Paul wilmott on quantitative finance vol 1 3, 2nd ed

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Paul wilmott on quantitative finance vol 1 3, 2nd ed

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Paul Wilmott On Quantitative Finance Paul Wilmott On Quantitative Finance Second Edition www.wilmott.com Copyright  2006 Paul Wilmott Published by John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wiley.com All Rights Reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to permreq@wiley.co.uk, or faxed to (+44) 1243 770620 Designations used by companies to distinguish their products are often claimed as trademarks All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners The Publisher is not associated with any product or vendor mentioned in this book This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the Publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Library of Congress Cataloging-in-Publication Data Wilmott, Paul Paul Wilmott on quantitative finance.—2nd ed p cm Includes bibliographical references and index ISBN 13 978-0-470-01870-5 (cloth/cd : alk paper) ISBN 10 0-470-01870-4 (cloth/cd : alk paper) Derivative securities—Mathematical models Options (Finance)— Mathematical models Options (Finance)—Prices—Mathematical models I Title HG6024.A3W555 2006 332.64 53—dc22 2005028317 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-470-01870-5 (HB) ISBN-10: 0-470-01870-4 (HB) Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production In memory of Detlev Vogel contents of volume one Visual Basic Code Prolog to the Second Edition xxv xxvii PART ONE MATHEMATICAL AND FINANCIAL FOUNDATIONS; BASIC THEORY OF DERIVATIVES; RISK AND RETURN 1 Products and Markets Derivatives 25 The Random Behavior of Assets 55 Elementary Stochastic Calculus 71 The Black–Scholes Model 91 Partial Differential Equations 101 The Black–Scholes Formulae and the ‘Greeks’ 109 Simple Generalizations of the Black–Scholes World 139 Early Exercise and American Options 151 10 Probability Density Functions and First-exit Times 169 11 Multi-asset Options 183 12 How to Delta Hedge 197 13 Fixed-income Products and Analysis: Yield, Duration and Convexity 225 14 Swaps 251 1366 index Longstaff & Schwartz model 2:586 for American options 3:1279–83 lookback-Asian options 2:464–7 lookback options 2:383, 425, 445–52, 464–7, 497–500 continuously-sampled maximum 2:445–7 dimensionality 2:448–9 discretely-sampled maximum 2:448–9 formulae 2:450–2 ladders 2:472–4 overview 2:445 payoff types 2:445, 450–2 similarity reductions 2:449 low-discrepancy sequences 3:1288–92 LTCM see Long- Term Capital Management LU decomposition, Crank-Nicolson method 3:1233–6 Macaulay duration 1:235–6 machine-abandonment example, real option theory 3:1152–3 maintenance margin, option writers 1:37 make-your-mind-up American options 1:163–5 marginal distributions 2:695–6 marginal values 3:760, 947 margins calls 2:725–7 CrashMetrics 2:725–7 hedging 1:136–7; 2:725–7 option writers 1:37 market frictions, dividends 3:1038 market makers 1:356; 3:1096 market movement strategy, transaction costs 3:802–4 market risk 2:675 markets backwardation concept 2:736–7 complete 1:270–1 contango concept 2:736 conversion premium ratio, CBs 2:556 conversion price, CBs 2:556 energy 3:1141–50 forecasting 1:343–58 liquidity 2: 669–72; 3:760, 992, 1126–8 microstructure modeling 1:356–7 overview 1:5–24 perfect market hypothesis 