Second Edition DE RI VAT IV ES PRINCIPLES and PRACTICE R a n g a r a j a n K S u n d a m Sanjiv R Das Derivatives: Principles and Practice The McGraw-Hill/Irwin Series in Finance, Insurance, and Real Estate Stephen A Ross Franco Modigliani Professor of Finance and Economics Sloan School of Management Massachusetts Institute of Technology Consulting Editor FINANCIAL MANAGEMENT Block, Hirt, and Danielsen Foundations of Financial Management Fifteenth Edition Brealey, Myers, and Allen Principles of Corporate Finance Eleventh Edition Brealey, Myers, and Allen Principles of Corporate Finance, Concise Second Edition Brealey, Myers, and Marcus Fundamentals of Corporate Finance Eighth Edition Brooks FinGame Online 5.0 Bruner Case Studies in Finance: Managing for Corporate Value Creation Seventh Edition Cornett, Adair, and Nofsinger Finance: Applications and Theory Third Edition Cornett, Adair, and Nofsinger M: Finance Third Edition DeMello Cases in Finance Second Edition Grinblatt (editor) 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Management and Insurance Second Edition Kapoor, Dlabay, and Hughes Focus on Personal Finance: An active approach to help you achieve financial literacy Fifth Edition Kapoor, Dlabay, and Hughes Personal Finance Eleventh Edition Walker and Walker Personal Finance: Building Your Future First Edition Derivatives: Principles and Practice Second Edition Rangarajan K Sundaram Stern School of Business New York University New York, NY 10012 Sanjiv R Das Leavey School of Business Santa Clara University Santa Clara, CA 95053 DERIVATIVES: PRINCIPLES AND PRACTICE, SECOND EDITION Published by McGraw-Hill Education, Penn Plaza, New York, NY 10121 Copyright © 2016 by McGraw-Hill Education All rights reserved Printed in the United States of America Previous edition © 2011 No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any 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All credits appearing on page or at the end of the book are considered to be an extension of the copyright page Library of Congress Cataloging-in-Publication Data Sundaram, Rangarajan K Derivatives : principles and practice / Rangarajan K Sundaram, Sanjiv R Das – Second edition pages cm ISBN 978-0-07-803473-2 (alk paper) Derivative securities I Das, Sanjiv R (Sanjiv Ranjan) II Title HG6024.A3S873 2016 332.64’57—dc23 2014037947 The Internet addresses listed in the text were accurate at the time of publication The inclusion of a website does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites www.mhhe.com To my lovely daughter Aditi and to the memory of my beautiful wife Urmilla RKS To my departed parents and Priya and Shikhar SRD Brief Contents Author Biographies Preface xvi xxi 19 Exotic Options II: Path-Dependent Options 467 20 Value-at-Risk Acknowledgments 18 Exotic Options I: Path-Independent Options 437 xv Introduction 495 21 Convertible Bonds 22 Real Options PART ONE Futures and Forwards Futures Markets PART THREE 21 Swaps Pricing Forwards and Futures II: Building on the Foundations 88 Hedging with Futures and Forwards 104 Interest-Rate Forwards and Futures 126 567 23 Interest Rate Swaps and Floating-Rate Products 569 24 Equity Swaps 614 25 Currency and Commodity Swaps 651 26 The Term Structure of Interest Rates: Concepts 653 155 157 27 Estimating the Yield Curve Options: Payoffs and Trading Strategies 173 29 Factor Models of the Term Structure 10 Early Exercise and Put-Call Parity 216 11 Option Pricing: A First Pass 231 261 13 Implementing Binomial Models 14 The Black-Scholes Model 671 28 Modeling Term-Structure Movements 688 No-Arbitrage Restrictions on Option Prices 199 12 Binomial Option Pricing 697 30 The Heath-Jarrow-Morton and Libor Market Models 714 PART FIVE 290 309 Credit Risk 753 31 Credit Derivative Products 755 15 The Mathematics of Black-Scholes 346 32 Structural Models of Default Risk 16 Options Modeling: Beyond Black-Scholes 359 33 Reduced-Form Models of Default Risk 816 17 Sensitivity Analysis: The Option “Greeks” 401 34 Modeling Correlated Default vi 632 PART FOUR Interest Rate Modeling PART TWO Options Markets 547 19 Pricing Forwards and Futures I: The Basic Theory 63 Options 516 850 789 Brief Contents Bibliography Index B-1 I-1 The following Web chapters are available at www.mhhe.com/sd2e: PART SIX Computation 35 Derivative Pricing with Finite Differencing 36 Derivative Pricing with Monte Carlo Simulation 23 37 Using Octave 45 vii Contents Author Biographies Preface xvi Acknowledgments Chapter Introduction 1.1 1.2 1.3 1.4 1.5 1.6 3.8 Futures Prices 75 3.9 Exercises 77 Appendix 3A Compounding Frequency 82 Appendix 3B Forward and Futures Prices with Constant Interest Rates 84 Appendix 3C Rolling Over Futures Contracts 86 xv xxi Forward and Futures Contracts Options Swaps 10 Using Derivatives: Some Comments The Structure of this Book 16 Exercises 17 Chapter Pricing Forwards and Futures II: Building on the Foundations 88 12 PART ONE Futures and Forwards Chapter Futures Markets 19 21 2.1 Introduction 21 2.2 The Functioning of Futures Exchanges 23 2.3 The Standardization of Futures Contracts 32 2.4 Closing Out Positions 35 2.5 Margin Requirements and Default Risk 37 2.6 Case Studies in Futures Markets 40 2.7 Exercises 55 Appendix 2A Futures Trading and US Regulation: A Brief History 59 Appendix 2B Contango, Backwardation, and Rollover Cash Flows 62 Chapter Pricing Forwards and Futures I: The Basic Theory 63 3.1 3.2 3.3 3.4 Introduction 63 Pricing Forwards by Replication 64 Examples 66 Forward Pricing on Currencies and Related Assets 69 3.5 Forward-Rate Agreements 72 3.6 Concept Check 72 3.7 The Marked-to-Market Value of a Forward Contract 73 viii 4.1 Introduction 88 4.2 From Theory to Reality 88 4.3 The Implied Repo Rate 92 4.4 Transactions Costs 95 4.5 Forward Prices and Future Spot Prices 96 4.6 Index Arbitrage 97 4.7 Exercises 100 Appendix 4A Forward Prices with Convenience Yields 103 Chapter Hedging with Futures and Forwards 104 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 Introduction 104 A Guide to the Main Results 106 The Cash Flow from a Hedged Position 107 The Case of No Basis Risk 108 The Minimum-Variance Hedge Ratio 109 Examples 112 Implementation 114 Further Issues in Implementation 115 Index Futures and Changing Equity Risk 117 Fixed-Income Futures and Duration-Based Hedging 118 5.11 Exercises 119 Appendix 5A Derivation of the Optimal Tailed Hedge Ratio h ∗∗ 124 Chapter Interest-Rate Forwards and Futures 6.1 6.2 6.3 6.4 6.5 Introduction 126 Eurodollars and Libor Rates 126 Forward-Rate Agreements 127 Eurodollar Futures 133 Treasury Bond Futures 140 126 Contents 6.6 Treasury Note Futures 144 6.7 Treasury Bill Futures 144 6.8 Duration-Based Hedging 144 6.9 Exercises 147 Appendix 6A PVBP-Based Hedging Using Eurodollar Futures 151 Appendix 6B Calculating the Conversion Factor 152 Appendix 6C Duration as a Sensitivity Measure 153 Appendix 6D The Duration