OPTIONS, FUTURES, & OTHER DERIVATI

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1 Fifth Edition OPTIONS, FUTURES, & OTHER DERIVATI John C. Hull Maple Financial Group Professor of Derivatives and Risk Manage, Director, Bonham Center for Finance Joseph L. Rotinan School of Management University of Toronto leg gg Spende DER VEREINIGUNG VON FREUNDEN DER TECHNISCHEN UNIVERSITAT ZU DARMSTADT E.V Ernst-Ludwigs-Hochschulgesellschaft gg «g «g Prentice. HaU PEARSON EDUCATION INTERNATIONAL

2 CONTENTS Preface 1. Introduction Exchange-traded markets Over-the-counter markets Forward contracts Futures contracts Options Types of traders Other derivatives 14 Summary 15 Questions and problems 16 Assignment questions Mechanics of 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 36 Summary 37 Suggestions for further reading 38 Questions and problems 38 Assignment questions Determination of forward and futures prices 41 " Investment assets vs. consumption assets Short selling Measuring interest rates Assumptions and notation Forward price for an investment asset Known income Known yield Valuing forward contracts Are forward prices and futures prices equal? Stock index futures Forward arid futures contracts on currencies Futures on commodities 58 xix ix

3 Contents 3.13 Cost of carry Delivery options Futures prices and the expected future spot price 61 Summary 63 Suggestions for further reading 64 Questions and problems 65 Assignment questions 67. Appendix 3A: Proof that forward and futures prices are equal when interest rates are constant Hedging strategies using futures Basic principles Arguments for and against hedging Basis risk Minimum variance hedge ratio Stock index futures Rolling the hedge forward 86 Summary 87 Suggestions for further reading 88 Questions and problems 88 Assignment questions 90 Appendix 4A: Proof of the minimum variance hedge ratio formula Interest rate 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 Ill 5.13 Duration Duration-based hedging strategies 116 Summary 118 Suggestions for further reading 119 Questions and problems 120 Assignment questions Swaps 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 145 Summary 146 Suggestions for further reading 147 ' Questions and problems 147 Assignment questions 149

4 Contents XI 7. Mechanics of 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 163 Summary 163 Suggestions for further reading 164 Questions and problems 164 Assignment questions Properties of 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 179 Summary 180 Suggestions for further reading.' 181 Questions and problems 182 Assignment questions Trading strategies involving options Strategies involving a single option and a stock Spreads T " Combinations Other payoffs 197 Summary 197 Suggestions for further reading 198 Questions and problems 198 Assignment questions Introduction to binomial trees A one-step binomial model Risk-neutral valuation Two-step binomial trees A put example American options Delta Matching volatility with it and d Binomial trees in practice 212 Summary 213 Suggestions for further reading 214 Questions and problems 214 Assignment questions 215

5 XII Contents 11. A model of the behavior of stock prices The Markov property Continuous-time stochastic processes The process for stock prices Review of the model The parameters Ito's lemma : The lognormal property 227 Summary 228 Suggestions for further reading 229 Questions and problems 229 Assignment questions 230 Appendix 11A: Derivation of Ito's lemma The Black-Scholes model Lognormal property of stock prices The distribution of the rate of return The expected return Volatility Concepts underlying the Black-Scholes-Merton differential equation Derivation of the Black-Scholes-Merton differential equation Risk-neutral valuation Black-Scholes pricing formulas Cumulative normal distribution function Warrants issued by a company on its own stock Implied volatilities The causes of volatility 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 Options on 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 288 Summary 289 Suggestions for further reading 290 Questions and problems 291 Assignment questions 294 Appendix 13A: Derivation of differential equation satisfied by a derivative dependent on a stock providing a dividend yield 295

6 Contents xiii Appendix 13B: Derivation of differential equation satisfied by a derivative dependent on a futures price The Greek letters Illustration Naked and covered positions A stop-loss strategy Delta hedging Theta Gamma Relationship between delta, theta, and gamma Vega ' Rho Hedging in practice Scenario analysis Portfolio insurance Stock market volatility 323 Summary 323 Suggestions for further reading 324 Questions and problems 326 Assignment questions 327 Appendix 14A: Taylor series expansions and hedge parameters Volatility smiles Put-call parity revisited Foreign currency options Equity options The volatility term structure and volatility surfaces Greek letters : When a single large jump is anticipated Empirical research 339 Summary 341 Suggestions for further reading 341 Questions and problems 343 Assignment questions 344 Appendix 15A: Determining implied risk-neutral distributions from volatility smiles Value at risk The VaR measure Historical simulation Model-building approach 350 ~ Linear model Quadratic model Monte Carlo simulation Comparison of approaches Stress testing and back testing Principal components analysis 360 Summary 364 Suggestions for further reading 364 Questions and problems 365 Assignment questions 366 Appendix l'6a: Cash-flow mapping 368 Appendix 16B: Use of the Cornish-Fisher expansion to estimate VaR 370