3:1035 portfolios 1:322 practice, barrier options 2:401–5 price of risk 2:512–13, 601, 602, 612; 3:758, 856–8 price of (volatility) risk 3:856–8 trading games 1:359–63 value models 3:962–5 view 3:954, 959–60 volatility 3:833–52 see also illiquid markets marking to market 1:201–2, 332 marking to model 1:332 Markov 1:75; 2:665, 693 HJM 2:611–13 overview 1:73 Markowitz model 1:319, 325, 327 marriage 3:1153, 1159 martingale 1:98 property 1:73, 75 variance reduction 3:1277 mathematics, requirements 1:7; 3:1317–27 matrices 2:665–7, 706 Cholesky factorization 3:1275–6 covariance 1:184 Crank-Nicolson method 3:1230–9 maturities bonds 1:225–6, 232, 242–8 forward contracts 1:21 maximization 1:153; 3:981–6 long-term growth 3:1057–8 utility 3:1010–11, 1019–23, 1030–2, 1052–3 maximum lookback options 2:445–9 lookback-Asian options 2:465–7 maximum likelihood estimation 3:820–4 MESs see mortgage-backed securities mean calculation, returns 1:58–65 mean square limit, overview 1:77–8 mean reversion 3:815–16 mean-reverting random walks 1:86–8 mean-variance analysis 3:758, 889–99 analysis 3:890–1 definitions 3:890–1 equations 3:891–2 interpretation 3:892–3 stochastic volatility 3:758, 829, 889–99 up-and-out call options 3:894–6 measurability, parameters/variables 3:869–71 Meriwether, J 2:740 Merton, RC 2:740, 744; 3:1051, 1059–60 Merton model 2:640–4 meshes, overview 3:1199–200 Metallgesellschaft 2:725, 735–7 Metropolis algorithm 3:986 MG Refining and Marketing (MGRM) 2:735–6 microstructure market modeling 1:356–7 migration, credit rating 2:663–7 index Milstein method 3:1264 model-dependent hedging 1:135 model-independent hedging 1:135 modeling approaches 3: 749–53, 1318–19 modern portfolio theory 1:319–22, 328–9; 3:954 modified duration 1:235–6 money market accounts 1:228 Monte Carlo simulation 1:167, 338; 2:469; 3:806, 1179, 1194–5, 1253, 1263–83 advantages 3:1267, 1278 American options 3:1278–83, 1314–15 antithetic variables 3:1277, 1292 basic integration 3:1286–8 Cholesky factorization 3:1275–6 cliquet option 2:501, 503 control variates 3:1277–8, 1293 convergence 3:1277–8, 1292 disadvantages 3:1278 Greeks 3:1274 HJM 2:613–14 Longstaff & Schwartz regression 3:1279–83 low-discrepancy sequences 3:1288–92 Martingale variance reduction 3:1277 overview 3:1263 pricing 2:387, 428–9; 3:1311–13 programs 3:1311–16 monthly payments, fixed rate mortgages 2:572 Moody’s 2:649, 663–4, 684 mortgage-backed securities (MBSs) 2:571–80 issuers 2:573–4 overview 2:571, 573 prepayments 2:572–3, 574–8 valuation 2:578–9 mortgages 2:571–80 prepayments 2:572–3, 574–8 types 2:571–3 mountain range options 2:479 moving averages 1:345, 347 cap/floor 2:551 exponentially-weighted 3:816 moving-window volatility 3:815 multi-asset options 1:183–95 crash modeling 2:715–23; 3:949–50 CrashMetrics 2:715–23 examples 1:191 hedging 1:191–3 pricing realities 1:194 problems 1:194 quantos 1:189–93 uncertain parameters 3:879 multi-dimensional lognormal random walks 1:183–6 multi-factor CIR model 2:591–3; 3:1307 multi-factor HJM 2:615 multi-factor interest rate modeling 2:581–93 general theory 2:591–3 overview 2:581 phase plane 2:587–90 popular models 2:584–7 theory 2:581–4 tractable affine models 2:591–3 yield curve swaps 2:590–1 multi-factor models 2:564–7, 581–93, 615 multi-factor Vasicek model 2:591–3 multi-index model CrashMetrics 2:723–4 portfolio management 1:327 multiple crashes, modeling 3:948–9 Musiela parameterization, HJM 