of a Futures Contract 154 PART TWO Options 155 Chapter Options Markets 157 7.1 7.2 7.3 7.4 7.5 Introduction 157 Definitions and Terminology 157 Options as Financial Insurance 158 Naked Option Positions 160 Options as Views on Market Direction and Volatility 164 7.6 Exercises 167 Appendix 7A Options Markets 169 Chapter Options: Payoffs and Trading Strategies 173 8.1 Introduction 173 8.2 Trading Strategies I: Covered Calls and Protective Puts 173 8.3 Trading Strategies II: Spreads 177 8.4 Trading Strategies III: Combinations 185 8.5 Trading Strategies IV: Other Strategies 188 8.6 Which Strategies Are the Most Widely Used? 191 8.7 The Barings Case 192 8.8 Exercises 195 Appendix 8A Asymmetric Butterfly Spreads 198 Chapter No-Arbitrage Restrictions on Option Prices 199 9.1 Introduction 199 9.2 Motivating Examples 199 9.3 Notation and Other Preliminaries 201 9.4 Maximum and Minimum Prices for Options 202 9.5 The Insurance Value of an Option 207 9.6 Option Prices and Contract Parameters 208 9.7 Numerical Examples 211 9.8 Exercises 213 Chapter 10 Early Exercise and Put-Call Parity 10.1 10.2 10.3 10.4 10.5 216 Introduction 216 A Decomposition of Option Prices 216 The Optimality of Early Exercise 219 Put-Call Parity 223 Exercises 229 Chapter 11 Option Pricing: A First Pass 231 11.1 Overview 231 11.2 The Binomial Model 232 11.3 Pricing by Replication in a One-Period Binomial Model 234 11.4 Comments 238 11.5 Riskless Hedge Portfolios 240 11.6 Pricing Using Risk-Neutral Probabilities 240 11.7 The One-Period Model in General Notation 244 11.8 The Delta of an Option 245 11.9 An Application: Portfolio Insurance 249 11.10 Exercises 251 Appendix 11A Riskless Hedge Portfolios and Option Pricing 255 Appendix 11B Risk-Neutral Probabilities and Arrow Security Prices 256 Appendix 11C The Risk-Neutral Probability, No-Arbitrage, and Market Completeness 257 Appendix 11D Equivalent Martingale Measures 260 ix www.downloadslide.net Subject Index Exchange-traded options, 169, 170–171 Ex-dividend date, 274 Ex-dividend price, 275 Exercise price, 9, 158 Exercise-time regret, 13 Exotic derivatives, Exotic options, 172, 437, 439 See also Path-dependent options; Path-independent options Expectations, unbiased expectations hypothesis, 96 Expected default frequency (EDF), 804 Expected shortfall (ES), 511 Expiration date, 158 Exponential GARCH model, 378 Exponential splines, 677–678, 685–687 F Factorials, in Octave, 46–47 Factor models of correlated default, 859–861 deriving fundamental PDE in, 712–713 of term structure, 697–713 affine, 697, 706–709 multifactor, 704–705 one-factor, 698–704 Federal funds transactions, 582 Fed funds effective rate (FFER), 582–583 Fed Funds rate, 582 Fill-or-kill (FOK) limit orders, 25 Final settlement price, 27 Financial forwards, 89 See also Forward contracts Financial futures, 22–23, 89 See also Futures contracts size of contracts, 32 Financial press, futures prices in, 30–31 Finite differencing, 3–22 Crank-Nicholson scheme and, 17–19 implicit, 13–17 pricing equity options and, 7–13 properties of numerical scheme and, 11–13 solution lattice for, 8–10 solution procedure for, 10 solving differential equations and, 4–7 for term-structure models, 19–21 First-to-default (FTD) basket, 775, 776, 854 Fixed-for-floating swaps, 573–574, 577, 646 cash flows from, 578 payment frequencies, 578 payoffs, 577–580 valuing and pricing, 580–586 forward method of, 584–586 principal method for, 580–583 swap spread and swap curve and, 583 Fixed-income futures, duration-based hedging and, 118 Fixed notional swaps, 624–628 Fixed-strike lookback options, 482, 484–485 Floating-for-floating commodity swaps, 647 Floating-rate notes (FRNs), 569–572 valuation of I-13 forward method of, 571–573 no-arbitrage approach to, 570–571 Floating-strike lookback options, 482, 484 Floor(s), 172 Black model for pricing, 605–610 controlling financing costs with, 602–603 defined, 601 payoffs, 606–607 put-call parity and, 604 swaptions and, 605 uses of, 601–602 Foreign currency forwards, Foreign exchange (FX) contracts, 633 forward, currency swaps vs., 638–640 spot, 633 Foreign exchange (FX) derivatives, 2, 3, Forward contracts credit spread, 758, 763 defined, 2, 5–6 as forward Libor rates, 607–608 hedging with, 3, See also Hedging interest-rate, 126–154 key characteristics of, long-term, 46 market-to-market value of, 73–74 vs options, 166 payoffs from, 6–7 speculation with, Forward default probabilities, 823, 824–825, 826 Forward Libor rate, 607–608 Forward markets, vs futures markets, 23 Forward price, 6, 7–8 Forward pricing, 63–81, 660–661 arbitrage and, 66–69, 71 backwardation and, 91–92 commodity, 644 with constant interest rates, 84–86 contango and, 91–92 with continuous compounding, 65–66 with convenience yields, 103 on currencies and related assets, 69–72 forward-rate agreements and, 72 future spot prices and, 96 implied repo rate and, 92–95 index arbitrage and, 96–100 interest rates and, 65 market-to-market value of forward contract and, 73–74 no-arbitrage assumption and, 63 option pricing compared to, 231–232 in real world, 88–92 by replication, 64–66 on stock indices, 72, 97–98 transaction costs and, 95–96 Forward-rate agreements (FRAs), 6, 72, 126, 127–133 deriving arbitrage-free rate, 130–131 vs eurodollar futures, 138–140 hedging with, 132–133 payoffs from, 128–129 pricing new, 130, 131 www.downloadslide.net I-14 Subject Index Forward-rate agreements (FRAs), (Cont ) swaps as portfolio of, 584–586 valuing existing, 131–132 Forward rates, 658–660 Forward spreads, defaultable HJM model and, 830–832 Forward-starting swaps, 589–590 Forward start options, 439–442 hedging, 442 pricing, 440–441 Sprint repricing scheme, 441 Forward yield curve, 715 France Telecom, 774–775 FRAs See Forward-rate agreements (FRAs) FRNs See Floating-rate notes (FRNs) FTSE 100, 97, 170 Fundamental partial differential equation, 351–352 Funding-cost arbitrage, 761–762, 768–769 Futures commission merchants (FCMs), 24 Futures contracts, 111 closing out, prior to maturity, 36 closing out positions, 35–36 defined, delivery options, 28–29, 34–35 duration of, 145, 154 fixed-income, and duration-based hedging, 118 GNMA CDR, 41–44 hedging with See Hedging interest-rate, 126–154 leverage and, 54 margins and, 37–40 mini, 32 vs options, 166 options on, 170–171, 327–329 rolling over, 86–87 short-term, 46 size of, 32, 33 specifications, 142 standard grade in, 32–34 standardization of, 32–35 volume of trading, 23 Futures exchanges, functioning of, 23–31 Futures Industry Association, 59 Futures markets, 8–9, 21–62 See also Futures contracts case studies in, 40–55 clearinghouse in, 30 closing out positions, 35–36 consolidation in, 21–22 contract performance in, 30 delivery and settlement procedures in, 28–29 vs forward markets, 23 growth of, 22–23 introduction to, 21–23 margin requirements, 37–40 order types, 25–27 players in, 24–25 position limits in, 29–30 prices in, 