-31 constant maturity swap (CMS), 599-600 constant maturity Treasury swap (CMT), 600-601 correlation, 605 covariance, 605 credit risk and, 145-46 currency, see Currency swaps deferred, 521 denned, 125 differential (diff swap), 601 equity, 601-2 forward, 521 Subject Index index amortizing rate, 605 indexed principal, 605 interest rate, see Interest rate swaps LIBOR-in-arrears, 599 step-up, 594 variance, 605 volatility, 605 Swaptions, 520, 712 Bermudan swaptions, 586 European swaption valuation, 521-22, 582-84 forward risk-neutral valuation and Black's model, 523-24 implied volatilities, 522-23 relation to bond options, 521 Swing option, energy market, 681, 712 Sydney Futures Exchange (SFE), 19 Synthetic option, 320, 712 Systematic risk, 61, 62, 488, 713 Tailing the hedge, 82 Take-and-pay option, energy market, 681, 713 Tax, 35-36, 161-62 Taxpayer Relief Act of 1997, 35, 162 Tenor, 515 Term repo, 94 Term structure models, interest rates, 537, 713 Term structure theories, shape of zero curve, 102 Terminal value, 713 Theta, 309-11,713 estimating, using binomial tree, 398 relationship with delta and gamma, 315-16 Time decay, 309, 713 Time-of-day order, 33 Time-to-expiration effects, 168 Time value, 154, 713 Timing adjustment, 527-29, 713 constant maturity swap (CMS), 600 accrual swap, 603 To-arrive contract, Tokyo International Financial Futures Exchange (TIFFE), 19 Tokyo Stock Exchange, 52 Top straddles, 195 Top vertical combinations, 196 Total return swap, 644-45, 713 Tradeable derivatives, prices of, 244 743 Traders, types of, 10-14, 32-33 Trading irregularities, 35 Trading strategies, involving options combinations, 194-96 for single option and stock, 185-87 spreads, 187-94 Tranches, 646 Transaction cost, 309, 713 Treasury bill, 104, 713 Treasury bill futures, 713 Treasury bond, 103-4, 713 Treasury bond futures, 104-10, 713 cheapest-to-deliver bond, 108 conversion factors, 106-7 quoted futures price, 109-10 quotations, 105-6 wild card play, 108-9 Treasury bond futures option, 279, 282 Treasury note, 713 Treasury note futures, 21, 22, 713 Treasury rate, 45, 93 zero rate, 94, 95, 96-98 Tree, 713 Trinomial tree, 713 for stock prices, 408-9, relation to finite difference method, 424-25 Triple witching hour, 713 U Underlying variable, 713 Unsystematic risk, 61, 488, 713 Up-and-in calls, 440, 713 Up-and-in puts, 440, 713 Up-and-out calls, 440, 713 Up-and-out puts, 440, 713 Uptick, 42, 713 U.S Department of Energy, 679 U.S dollar index, 53 U.S Treasury Department, 34 V Value additivity, 618 Value at risk (VaR), 346, 713 conditional VaR (C-VaR), 347-48 diversification and, 352 historical simulation, 346, 348-50, compared with model building, 359-60 linear model, 352-55 744 model building approach, 346, 350-52, compared with historical simulation, 359-60 Monte Carlo simulation, 359 principal components analysis, 360-63 quadratic model, 356-59 RiskMetrics and, 375 time horizon, 348 variance-covariance approach, 350-52 volatilities and, 350-52 Variance-covariance matrix approach, value at risk, 350-52, 713 Variance rate, 372, 713 estimating constant variance, maximum likelihood methods, 378-79 Variance reduction procedures, 414-18, 713 antithetic variable technique 414-15 control variate technique, 415 importance sampling, 415 moment matching, 416 quadratic resampling, 416 quasi-random sequences, 416-17 representative sampling through a tree, 417-18 stratified sampling, 415-16 Variance swap, 605 Variance targeting, 380 Variation margin, 24, 713 Vega, 316-18, 713 estimating, using binomial tree, 398 interest rate derivatives, 531 Vega neutral portfolio, 318, 713 Volatility, stock prices, 713 Black-Scholes model and, 238-41, 330-31 causes of, 251-52 defined, 168, 211-12 estimating, 239-41, 372-74 See also EWMA, GARCH forecast future volatility, 382-85 implied, 250-51, 286 See also Volatility smile portfolio insurance and stock market volatility, 323 term structure, volatility of stock return, 336-37, 384, 714 Subject Index volatility skew, 334 volatility surface, 336-37, 460 Volatility, interest rate derivatives flat volatility, 518-19 forward rate volatility, 579 spot volatility, 518-19 volatility skew, 585, Volatility matrix, 714 See also Volatility surface Volatility skew, 334, 585, 713 Volatility smile, 330, 714 equity options, 334-36 foreign currency options, 331 -34 Volatility surface, 336-37, 460 See also Volatility matrix Volatility swap, 605, 714 w Wall Street Journal, 27-30, 53, 57, 104, 105-6, 155-57, 270-72, 276-77, 279-82 Warrant, 162-63, 249-50, 714 Wash sale rule, 161-62 Weakening of the basis, 75 Weather derivatives, 15, 679-80, 714 Weather Risk Management Association (WRMA), 679 Wiener process, 218-21, 714 Wild card play, 108-9, 714 Writing a covered call, 185 Writing an option, 8, 714 Y Yield, 714 Yield curve, 714 Zero-cost collar, 435 Zero-coupon bond, 714 Zero-coupon interest rate, 714 See also Zero rate Zero-coupon yield curve, 714 See also Zero curve Zero curve, 97-98 shape of zero curve, theories, 102 Zero rate, 94, 96-98 ... indices Currency options Futures options Valuation of futures options using binomial trees Futures price analogy Black's model for valuing futures options Futures options vs spot options Summary... Nonstandard American options 19.3 Forward start options 19.4 Compound options 19.5 Chooser options 19.6 Barrier options 19.7 Binary options 19.8 Lookback options 19.9 Shout options 19.10 Asian options. .. "Questions and Problems" and "Assignment Questions" Solutions to the Questions and Problems are in Options, Futures, and Other Derivatives: Solutions Manual, which is published by Prentice Hall and