2:614–15 negative option prices, transaction costs 3:798 Newton-Raphson method 1:131–3 Nikkei 2:718, 739–40 no arbitrage see arbitrage ‘no free lunch’ argument 1:91 nodes grids 3:1200 singularities 2:588 noise traders 1:356; 3:989, 991 non-anticipatory integration 1:76 non-attainable origins 1:90 non-infinitesimal short rates, HJM 2:620–1 non-normal returns, transaction costs 3:800 non-optimal trading 2:459 non-probabilistic model bond options 3:1103–8 crash modeling 3:1122–6 economic cycles 3:1118–19 embedded-decision contracts 3:1108–10 forward-rates 3:1117–18 hedging 3:1077–128 index amortizing rate swaps 3:1110–13 interest rates 3:1077–128 liquidity 3:1126–8 portfolios 3:1099–102 pricing 3:1082–116 real portfolios 3:1099–102 swaps 3:1108–13 non-single-signed gammas 3:790–1 non-linear equations 3:966 diffusion equations 3:966 interest rate modeling 3:1077–97 linear conversions 3:792–3 parabolic partial differential equations 3:787 1367 1368 index non-linear equations (continued ) pricing 3:973–6 solution existence 3:798–9 spreads 3:973–6 static hedging 3:969–87 uncertain parameters 3:870–80, 972 worst-case scenarios 3:1079–81 non-linear models benefits 3:759–60 Crash Metrics 2:709–29 crash modeling 3:939–51, 972 Epstein-Wilmott model 3:1079, 1101, 1103, 1115, 1117–18, 1305–7 Hoggard Whalley & Wilmott model 3:785–90, 798, 809–11 summarization 3:972–4 transaction costs 3:783–811 uncertain parameters 3:757, 869–80, 881–2, 972 Whalley & Wilmott & Henrotte model 3:793–4 non-linearity American options issue 1:165 model interpretations 3:797–9 Normal distribution 3:1133–4 accuracy 1:295–9 fat tails 1:297–8, 299 returns 1:60–2, 69 ‘normal event’ risks 3:939–40 notes 1:229 nth to default 2:683 numerical methods finite difference program code 2:1296–309 Monte Carlo simulation 1:167, 338; 2:469; 3:806, 1253, 1263–83 one-factor model finite-difference methods 3:1199–251, 1296–7 overview 3:1199–200 program code 3:1296–309 simulations 2:1263–83 two-factor model finite-difference methods 3:1253–62 see also Black-Scholes model offers, trading games 1:359–63 oil see energy one-day options, energy 3:1148 one-factor interest rate modeling 2:509–24, 603 bond pricing 2:510–12, 513–16 futures contracts 2:523–4 implied modeling 2:597–8 market price of risk 2:512–13, 587 named models 2:517–21, 530–1, 596–7, 616–17 overview 2:509 stochastic rates 2:521–2 one-factor models, finite-difference methods 3:1199–251, 1296–7 one-sided difference 3:1224–6 one-touch options 1:161–2; 3:971 one-way floaters 2:551 open interest, charting 1:356 open low-discrepancy sequences 3:1288 optimal close down, firm’s value 2:645 optimal exercise point 1:154–61, 178–9; 3:761, 1013–33 optimal rebalance point 3:796–7 optimal static hedging 3:871, 973 barrier options 3:894–7, 979–81 crash modeling 3:946–7, 1122–6 CrashMetrics 2:713–15 definition 3:976–7 interest rate modeling 3:1082–116 non-probabilistic model 3:1082–116 optimization problem 3:981–6 overview 3:759–60, 976–8 path dependent options 3:978–81 Platinum Hedging 1:137; 2:709, 713–15, 724, 727; 3:946–7 portfolios 3:894, 980–1 vanilla options 3:978–81 optimal stopping, confusions 3:1026 optimal trades 2:457–9 optimization problem 3:981–6, 1010–11, 1152–3 options 1:25–53 anteater options 2:437 on baskets 1:186–7 Black-Scholes model 1:91–9 calculation time 3:1276–7 cliquet 2:500–4 compounds and choosers 1:375–8 converts as 2:556–8 convexity 1:58 