27 price ticks and price limits in, 28 technology, 22 trading platform, 22 Futures pricing, 75–77 backwardation and, 91–92 with constant interest rates, 84–86 contango and, 91–92 currency, 89 daily marking-to-market and, 76–77 delivery options and, 75–76 in financial press, 30–31 gold, 90–91 implied repo rate and, 92–95 index arbitrage and, 96–100 oil, 91–92 in real world, 88–92 spot, forward prices and, 96 transaction costs and, 95–96 Futures trading, US regulation and, 59–61 G Gamma Black-Scholes, 434–435 cash-or-nothing options and, 445–447 compound, 454–455 convertible, 533, 534–535 option See Option gamma position, 427–428 Gap options, 442–443 GARCH (generalized autoregressive conditional heteroskedasticity) models, 376–380 calibration and simulation of, 379–380 description of approach, 377 option pricing and, 378–379 program code for simulating stock price distributions, 399–400 varianets and extensions of, 377–378 GARCH (p,q) model, 378 Gaussian copula, 873–874 General n-period trees, European option pricing in, 271 Geometric average, 478–479 Geometric Brownian motion, 309–311, 346–350, 359, 380, 530 drift and variance and, 347 Ito processes and Ito’s lemma and, 347–349 Merton model and, 790 stochastic volatility models and, 375 Wiener processes and, 346–347 GJR GARCH model, 378 Global financial crisis, 61 Globalization, 61 GLOBEX electronic platform, 22 GNMA CDR futures contract, 41–44, 134 Goldman Sachs Commodity Index, 646 Gold market, 90–91 Gold prices, 60 Good-for-day (GFD) limit orders, 25 Good-till-canceled (GTC) limit orders, 25 Government National Mortgage Association (GNMA), 41–44 www.downloadslide.net Subject Index Grain Futures Act, 60 Gram-Schmidt decomposition, 559–560 Greeks See Option greeks Group of Thirty, 495 Guarantee fund, 30 Gumbel copula, 876 H Heath-Jarrow-Morton (HJM) model, 529, 704, 714–732 defaultable, 830–832 one-factor, 716–724 advantage over spot-rate models, 724 evolution of forward curve and, 717–719 extending to multiple periods, 721 lower subtree and, 722–723 numerical example of, 717 risk-neutral risk identification and, 719–721 tree recombination and, 723–724 two-factor models vs., 728–729 upper subtree and, 721–722 risk-neutral drifts and, 729–732 risk-neutral drifts and volatilities in, 748–751 two-factor, 725–729 one-factor models vs., 728–729 Hedge portfolios riskless, 240 option pricing and, 255–256 Hedge ratio, 106 minimum-variance, 109–114 Hedgers, 24 Hedging, 3, 6, 104–125 alternative of not, 111 Asian options, 481 barrier options, 475–476 basis risk and, 105–106, 108 cash flow from hedged position, 107–108 cash-or-nothing options, 443–444 choosers, 449 cliquets, 486–487 compound options, 454 cross-currency borrowing, 635–637 cross-hedging, 105 with currencies, 112–113 with equities, 113–114 defined, 3, 11–12 derivation of the optimal tailed hedge ratio h, 124–125 derivatives in, 12–13 duration-based See Duration-based hedging with forward contracts, 3, with forward-rate agreements, 132–133 forward starts, 442 hedge ratio and, 106, 109–112 implementing, 114–117 index futures and changing equity risk and, 117–118 interest-rate risk using eurodollar futures, 135–138 lookback options, 483–484 margining and, 40 minimized cash-flow variance and, 110–111 I-15 minimum-variance, 105–106 implementation of, 114–117 minimum-variance hedge ratio and, 108–114 with multiple futures contracts, 115–116 multiple risks simultaneously, 116 one-for-one, 111–112 options, using delta, 247 PVBP-based, using eurodollar futures, 151 risk decomposition and, 507–508 “stack-and-roll” strategy, 45 swaps, 598–600 tailing the hedge, 116–117 Help function, in Octave, 48–9 Histograms, of joint default frequencies, 857–858 Historical simulation, 500–501 Historical volatility, 292 estimation of, 307–308 HJM model See Heath-Jarrow-Morton (HJM) model Holder, of option, Holding costs, 64 Ho-Lee model, 693, 698 Horizontal spreads, 177 payoffs from, 183–184, 185 reasons to use, 184 using calls, 183–184 using puts, 184–185 Hull-White model, for term-structure modeling, 694 I ICE Futures Canada, 22 ICE Futures Europe, 21, 30 ICE Futures US, 22 ICE swaps, 48–49 Immediate-or-cancel (IOC) limit orders, 25 Implicit finite differencing, 13–17 Implied binomial trees, 360, 381–391 calibrating first period, 385–386 construction of, 384–391 notation and preliminaries, 382–384 second period, 386–388 state prices and, 386 third period, 388–389 Implied copulas, 877–879 Implied repo rate, 88, 92–95 arbitrage and, 94–95 as synthetic borrowing/lending rate, 93–94 Implied volatility, 184, 292, 311 Black-Scholes model and, 329–334 skew under jump-diffusions, 367 Incremental risk approach, 503–504 Independently and identically distributed (i.i.d.) returns, 293–294 Index arbitrage, 96–100 Index futures, changing equity risk and, 117–118 Index rolls, 778 Index tracking, with equity swaps, 615 Index tranches, 779–781 Initial margin, 37 Innovation, www.downloadslide.net I-16 Subject Index Insurance options as, 3–4, 207–208 portfolio, 249–250 Insurance value of call options, 217 depth-in-the-money and, 219 of options, 218–219, 422 put-call parity and, 227–228 of put options, 218 Intensity processes Litterman-Iben model and, 827 reduced-form models of default risk and, 817–821 constant intensity processes and, 817–819 limitations of constant intensities and, 818–819 mathematical context of, 817–818 non-constant intensity processes and, 819–820 spread curves with non-constant intensities and, 819–820 stochastic intensities and, 820–821 Inter-arrival time, 818 InterContinental Exchange (ICE), 21, 48–49 Interest compounding frequency and, 82–83 converting between frequencies and, 83–84 present values under different, 83 continuous compounding and Black-Scholes model and, 310 currency forward prices under, 70–71 forward pricing with, 65–66 risk-free rate of, 310 Interest-rate convention, 65 Interest-rate derivatives, 2, 3, Interest-rate forwards and futures, 6, 22, 126–154 duration-based hedging and, 144–147 eurodollar futures, 133–140 eurodollars and libor rates and, 126–127 forward-rate agreements and, 127–133 Treasury bill futures, 144 Treasury bond futures, 140–143 Treasury note futures, 144 Interest-rate models, vs equity models, 688–689 Interest-rate options, 158, 172 Interest-rate risk, 1, 541 hedging, using eurodollar futures, 135–138 Interest rate(s), 233 Black-Scholes model and, 310 call values and, 424–425 compounding conventions and, 660 compounding frequency and, 82–84 constant, forward and futures prices with, 84–86 convertible bonds and, 522–523, 539 delta and, 321 estimating historical volatility for, 32–33 processes, 30–32 put values and, 424–425 reduced-form models of default risk and spread curves with non-constant intensities and, 822 sensitivity of portfolio value to, 153–154 term structure of, 655–666 yield-to-maturity and, 653–655, 660–661 construction of yield-to-maturity curve