7 xiv Contents 17. Estimating volatilities and correlations Estimating volatility The exponentially weighted moving average model The GARCH(1,1) model Choosing between the models Maximum likelihood methods Using GARCH(1, 1) to forecast future volatility Correlations 385 Summary 388 Suggestions for further reading 388 Questions and problems 389 Assignment questions Numerical procedures Binomial trees Using the binomial tree for options on indices, currencies, and futures contracts Binomial model for a dividend-paying stock Extensions to the basic tree approach Alternative procedures for constructing trees Monte Carlo simulation Variance reduction procedures Finite difference methods Analytic approximation to American option prices 427 Summary 427 Suggestions for further reading 428 Questions and problems 430 Assignment questions 432 Appendix 18A: Analytic approximation to American option prices of MacMillan and of Barone-Adesi and Whaley Exotic options Packages Nonstandard American options Forward start options Compound options Chooser options.: Barrier options Binary options Lookback options Shout options Asian options Options to exchange one asset for another Basket options vrr." Hedging issues Static options replication 447 Summary 449 Suggestions for further reading 449 Questions and problems 451 Assignment questions _. 452 Appendix 19A: Calculation of the first two moments of arithmetic averages and baskets More on models and numerical procedures 'The CEV model The jump diffusion model 457

8 Contents xv 20.3 Stochastic volatility models The 1VF model Path-dependent derivatives Lookback options Barrier options Options on two correlated assets Monte Carlo simulation and American options 474 Summary 478 Suggestions for further reading 479 Questions and problems 480 Assignment questions Martingales 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 499 Summary 500 Suggestions for further reading 500 Questions and problems 501 Assignment questions 502 Appendix 21 A: Generalizations of Ito's lemma 504 Appendix 2IB: Expected excess return when there are multiple sources of uncertainty ' Interest rate 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 530 Summary 531 Suggestions for further reading 531 Questions and problems 532. Assignment questions 534 Appendix 22A: Proof of the convexity adjustment formula Interest rate 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 546

9 xvi Contents Options on coupon-bearing bonds Interest rate trees A general tree-building procedure Nonstationary models Calibration Hedging using a one-factor model Forward rates and futures rates 566 -Summary 566 Suggestions for further reading 567 Questions and problems 568 Assignment questions Interest rate derivatives: more advanced models Two-factor models of the short rate The Heath, Jarrow, and Morton model The LIBOR market model Mortgage-backed securities 586 Summary 588 Suggestions for further reading 589 Questions and problems 590 Assignment questions 591 Appendix 24A: The A(t, T), a P, and 9(t) functions in the two-factor Hull-White model Swaps revisited Variations on the vanilla deal Compounding swaps Currency swaps ; More complex swaps Equity swaps i6 Swaps with embedded options Other swaps Bizarre deals 605 Summary 606 Suggestions for further reading 606 Questions and problems 607 Assignment questions 607 Appendix 25A: Valuation of an equity swap between payment dates Credit risk Bond prices and the probability of default Historical data Bond prices vs. historical default experience Risk-neutral vs. real-world estimates Using equity prices to estimate default probabilities The loss given default Credit ratings migration Default correlations Credit value at risk 630 Summary 633 Suggestions for further reading 633 Questions and problems 634 ' Assignment questions 635 Appendix 26A: Manipulation of the matrices of credit rating changes 636

10 Contents xvu 27. Credit derivatives Credit default swaps Total return swaps Credit spread options Collateralized debt obligations Adjusting derivative prices for default risk Convertible bonds 652 Summary.._ 655 Suggestions for further reading 655 Questions and problems 656 Assignment questions Real options Capital investment appraisal Extension of the risk-neutral valuation framework Estimating the. market price of risk Application to the valuation of a new business Commodity prices Evaluating options in an investment opportunity 670 Summary 675 Suggestions for further reading 676 Questions and problems 676 Assignment questions Insurance, weather, and energy derivatives Review of pricing issues Weather derivatives Energy derivatives Insurance derivatives 682 Summary '. 683 Suggestions for further reading 684 Questions and problems 684 Assignment questions Derivatives mishaps and what we can learn from them Lessons for all users of derivatives Lessons for financial institutions Lessons for nonfinancial corporations 693 Summary 694 Suggestions for further reading 695 Glossary of notation 697 Glossary of terms 700 DerivaGem software 715 Major exchanges trading futures and options 720 Table for N(x) when x < Table for N(x) when x ^ Author index 725 Subject index 729

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