correlation 1:92 decision features 2:374; 3:1108–10 dividend dates 1:140–3; 3:1037–44 energy 3:1148–50 holders 3:1014–27 indices 1:28, 29 margins 1:37 overview 1:25–53 parameters 1:38 pre-expiry valuations 1:38 price factors 1:38–9 real option theory 3:1151–7 replication 1:94 , 270, 271 swing options 3:1150 index trading games 1:359–63 value 1:91–2 writers 1:37, 40, 156; 2:459; 3:1014–27 Orange County, California 2:731–3 order, definition 2:373 ordinal utility 3:1011 Ornstein-Uhlenbeck process 1:87–8; 3:861–2, 903 oscillators 1:345, 348 OTC see over the counter options out barrier options 2:371–2, 373, 381, 385–96, 402, 408–9 best/worst prices 3:873–7 pricing 2:388; 3:873–7 out of the money definition 1:31 strangles 1:46 volatility 1:129–30 outside barrier options 2:400 over the counter options (OTC) 1:38, 51–3, 332; 3:639, 676, 725, 726, 736, 737, 976, 1016, 1029 over-relaxation parameter, SOR method 3:1238 P&G see Proctor & Gamble par bonds 2:653–4, 658–61 par swap 1:254 parabolic partial differential equations, nonlinear models 3:787 parallel shifts, yield curves 1:241 parameters 1:91–2; 3:1238–9 Black-Scholes assumption 3:757, 869–80 crash effects 2:727 credit risks 2:642–3 fudgeability 3:835 Hoggard, Whalley & Wilmott model 3:790 jump diffusion 3:936–7, 939 measurability 3:869–71 RiskMetrics 2:702–5 time-dependent parameters 1:147–8 variables 1:38 see also uncertain parameters parameterization, local volatility surface 3:845 Parasian contracts 2:474 Parisian options 2:400 finite difference program code 3:1299–300 overview 2:474–8 Parkinson measure 3:819 partial barrier options 2:398–9 partial differential equations 1:94, 101–8; 2:388; 3:1016 cliquet option 2:501–3 history 1:101–2 non-linearity 3:973–6 overview 1:101 pricing 2:430 solutions 1:104–6 see also Black-Scholes model partnerships, firm’s value 2:645 passport options 2:453–60, 504, 505; 3:1029–33, 1175 decision features 2:374 program 3:1300–1 utility maximization 3:1030–2 path dependency 1:367–84; 2:385, 388, 400; 3:1155–7 CBs 2:568 cliquet option 2:500–4 combined quantities 2:465–7 constant volatility 3:917–18 definition 2:371 finite-difference methods 3:1249–50 hedging errors 3:770–2, 776–7 historical volatility 2:467–9 jump conditions 3:1249–50 lookback-Asian options 2:464–7 miscellaneous exotics 2:461–80 optimal static hedging 3:978–81 order 2:373–4 weak dependency 2:371–2 see also strong path dependency payer swaptions 2:541 payoff Asian options 2:427–43 barrier options 2:386–7 Black-Scholes model 1:110–21 bond options 2:534–6 credit rating changes 2:693–4 delta hedging 3:974–6 expected present value 3:954–5 finite-difference methods 3:1207 formulation 1:156–9 log contracts 1:149–50 lookback options 2:445–52 make-your-mind-up American options 1:163–5 optimal static hedging 3:976–8 overview 1:26, 32–6 path dependency 2:425 perpetual American puts 1:151–5 power options 1:149 programs 1:286 put-call parity 1:41–2 payoff diagrams bear spreads 1:44–5 binary calls 1:43–4 binary puts 1:43–4 1369 1370 index payoff diagrams (continued ) bull spreads 1:44–5 butterfly spreads 1:49 condors 1:49 overview 1:32, 33–6 risk reversal 1:47–9 straddles 1:46–7 strangles 1:46, 48 perfect market hypothesis 3:1035 perfect trader options 2:453–60 performance measurement portfolio management 1:329–30 VaR 1:339 periodic floors 2:738 perpetual American calls 1:155 perpetual American puts 1:151–5 perpetual American straddles 