and, 661–665 discount functions and, 656–657 forward rates and, 658–661 zero-coupon rates, 660–661, 657–658 Interest-rate swaps, 573–574 cross-currency See Currency swaps equity swaps and, 615 fixed-for-floating See Fixed-for-floating swaps zero-coupon rates and, 10 Intermediaries, 24–25 International Petroleum Exchange, 21 International Swaps and Derivatives Association (ISDA), 1, 582, 755, 764 In-the-money options, 160, 162 Intrinsic value of call options, 217 depth-in-the-money and, 219 of put options, 218 Introducing brokers (IBs), 24 Inverted market, 91 Ito processes, 347–348 Ito’s lemma, 347–349 heuristic motivation of, 348–349 ITraxx indices, 759, 767, 777–781, 851 J Japanese government bond (JGB) futures, 141, 142 J Aron, 477–478 Jarrow-Lando-Turnbill (JLT) model, 832–840 adjustment factors and, 839–840 identifying Q = Q(0, 1) and, 834–835 identifying Q(0, t) for t > 1, 835–838 notation and, 834 Jarrow-Rudd (JR) solution, 297, 298–299 J.P Morgan, 495 JP Morgan Chase, 54 Jump-diffusion models, 359, 360–369 bias from ignoring jumps and, 363–364 binomial trees with jump-to-default and, 361–362 calibration of, and empirical performance, 368–369 comparison of Black-Scholes prices and, 369 depicting jumps in binomial models and, 361 implied volatility skew and, 367 implied volatility smiles and, 368 Merton option pricing formula and, 366–367 moment implications of, 365–366 multiperiod example, 362–363 one-period example, 362–363 Poisson distribution and, 364 pricing bias from ignoring jumps and, 367–368 program code for, 395–396 returns specification and, 365 Jump risk, gamma as view on, 414 Jump-to-default binomial model, 529–530 www.downloadslide.net Subject Index K Kansas City Board of Trade, 59 Knock-in options, 467–469 Knock-out options, 467–469 KOSPI 200, 97, 170 Kurtosis, 331–333, 374–375 L LCH Clearnet, 30 Leptokurtosis, 331, 332 Leverage expected returns from options and, 239 futures contracts and, 54 LTCM and, 596 margin sizes and, 38–39 Merton model and, 797–798 Levy processes, 380 Libor (London Interbank Offered Rate), 128, 581, 582, 583 eurodollar futures and, 135 with fixed notional principal, 617–618 forward Libor rate, 607–608 spread to, 588 with variable notional principal, 619–620 Libor Market Models (LMMs), 732–741 calibration and, 740–741 martingales and, 733–734, 737–740 risk-neutral pricing in, 736–740 simulation of, 740 Libor rates, 126–127 notation for, 734–736 Limit orders, 25–26 Lincoln Rule, 61 Linear homogeneity, 504, 508 Linear payoffs, 166 Linear splines, 674 Liquidity, 55 Liquid Yield Option Notes (LYONs), 519 Litterman-Iben model, 820, 824–827 evolution of spreads and, 823, 825–827 forward probabilities of default and, 823, 824–825, 826 intensity processes and, 827 LMMs See Libor Market Models (LMMs) Ln function, 317–318 Local volatility models, 381–391 Lognormal density function, 292 Lognormal distribution, 264, 290–299 actual and risk-neutral distributions, 295 binomial approximations of, 295–299 deviations from, 331–333 estimating historical volatility and, 307–308 as model of bond returns, 294 working with, 294 Log-returns, 291 deviations from normality, 331–333 i.i.d returns, 293–294 simple returns Sr /S0 and, 292–293 volatility and, 291–292 Log-stable models, 380–381 I-17 London Clearing House, 30 London Interbank Offered Rate See Libor (London Interbank Offered Rate) London International Financial Futures and Options Exchange (LIFFE), 21 London Stock Exchange, 97 Long gilt futures, 142 Long horizontal call spreads, 183, 184 Long positions, 5, 35, 158 call positions, 164, 166, 800 payoffs from, 160–162 futures position, 110 put positions, 164 payoffs from, 162–163 synthetic, 757 Longstaff and Schwartz approach, 38–40 Longstaff-Rajan model, 880–881 Long-Term Capital Management (LTCM), 595–597 Long-term forwards, 46 Lookback options, 482–485 fixed-strike, 482, 484–485 floating-strike, 482, 484 pricing and hedging, 483–484 pricing formulae for, 484–485 reasons to use, 482–483 LOR Associates, 250 M Maintenance margin, 37 Mandatory convertibles, 520–521 Margin accounts, 30 Margin calls, 37 Margins clearinghouse, 40 default risk and, 38 hedging and, 40 initial, 37 maintenance, 37 procedure for, 37–38 requirements, 37–40 sizes and leverage, 38–39 SPAN system, 39–40 valuation and, 40 variation, 37 Market direction, options as views on, 164–166 Market expectations, 96 Market-if-touched (MIT) orders, 26 Market Models, 732 Libor See Libor Market Models (LMMs) Swap, 741–743 Market-on-close (MOC) orders, 27 Market-on-open (MOO) orders, 27 Market orders, 25 Market price of risk, 691 Market risk, 767 Marking-to-market, 37, 38, 65, 138 Martingale measures, 260 Martingale probabilities, 242 See also Risk-neutral probabilities Martingales, 733–734, 737–740 www.downloadslide.net I-18 Subject Index Mathematica, 23 Matlab, 23 Matrix inversion, bootstrapping by, 684–685 Maturity date, 6, 158 Maximum limits, 28 Merchant Association of St Louis, 59 Merton/Ho-Lee model, 699, 706 Merton model, 790–799 actual default probability and, 793–794 complex capital structures and, 801–803 extensions of, 806–807 implementation issues with, 799–803 leverage and, 797–798 risk-free rate and, 798 risk-neutral probability of default and, 793 risk-neutral recovery rates and, 794–795 risky debt as option and, 791–792 for term-structure modeling, 693 term structure of credit spreads and, 795–797 unobservability of the Vt process and, 799–801 valuing risky debt and, 792–793 volatility changes and, 798–799 Merton option pricing formula, 366–367 Metallgesellschaft AG, 1, 38, 44–47 Microsoft, 171, 189 Milwaukee Chamber of Commerce, 59 Mini futures, 32 Minimized cash-flow variance, 110–111 Minimum-variance hedge ratio, 108, 109–112 examples, 112–114 Minimum-variance hedges, 105–106 implementation of, 114–117 MIPS (Monthly Income Preferred Securities), 519–520 Mispricing, 99–100 MKMV model, 803, 804–806 Model dependence, 232 Modeling term structure See Term-structure modeling Mod-Mod-R, 766 Mod-R, 766 Monotonicity, VaR and, 508–509 Monte Carlo simulation, 23–44, 501 for American options, 38–42 Longstaff and Schwartz approach for, 40–42 polynomial boundary technique and, 38–40 ARCH models and, 29–30 bivariate random variables and, 25 Cholesky decomposition and, 25–27 estimating historical volatility and for equities, 32 for interest rates, 32–33 interest-rate processes and, 30–32 introduction to, 23 path-dependent options and, 33–35 simulating normal random variables and, 24–25 stochastic processes for equity prices and, 27–29 variance reduction and, 35–38 antithetic variate method and, 35–37 control variate techniques and, 37–38 Moody’s KMV vendor model, 803, 804–806 Multifactor models of term structure, 704–705 Multiname credit derivatives, 758–759 Multinational corporations (MNCs), real options and, 562 Multiperiod correlated default, 862–865 N N(·) function, 317–318 Naked option positions, 