1:157–8 perpetual bonds 1:230 perpetual warrants 1:51 phase plane, multi-factor interest rate modeling 2:587–90 Pilopovi two-factor model 3:1146–8 Pindyck, RS 3:1159–60 plateauing, volatility 2:702–3 Platinum Hedge 1:137; 2:709, 713–15, 724, 727; 3:946–7 plotting 1:344 point and figure charts 1:353 Poisson processes dividends 3:1040 instantaneous risk of default 2:651–4, 669 jump diffusion 3:931, 936–7 jump drift 3:959–62 portfolio insurance 3:989, 990, 993–6 portfolio management 1:317–30 CAPM 1:99, 325–7 cointegration 1:328–9 diversification 1:318–19 efficient frontiers 1:321, 323–4 Markowitz model 1:319, 325, 327 modern theory 1:319–22, 328–9 multi-index model 1:327 overview 1:317 performance measurement 1:329–30 single-index model 1:325–7 uncorrelated assets 1:319 VaR 1:331–42 see also CrashMetrics; CreditMetrics; RiskMetrics portfolios barrier options 3:979–81 changes 1:91–2 growth-optimum portfolios 3:1057–8 hedging with implied volatility 1:212–14 non-probabilistic model 3:1099–102 Platinum Hedge 3:946–7 pricing 3:1170–3 singles distinction 3:787–8, 880, 974 static hedging 3:894, 974–6 static replication 3:969–71 stochastic volatility 3:855–6 theory 1:319–22 see also assets POs see principal only MBSs positive interest rates 2:514, 519 positive recovery, credit risks 2:657 power method, HJM 2:619–20 power options, formulae 1:149 predictions crises 1:357 markets 1:343–58 premium 1:31, 37 premium payback period, CBs 2:556 prepayments, mortgages 2:572–3, 574–8 present values 1:7 bonds 1:231 debt 2:643 equation 1:232 expected payoff 3:954–5 swaps 1:254 price elasticity of demand 3:992 price factors, options 1:38–40 price/yield relationship, bonds 1:233, 235 pricing 2:388–96 American options 3:1016–27 CAPM 1:99, 325–7 CBs 2:559–69; 3:1114–16 credit derivatives 2:689 credit risks 2:667–8 delta hedging and 3:1027 discrete hedging equation 3:772–3 dividend effects 3:1037–40 energy derivatives 3:1141–50 HJM 2:613 incorrect 1:269–70 index amortizing rate swaps 3:1110–13 inflation-linked products 3:113, 1131–3 market 1:267 mean-variance analysis 3:890–1 Monte Carlo simulation 2:387, 429–30; 3:1311–13 non-probabilistic model 3:1082–116 non-linear equations 3:973–6 non-linear methods 3:1138 index one-factor interest rate modeling 2:510–12, 513–17, 525–32 partial differential equations 2:430 portfolios 3:1170–3 predictions 1:343–58 quasi Monte Carlo simulation 3:1313–14 risk-neutral models 3:858, 889–99 serial autocorrelation 3:1049 single life policy 3:1166–9 stochastic volatility 3:855–8, 886–7 technical analysis 1:343–53 theoretical 1:267 two-factor interest rate modeling 2:564–7 principal amortization 1:229 zero-coupon bonds 1:225–6 principal component analysis, HJM 2:617–20 principal only MBSs (POs) 2:573, 579 probabilistic modeling 3:1318 probability density functions 1:169–81 asset price distribution 1:280 blackjack 305 for chi-squared distribution 3:782 empirical analysis 3:884, 887 interest rate modeling 2:517–21 jump diffusion 3:927–8, 933 local volatility surface 3:842–5 Monte Carlo simulation 3:1270–2 spot interest rates 2:600–1, 602 trading strategy 3:996–1002 probability of death 3:1163–6 Proctor & Gamble (P&G) 2:733–5 producers 1:356 product copula 2:696 products, overview 1:5–24 profits American options 3:1023–5 diagrams 1:35, 36 programs American options 1:290 binomial model 1:286–7 Chooser Passport Option 3:1301–3 cliquet option 3:923–5 crash modeling 3:1303–4 downhill simplex method 3:982–6 explicit convertible bond model 3:1296–7 explicit Epstein- Wilmott model 3:1305–7 explicit finite-difference methods 3:1211–13, 1215–22, 1225, 1296–7, 1299–300, 1303–4 explicit Parisian option model 3:1299–300 explicit stochastic volatility 3:1303–4 finite-difference methods 3:1295–309 implicit American option model 3:1297–9 index amortizing rate swap 2:634–5 Monte Carlo 3:1311–16 Passport Options 3:1300–1 payoff 1:286 risky-bond calculator 3:1307–9 uncertain volatility 3:1304 projected SOR 3:1246 protected barrier options 2:398–9 public securities association model (PSA) 2:575–7 pull to par, bonds 2:536, 537 put-call parity 1:41–2, 118; 2:439–40 put-call symmetry 2:405 put features, CBs 2:561–3 put options 1:97 Black-Scholes formula 1:118–21 definition 1:26 delta 1:122–3 gamma 1:124–5 history 1:25 payoff diagrams 1:32, 33–6 put-call parity 1:41–2, 118; 2:439–40 replication 3:990–1, 993, 1001 rho 1:130 speed of 1:126 theta 1:126 vega 1:128 puttable swaps 1:257; 2:551 Q-Q plot 3:928–30 quadratic variation 1:73, 75 quantile-Quantile 3:928–30 quantitative analysis 1:56 quantos 1:189–93 quants 3:1173–4 quants’ salaries 3:823–4 quasi-random sequences 3:1288–92 rainbow options 1:186–7; 2:400 RAND function 1:12 random numbers 3:1267–8, 1277 random volatility 3:853 random walks 1:88–90; 3:1264, 1270 Asian options 2:430–7 binomial model 1:262 Black-Scholes assumption 1:95–6; 3:761 ergodic property 3:884 lognormal 1:85 Markov property 1:73 mean-reverting 1:86–8 model 1:64 1371 1372 index random walks (continued ) multi-asset options 1:183–95 Poisson process 3:931 probability density functions 1:169–81 quadratic variation 1:73, 75 risk-neutral 1:180; 3:858 speculation 3:954 spreadsheet calculations 1:67 steady-state distribution 1:173–4 stochastic calculus examples 1:84–90 trinomial model 1:170–1, 291–2 randomness analysis 1:169–81 asset behavior 1:55–70 credit risks 2:655–7 hedging 1:93 importance 1:169 Jensen’s inequality 1:56–8 multi-factor interest rate modeling 2:581–93 one-factor interest rate modeling 2:509–24 phase plane 2:587–90 stochastic volatility 3:853–67 stock prices 1:8–12 variance 1:58 volatility 1:39–40 range crash modeling 3:948 notes 2:379–80, 540 Range-based Exponential GARCH 3:862–3 range notes 2:493–6 ranking, utility theory 3:1005–7 ratchets 2:551 rate options 2:428, 439–40, 445 rating protected notes 2:694 ratio swaps 2:737–8 reaction-convection-diffusion equations 1:102–3 real expected value 3:967 real option theory 3:1151–7 examples 3:1152–4 financial options 3:1151 machine-abandonment example 3:1152–3 optimal investment example 3:1153–4 overview 3:1151 real world 1:272–4 rebates, barrier options 2:387, 396 recursive stratified sampling 3:1293 reduction of variance, Monte Carlo simulation 3:1293 reflection principle 2:405 reflex cap/floor 2:551 reflexivity, utility theory 3:1006 REGARCH 3:862–4 rehedging 1:122; 2:800–6 relative growth, concept 1:58 relative risk aversion function 3:1008, 1054–5 relative strength index 1:345 repeated hits, barrier options 2:399 replication 1:270, 271; 3:990–1003 boundaries 3:997–8 excess demand function 3:991 forward equation 3:996–8 influence 3:993–6 options 1:94 put options 3:990–1, 993, 1001 static replication 3:969–71 Tulip curves 3:997, 1001 see also trading strategy repos 1:145–6, 229 resets, barrier options 2:399 residual payoff delta hedging 3:975–6 optimal static hedging 