160–163 Nasdaq 100, 97 NASDAQ 100 volatility index (VXN), 335 National Stock Exchange (NSE), 22 Natural gas derivatives, 48–49, 50–51 Natural gas market, 48 Natural log function (ln), 317–318 NDP assets See Non-dividend paying (NDP) assets Nelson-Siegel formulation, 663, 665, 678–679 Nelson-Siegel-Svensson formulation, 663 yield curve and, 678–680 Net-of-dividends stock price, 278, 280 New York Board of Trade, 22 New York Butter and Cheese Exchange, 59 New York Cotton Exchange, 59 New York Gold Exchange, 59 New York Mercantile Exchange (NYMEX), 21, 59 New York Produce Exchange, 59 New York Stock Exchange (NYSE), 21 Nikkei 225, 97, 170 Nikkei index, 194 No-arbitrage assumption, 63, 94 No-arbitrage models vs equilibrium models, 692–693 for term-structure modeling, 691–693 No-arbitrage restrictions, on option prices, 199–215 No-dividends assumption, 93 Non-constant intensities, 819–820 Non-dividend paying (NDP) assets, 201 American options on, 225–226 calls, 220 puts, 222, 272–274 European options on, 223–224 Non-inversion note, 442 Non-normal models, 360 Normal distribution, deviations from, 331–333 Normal market, 91 Numeraires, 741–742 NYMEX futures, 48–49 NYMEX Holdings, 21 NYSE Euronext, 21, 141 NYSE Liffe, 170 O Octave, 23, 45–57 commands in, 45–47 binomial trees, 47 factorials, 46–47 “for” loops and, 46 “while” loops and, 46 equation solving and, 55 finding and, 53–54 www.downloadslide.net Subject Index reading in data and, 50–52 regression and integration and, 48–50 help function and, 48–9 screenshots, 55–57 sorting and, 52–53 Official close, 27 Off-market swaps, 586, 587 Offsetting positions, 35–36 Oil futures prices, 91–92 Oil prices, 45–46, 47 Old-R, 765–766 Omicron, convertible, 539–540 One-cancels-the-other (OCO) orders, 26–27 One-factor models of term structure, 698–704 CIR model, 699–700, 701–702 Merton/Ho-Lee model, 699 solving PDE approach, 700–703 risk-neutral approach, 703–704 Vasicek model, 699, 701–702 Open and shut options, 552–553 Opening prices, 27 Opening range, 27 Optimal tailed hedge ratio h, 124–125 Optionality, 569 Option delta, 231, 245–249, 402, 405–409 behavior in r , 341 behavior in σ , 340 behavior in T − t, 341 binomial pricing and, 262 Black-Scholes model and, 320–321, 340–341 cash-or-nothing options and, 443–444 chooser, 450 compound options and, 454–455 computing, 405 curvature and, 408–409 defined, 245 gamma as predictor of changes in, 414–415 as probability, 248 properties of, 245–246, 405–406 puts, 269–270 straddle, 450 uses of, 246–249 using, 407 ways of defining, 248 Option gamma, 403, 409–415 cash-or-nothing options and, 445–447 compound options and, 454–455 computing, 409–410 curvature and delta-hedging and, 412–414 as curvature correction, 411–412 gamma-theta trade-off, 420, 421 as indicator of hedge rebalancing, 415 as predictor of changes in delta, 414–415 properties of, 410–411 using, 411–415 as view on jump risk/volatility, 414 Option greeks, 401–436 binary, 443 cash-or-nothing options and, 444–447 I-19 deriving Black-Scholes and, 433–436 interpreting, 401–404 need for, 403–404 option delta See Option delta option gamma See Option gamma option rho See Option rho option theta See Option theta option vega See Option vega portfolio greeks and, 426–429 Option premium, 159 Option pricing, 10, 159, 199–215, 231–260 arbitrage-free, 315 Asian options arithmetic-average price options, 479–480 binomial example, 480 on geometric average, 478–479 asset-or-nothing options, 444 barrier options, 469–471, 472–475, 493–494 binomial, 261–289 binomial model and See Binomial option pricing Black-Derman-Toy model and, 697–698 in Black-Scholes setting, 311–315 See also Black-Scholes model call options bounds on, 202–204 decomposition of, 217 by replication, in a one-period binomial model, 234–236 strike price and, 208–209 time to maturity and, 209–210 CAPM approach to, 238–240 cash-or-nothing options, 443 choosers, 449 cliquets, 486 compared to forward pricing, 231–232 compound options and, 452–454 contract parameters and, 208–210 cum-dividend prices, 282 decomposition of, 216–219 depth-in-the-money and, 219 early exercise and, 219–223 equity, finite differencing and, 7–13 European options general representation of, 287–289 two-period, 264–271 examples, 199–200 exchange options, 455–456 exotic option, 439 forward start options, 440–441 GARCH models and, 378–379 general procedure for, 279–281 homogeneity of degree and, 440 implied binomial trees and, 384 insurance value of option and, 207–208 lookback options, 483–485 maximum and minimum prices, 202–207 Merton option pricing formula and, 366–367 no-arbitrage restrictions and, 199–215 notation for, 201–202 numerical examples, 211–213 www.downloadslide.net I-20 Subject Index Option pricing (Cont ) option type and, 201 plotting option prices and, 316–317 put-call parity and, 223–228 put options bounds on, 205–207 decomposition of, 217–218 by replication, in a one-period binomial model, 236–238 strike price and, 209 time to maturity and, 210 quanto options, 457–458 by replication, in a one-period binomial model, 234–238 riskless hedge portfolios and, 255–256 shout options, 487–489 under stochastic volatility, 375–376 in stochastic volatility model, 371–372 using risk-neutral probabilities, 240–244 via replication, 234–241 volatility and, 199 Option rho, 402, 423–426 cash-or-nothing options and, 446–447 properties of, 424–425 using, 425–426 Option(s), 3, 5, 9–10 See also Forward contracts; Futures contracts; Swap(s) American See American-style options barrier See Barrier options Bermuda, 158 call, 3, See also Call option(s) cash-or-nothing See Cash-or-nothing options chooser (as-you-like-it or U-Choose), 447–450 compound See Compound options credit spread, 758, 763 on currencies, 170, 326–327 defined, 3, 157–158 directional views and, 164 early-exercise premiums on, 208 embedded, 172 on equities, 170 European See European-style options exchange, 455–456 exchange-traded, 169, 170–171 exotic, 172, 437, 439 See also Path-dependent options; Path-independent options expected returns from, and leverage, 239 as financial insurance, 158–159 vs forwards/futures/spot, 166 on futures, 170–171, 327–329 greeks on See Option greeks holder of, importance of replicability and, 239 on indices, 326 as insurance, 3–4, 207–208, 218–219 maximum-of-three-assets, 461 maximum-of-two-assets, 459 maximum-of-two-assets-and-cash, 459–460 on maximum or minimum, 460–461 minimum-of-two-assets, 459 naked option positions and, 160–163 over-the-counter, 169–170, 171–172 path-independent See Path-independent options payoffs from, 160–163 plain vanilla, 172 premium on, put, 3, See also Put options rainbow, 437 real See Real options strike price of, 158 terminology, 157–158 time decay and, 184 values, as maturity approaches, 320 as views on market direction and volatility, 164–166 volatility and, 160, 164–166, 199, 232 volume of trading, 23 Options markets, 157–172 embedded options and, 172 exchange-traded options and, 170–171 over-the-counter options and, 171–172 size and composition of, 169–170 Options modeling, 359–400 ARCH/GARCH models, 360, 376–380 Black-Scholes model See Black-Scholes model GARCH models, 376–380 implied binomial trees/local volatility models, 360, 381–391 jump-diffusion models, 359, 360–369, 395–396 log-stable models, 380–381 non-normal models, 360 other approaches, 380–381 stochastic volatility models, 359–360, 370–376, 396–398 variance-gamma models, 381 Options trading strategies, 173–198 Barings case and, 192–195 box spreads, 190 butterfly spreads, 181–183 asymmetric, 198 collars, 188–190 combinations, 185–188 condors, 191 covered calls, 173–175 protective puts, 175–177 ratio spreads, 190–191 spreads, 177–185 straddles, 185–186 strangles, 186–187 straps, 188 strips, 187 widely used, 191–192 Option theta, 402, 415–420 cash-or-nothing options and, 446–447 decomposing, 417–419 depth-in-the-money and, 417 gamma-theta trade-off, 420, 421 sign of, 416–417 using, 419–420 Option vega, 402, 420–423 call vega, 422 www.downloadslide.net Subject Index cash-or-nothing options and, 445–446 properties of, 421–423 using, 423 Orders limit, 25–26 market, 25 market-if-touched, 26 market-on-close, 27 market-onopen, 27 one-cancels-the-other, 26–27 spread, 26 stop, 26 stop-limit, 26 types of, 25–27 Ornstein-Uhlenbeck (O-U) process, 375, 530 Osaka Exchange (OSE), 192 Out-of-the-money options, 160, 162 Over-the-counter (OTC) derivatives, 1, gross market value, growth of, 61 notional outstanding, 2, Over-the-counter (OTC) options, 169–172 P Parity, 517 See also Put-call parity Par spreads, 784 Partial differential equation (pde), 700–701 Path-dependent options, 437–438, 467–494, 548 Asian, 476–482 arithmetic and geometric averages and, 478 exposure to average and, 476–477 hedging, 481 pricing, 478–480 put-call parity for, 478 smoothing the data and, 477 for speculation, 477–478 vs vanilla puts, 481–482 barrier, 467–476 hedging, 475–476 knock-out and knock-in, 467–469 pricing, 469–471, 472–475, 493–494 reasons to use, 469 vanilla, 469 volatility and, 471–472 cliquets, 485–487 hedging, 486–487 pricing, 486 reverse, 487 lookback, 482–485 fixed-strike, 482, 484–485 floating-strike, 482, 484 pricing and hedging, 483–484 pricing formulae for, 484–485 reasons to use, 482–483 Monte Carlo simulation and, 33–35 shout, 487–489 Path-independent options, 437–466 asset-or-nothing, 442, 443 greeks and, 443 payoffs, 443 I-21 pricing, 444 binary/digital, 442–447 cash-or-nothing, 442, 443 greeks and, 443, 444–447 hedging, 443–444 payoffs, 443 pricing, 443 chooser (as-you-like-it or U-Choose), 447–450 hedging, 449 pricing, 449 straddles and, 448–449 compound, 450–455 delta and gamma and, 454–455 hedging, 454 as installment options, 453 payoffs from, 451–452 pricing, 452–453 pricing formulae for, 453–454 exchange, 455–456 hedging, 456 pricing, 455–456 variants on, 458–461 forward start options, 439–442 framework for, 438–439 gap, 442–443 introduction to, 437–439 quanto, 456–458 pricing, 457–458 Payer swaption, 604 Payoffs See also specific instruments from compound options, 450–451 from forward contracts, 6–7 from forward-rate agreements, 128–129 from long and short positions on call options, 160–162 Performance bonds, 37 Perpetual options, 158 Physical settlement, 764–765 Plain vanilla derivatives, Plain vanilla options, 172 Platykurtosis, 331 Poisson distribution, 364 Poisson processes, 380 Polynomial splines, 674–677 estimating parameters by OLS and, 677 parameters to be estimated, 675 reduced-form representation of the discount function and, 675–677 Portfolio greeks, 426–429 Portfolio insurance, 231, 249–250 Portfolio optimization, risk decomposition and, 508 Position delta, 426–427 Position gamma, 427–428 Position limits, 29–30 Position rho, 429 Position theta, 428–429 Position vega, 429 Premium on convertible bond, 517 option, www.downloadslide.net I-22 Subject Index Present values, 151 of call options, 234 under different compounding frequencies, 83 Price limits, 28 Prices See also specific instruments ask, 95 bid, 95 closing, 27 conversion, 516 delivery, forward, opening, 27 settlement, 27 stock, 280, 383 strike, 158 Price ticks, 28 Probability, 236 option delta as, 248 Procter & Gamble (P&G) case study, 591–594 Protection, 756, 757 Protective put portfolio (PPP), 175–177 Protective puts, 175–177 Pseudo-probabilities, 242 See also Risk-neutral probabilities Put bear spreads, 180 Put bull spreads, 178–179 Put butterfly spreads, 198 Put-call parity, 223–228, 604 American options and, 225–226 Asian options and, 478 Black-Scholes formulae and, 319 with continuous dividend yield, 228 European options and on dividend-paying assets, 224–225 on non-dividend-paying assets, 223–224 insurance value and, 227–228 uses of, 224 Put options, 3, 4, 9, 158 American-style on dividend-paying assets, 222–223, 277 early exercise of, 221–223 on non-dividend-paying assets, 222, 272–274 arbitrage from overvalued, 238 arbitrage from undervalued, 237–238 for bearish vertical spreads, 180 for bullish vertical spreads, 178–179 for butterfly spreads, 182–183 delta of, 246, 269–270 depth-in-the-money and, 319–320 European, 268–270 for horizontal spreads, 184–185 as insurance for sellers, 159 insurance value of, 218 interest rates and, 424–425 intrinsic value of, 218 payoffs from long and short positions, 162–163 payouts of, 218 premium on, prices, 201 bounds on, 205–207 decomposition of, 217–218 impact of maturity on, 270 by replication, in a one-period binomial model, 236–238 strike price and, 209 time to maturity and, 210 protective puts, 175–177 time value of, 218 vanilla, vs Asian options, 481–482 writing, 164 Puttable bonds, 172 PVBP (present value of a basis point) analysis, 138 PVBP-based hedging, using eurodollar futures, 151 Q Quadratic splines, 674 Quadrinomial stochastic volatility tree, 372 Quality options, 142 Quanto options, 456–458 pricing, 457–458 R R, 23 Rainbow options, 437 Rainbow swaps, 614 Random variables bivariate, 25 normal, simulating, 24–25 Ratchets, 485–487 Ratings migration matrix, 832 Ratings transition matrix, 832–834 Ratio spreads, 190–191 Real options, 547–566 applications of, 562–563 case study of, 553–559 introduction to, 547–548 open and shut, 552–553 quantity risk and, 549–550 state-spaced approach and, 559–562 with stochastic cash flows, 549 types of, 548 “waiting-to-invest” setting and, 550–552 Rebates, 468 Receiver swaptions, 604 Recombination, two-period binomial tree and, 263–264 Recovery of Face Value (RFV), 823 Recovery of Market Value (RMV), 823 Recovery of Par (RP), 823 Recovery of Treasury (RT), 823 Recovery rate conventions, 822–823 Recursion, 299–300 implementation of, 300–301 problem with, 301 Reduced-form models, 790 of correlated default, 861–862 of default risk, 816–849 Duffie-Singleton result and, 828–830, 846–847 intensity processes and, 817–821 introduction to, 816–817 www.downloadslide.net Subject Index JLT model, 832–840 Litterman-Iben model, 824–827 pricing CDS and, 840–842 Regulations, futures, 59–61 Regulatory restrictions, equity swaps and, 616 Relative value, 595–596 Replicability, importance of, 239 Replicating portfolios, 64 Replication, 63, 231 Black-Scholes formula and, 312–313, 314, 350–353 dynamic, 240, 267–268 option pricing via, 234–241, 267–268 pricing by, in a one-period binomial model, 234–238, 262 pricing forwards by, 64–66 principle of, 64 Reset convertibles, 518 Return, 495, 502 Reversal, valuation by, 73–74 Reverse cash-and-carry arbitrage, 65 Reverse cliquets, 487 Rho Black-Scholes, 436 convertible, 533, 537–538, 539 option See Option rho position, 429 Risk, 495 See also Value-at-Risk (VaR) basis, 12, 45, 105–106, 107, 108 call, 541 credit See Credit risk default See Default risk downside, 498 equity, 1, 117–118 interest-rate, 541 jump, 414 market, 767 market price of, 691 Risk-adjusted probabilities, 242 See also Risk-neutral probabilities Risk aggregation, using option delta, 247 Risk budgeting, 495, 497 Risk-contributions calculating, 505–506 computing, 507 in delta-normal method, 506–507 Risk decomposition, 502–508 challenge of, 502–503 delta-normal method and, 506–507 Euler’s theorem and, 504–505 hedging and, 507–508 incremental risk approach and, 503–504 linear homogeneity and, 504 portfolio optimization and, 508 uses of, 507–508 Risk-free rate Litterman-Iben model and, 823–824 Merton model and, 798 Riskless hedge portfolios, 240 option pricing and, 255–256 Risk management, using derivatives, I-23 Risk measures coherent, 508–511 linear homogeneity and, 504, 508 VaR as, 496–497 RiskMetrics, 807 Risk-neutral distribution, 295 Risk-neutral pricing, 231, 240–244 binomial option pricing and, 262 Black-Scholes formula via, 353–356 examples, 242–243 historical underpinnings of, 241 steps in, 241–242 terminology, 242 workings of, 242–243 Risk-neutral probabilities, 257–258, 262 arbitrage and nonexistence of, 258–259 Arrow security prices and, 256–257 binomial approximations of lognormal and, 297–298 Black-Scholes via, 313–315 cash dividends and, 279 of default, 793 equivalent martingale measures and, 260 European option pricing and, 270–271, 287–289 implied binomial trees and, 383 JLT model and, 834 Litterman-Iben model and, 824–825 other uses of, 243 pricing using, 240–244 uniqueness of, completeness and, 259–260 Risk-neutral recovery rates, Merton model and, 794–795 Risk-neutral valuation, 282–283 Risky debt as option, in Merton model, 791–792 valuing, 792–793 Roller-coaster swaps, 590 Rollover cash flows, 62 Russell 2000 volatility index (RVX), 335 S S&P 100 index, 170 S&P 500, 97, 117, 170, 334, 335, 442 Sao Paofo Stock Exchange (Bovespa), 22 SARON, 582 Second-to-default (STD) basket, 775, 854 Securities and Exchange Commission (SEC), 60, 61 Securities-based swaps, 61 Self-exciting default models, 881–882 Self-financing, 268 Sellers, puts as insurance for, 159 Sensitivity analysis, 401–436 Black-Scholes model and, 404 interpreting the greeks and, 401–404 introduction to, 401 need for greeks and, 403–404 option delta and, 402, 405–409 option gamma and, 403, 409–415 option rho and, 402, 423–426 option theta and, 402, 415–420 www.downloadslide.net I-24 Subject Index Sensitivity analysis, (Cont ) option vega and, 402, 420–423 portfolio greeks and, 426–429 Sensitivity measure, option delta as, 247–248 Settlement prices, 27 Settlement procedures, in futures market, 28–29 SGX, 97 Short forwards, 207 Short horizontal call spreads, 183 Short positions, 5, 35, 158 calls, 164 payoffs from, 160–162 futures, 110 puts, 164, 166 payoffs from, 162–163 synthetic, 757 Short straddles, 186 Short-term futures, 46 Shout options, 487–489 Silver Crisis (1980), 38, 59–60 Single-name credit derivatives, 758 Skewness, 331–333, 374–375 SONIA, 582 SOQ See Special Opening Quotation (SOQ) SPAN (Standard Portfolio Analysis of Risk), 37 SPAN system, 39–40 Special Opening Quotation (SOQ), 97 Special-purpose entities (SPEs), 519 Special-purpose vehicles (SPVs), 772 Speculation, defined, 12 with derivatives, 13–15 swaps and, 576–577 Speculators, 24 SPEs See Special-purpose entities (SPEs) Splines, 673–674 cubic, 674 exponential, 677–678, 685–687 implementation issues, 678 linear, 674 polynomial, 674–677 quadratic, 674 Spot curve, 657–658 Spot FX contracts, 633 Spot rate, 656 Spots, vs options, 166 Spot yield curve, 715 Spread orders, 26 Spread(s), 177–185 box, 190 butterfly, 181–183, 198 credit See Credit spreads defined, 177 horizontal (calendar), 177, 183–185 ratio, 190–191 vertical, 177–180, 191 volume traded, 191 Sprint repricing scheme, 441 SPVs See Special-purpose vehicles (SPVs) Squeezes, 34 “Stack-and-roll” strategy, 45 Standard and Poor Depository Receipts (SPDRs), 99 Standard Brownian motion, 346–347 Standard normal distribution N(·), 317–318 State prices, 256–257, 258 State space, 559–562 State variables, 689 Stochastic intensities, 820–821 Stochastic processes, 861–862 Stochastic volatility models, 359–360, 370–376 binomial-based, 370–371 calibration and empirical performance of, 376 continuous-time, 373–375 option pricing, 371–372 comparison of Black-Scholes model and, 372–373 under stochastic volatility, 375–376 program code for, 396–398 Stock indices defined, 96 dividend yield on, 97, 99 forwards pricing on, 72 futures contracts on, 96–97 options on, 326 pricing forwards on, 97–98 Stock prices implied binomial trees and, 383 net-of-dividends, 280 total, 280 Stock price tree, 278 Stop-buy orders, 26 Stop-limit orders, 26 Stop orders, 26 Stop-sell orders, 26 Straddles, 166, 185–186, 191 choosers and, 448–449 delta, 450 payoffs from, 186 short, 186 Strangles, 186–187, 191 Straps, 188 Strike prices, 158 call prices and, 208–209 put prices and, 209 Strips, 187 Strong backwardation, 91 Structural models of correlated default, based on asset values, 855–861 of default risk, 789–915 Delianedis-Geske model, 813–815 empirical performance of, 808–809 evaluation of approach, 808–810 extensions of Merton model, 806–807 Merton model, 790–803, 806–807 Moody’s KMV vendor model, 803–806 vs reduced-form models, 790 Subadditivity, 509–510, 511 Sumitomo, Surplus fund, 30 www.downloadslide.net Subject Index Swap curve, 583 Swap Market Models (SMMs), 733, 741–743 swaptions and, 743–744 Swap(s), 10–11, 61 “5/30,” 591–594 accreting, 590 amortizing, 590 asset, 770–771 basket default, 758, 775–776 changing fixed rates and, 587–588 commodity See Commodity swaps credit default See Credit default swaps (CDS) credit risk and credit exposure and, 597–598 cross-currency, 573 currency, 10–11 defined, 5, 569 equity See Equity swaps equity-for-equity, 627 fixed-for-floating, 573–574, 577, 646 fixed notional, 624–628 floating-for-floating, 647 forward-starting, 589–590 hedging, 598–600 ICE, 48–49 interest rate, 10, 573–574 Long-Term Capital Management case study of, 595–597 off-market, 586, 587 overnight index, 582–583 P&G case study of, 591–594 payoffs, 577–580, 592–593 as portfolio of bonds, 580–583 as portfolio of FRAs, 584–586 rainbow, 614 roller-coaster, 590 securities-based, 61 spread to Libor and, 588 total return See Total Return Swaps (TRSs) uses of, 11, 574–577 analyzing funding costs, 576 comparative advantage, 574–576 risk/maturity management, 576, 577 speculation and other uses, 576–577 valuing and pricing, 580–590 zero-coupon, 586–587 Swap spread, 583 Swaptions, 172, 604–605 Black model for pricing, 605–610 Swap Market Model and, 743–744 Swiss Options and Financial Futures Exchange (SOFFEX), 21, 22 Synthetic calls, 235 Synthetic long forwards, 207 Synthetic long positions, 757, 760–761 Synthetic short positions, 757, 760–761 Synthetic zero-coupon bonds, 190 T Target forwards, 14–15 Technology, futures markets, 22 Term structure, 54 of credit spreads, 795–797 Term-structure modeling, 688–696 arbitrage violations and, 689–690 equilibrium approach to, 691–693 factor models, 697–713 affine, 706–709 Black-Derman-Toy, 697–698 deriving fundamental PDE in, 712–713 Ho-Lee, 698 multifactor, 704–705 one-factor, 698–704 finite differencing for, 19–21 no-arbitrage approach to, 691, 692–693 Theta Black-Scholes, 435 convertible, 533, 536 option See Option theta position, 428–429 Ticks, 28 Time decay, 184, 219, 415, 416, 428–429 Time deposits (TDs), 135 Time to maturity behavior of option deltas and, 341 call prices and, 209–210 delta and, 321 put prices and, 210 theta and, 416–417 Time value of call options, 217 depth-in-the-money and, 219 of put options, 218 Tokyo Stock Exchange, 141, 170 TONAR, 582 “Too big to fail,” 61 Total Return Swaps (TRSs), 758, 759–763 as form of financing, 761 funding-cost arbitrage and, 761–762 as synthetic long/short positions, 760–761 Total stock price, 280 Trading platform, futures markets, 22 Trading strategies See Options trading strategies Traditional cash-pay convertibles, 519 Tranche correlation, 781 Transaction costs, 88, 95–96, 99 Translation invariance, 509 Treasury bill futures, 22, 134, 144 “tailing” the hedge and, 141 Treasury bond futures, 22, 134, 140–143 calculating conversion factor, 152 pricing, 143 quality option, 142 specification of, 141–143 Ultra, 143 wild card option, 143 Treasury note futures, 141, 144 5-year, 144 specifications, 142 10-year, 144 2-year, 144 I-25 www.downloadslide.net I-26 Subject Index Trinomial model, 259–260 TSRs See Total Return Swaps (TRSs) Tulipmania, 157 Two-equity swaps, 627–628 Two-period binomial tree, 263–264 European options and, 264–271 recombination and, 263–264 U UBS, 195 U-Choose options See Chooser options (as-you-like-it or U-Choose) Ultra Treasury bond futures, 143 Unbiased expectations hypothesis, 96 Uncertainty, cash-flow, 13 Underlying assets, 2, 4, 5–8 Up-front cost, 13 Upsilon, convertible, 540 US regulations, on futures, 59–61 V Valero Energy Corporation, 520–521 Valuation See also specific instruments of existing FRA, 131–132 margining and, 40 of options, 160 by reversal, 73–74 risk-neutral, 282–283 Value-at-Risk (VaR), 495–515 assessment of, 501 calculating, 498 coherent risk measures and, 508–511 worst-case scenario analysis and expected shortfall and, 511 component, 505–506 concentrated tail risks and, 510–511 estimation of delta-normal method, 498–500 historical simulation, 500–501 Monte Carlo simulation, 501 failure of subadditivity and, 509–510 introduction to, 495 limitations of, 497–498 marginal, 505 risk decomposition and, 502–508 challenge of, 502–503 delta-normal method and, 506–507 Euler’s theorem and, 504–505 incremental risk approach and, 503–504 linear homogeneity and, 504 uses of, 507–508 as risk measure, 496–497 signs and, 496 Vanilla options, 469, 481–482 VaR See Value-at-Risk (VaR) Variance, 6, 105, 107, 347 Variance-gamma (VG) models, 381 Variance reduction, 35–38 antithetic variate method and, 35–37 control variate techniques and, 37–38 Variance swaps, 341–345 Variation margin, 37 Vasicek model, 699, 706 bond prices in, 701–703 for term-structure modeling, 694 Vega Black-Scholes, 435 convertible, 533, 536–537, 538 option See Option vega position, 429 Vertical spreads, 177–180, 191 bearish motivation for, 179 using calls, 179–180 using puts, 180 bullish motivation for, 177 using calls, 177–178 using puts, 178–179 VIX, 334–336 futures, 336 options, 336 Volatility, 55 annualized, 233 barrier options and, 471–472 in binomial model, 233 delta and, 321 estimating historical for equities, 32 for interest rates, 32–33 gamma as view on, 414 historical, 292 impact on call values, 422 implied, 184, 292, 311 Black-Scholes model and, 329–334 Merton model and, 798–799 notion of, 291–292 option pricing under stochastic, 375–376 options and, 160, 164–166, 199, 232 risk neutral drifts and, in HJM, 748–751 straddles and, 185–186 swaps, 341, 345 trading, 336 VIX and its derivatives and, 334–336 Volatility indexes, 334–336 Volatility skew, 329–334 implied, under jump-diffusions, 367 source of, 330–333 Volatility smiles, 329, 333, 368 implied, in stochastic volatility model, 374 Volatility swaps, Volcker Rule, 61 W Waiting-to-invest options, 550–552 Weak backwardation, 91 Wiener processes, 346–347 Wild card option, 143 Worst-case scenario analysis, 511 www.downloadslide.net Subject Index X Yield, 278 Yield curve, 656, 657–658, 660–661, 671–687 bootstrapping and, 671–673, 684–685 forward, 715 Nelson-Siegel-Svensson approach for, 678–680 pricing bonds using, 658 splines and, 673–674 exponential, 677–678, 685–687 implementation issues, 678 polynomial, 674–677 spot, 715 zero-coupon, 656, 657–658, 660–661 Yield-to-maturity, 653–655 construction of yield-to-maturity curve and, 661–665 forward rates and, 660–661 problems with, 654–655 I-27 raw data, 668–670 zero-coupon rate and, 660–661 Yield-to-maturity curve constructing, empirical illustration of, 661–665 zero-coupon rates and forward rates and, 660–661 Y Zero-cost collars, 189 Zero-coupon bonds, 154, 190 Zero-coupon convertibles, 519 Zero-coupon debt, 804 Zero-coupon rate, 656, 657–658, 660–661 Zero-coupon swaps, 586–587 Zero-coupon yield curve See Yield curve Zero-dividends, 310 Zero recovery, Duffie-Singleton result and, 829 Zero-sum 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Investments: Valuation and Management Seventh Edition Stewart, Piros, and Heisler Running Money: Professional Portfolio Management First Edition Sundaram and Das Derivatives: Principles and Practice Second.. .Derivatives: Principles and Practice The McGraw-Hill/Irwin Series in Finance, Insurance, and Real Estate Stephen A Ross Franco Modigliani Professor of Finance and Economics Sloan... FINANCIAL INSTITUTIONS AND MARKETS Rose and Hudgins Bank Management and Financial Services Ninth Edition Rose and Marquis Financial Institutions and Markets Eleventh Edition Saunders and Cornett Financial