3:976–8 resistance concept 1:345, 346 Retail Price Index (RPI) 1:20, 230; 3:1129, 1131–8 retired options 3:980–1 returns 2:701–8 correlations 1:184–5 definition 1:58 examinations 1:58–62 hedging error 3:774–6 jump diffusion 3:927–31 Leland model 3:784–5 non-normal returns 3:800 portfolio management 1:318–30 speculation 3:955–62 spreadsheet calculations 1:60 timesteps 1:62–5 VaR 1:331–42 reverse floater 2:551 reward to variability, Sharpe ratio 1:329–30, 339; 3:1175–80 reward to volatility, Treynor ratio 1:329–30 rewards, risks 1:319–24; 3:956–7 rho binomial model 1:287–9 formulae 1:130 Richardson extrapolation 3:1243–4 risk aversion 3:1008, 1016–27, 1054–6 risk-free interest rates 3:870, 966 risk-free investments 1:322, 328–9 risk-neutrality 3:1303–4 drift rate 3:857, 865–6, 1031 expectation 1:273 forward-rates 2:613 index jump diffusion 3:933–4 probabilities 1:273 random walks 1:180; 3:858, 1264, 1270 risk-neutral world 1:272–4 spot rates 2:513, 604–6 stochastic volatility 3:889–99, 1303–4 valuation 3:955 volatility 3:857, 865–6, 889–99, 1303–4 risk of default see credit risks risk-reversals Black-Scholes model 3:847 overview 1:47–9 skews and smiles 2:824–5 volatility 3:847 RiskMetrics 2:701–8 datasets 2:702–5 overview 2:701 parameter calculation 2:702–5 volatility estimates 2:702–3 risks basis risk 2:736–7 continuous-time investments 3:1051–60 counterparty risks 2:727 CrashMetrics 1:137; 2:709–29 credit derivatives 2:675–99 CreditMetrics 2:701, 705–7 delta hedging 1:92–3, 135 diversifiable risks 1:327; 3:933 hedging 1:317; 3:800–6 HJM 2:612–13 interest rates 1:180, 196; 2:649–50, 667–8; 3:870, 966, 1307–9 market price of risk 2:512–13, 587, 601–3,612; 3:758, 856–8, 1138 no arbitrage 1:93–4 portfolio management 1:317–30 preferences 1:319–22 rewards 1:319–24; 3:956–7 risk-free investments 1:322 RiskMetrics 2:701–8 seeking 1:267 spot interest rates 2:601–5 systematic risks 1:327 utility theory 3:1005–12, 1016–27 writers 1:40 see also credit risks; value at risk risky bonds 2:649–50, 654–5, 667–8, 683–5, 705–8 program code 3:1307–9 Rogers & Satchell measure 3:820 rolling cap/floor 2:551 roulette 1:309–10 rounding tops and bottoms 1:348, 350 RPI see Retail Price Index Russian GKOs 2:742 S&P500 2:481–3, 716 sampling dates jump conditions 2:468 path dependency 2:421–2 Samurai bonds 1:230 saucer tops and bottoms 1:348, 350 Scholes, M 1:94; 2:740 Schăonbucher model, stochastic implied volatility 3:865–6 Scott’s model 3:904 second order options 2:373, 389 securities fixed-income 1:17–19 mortgage-backed securities 2:571–80 seniority, debt 2:649–50, 657 Serial Autocorrelation 1:299; 3:1045–50 series 1:37 sex, life expectancy 3:1163 shareholders 1:8 shares see equities Sharpe ratio bonuses 3:1175–80 reward to variability 1:329–30, 339; 3:1175–80 short positions definition 1:31 distinction 3:974 static hedging 3:974–6 uncertain parameters 3:879 short-term interest rates 1:241 shout options, overview 2:463–4 similarity reductions 1:106–7 Asian options 2:437–8 lookback options 2:449 similarity solution asset exchanges 1:188–9 CBs 2:567 index amortizing rate swap 2:633–4 perfect trader options 2:455–6 simplex, downhill simplex method 3:982–6 simulated annealing 3:986 simulations 3:986, 1253, 1263–83, 1319 HJM 2:613–14 VaR 1:338–9 single assets, CrashMetrics 2:711–13 single-index model CrashMetrics 2:715–22,727 portfolio management 1:325–7 single monthly mortality (SMM) 2:575 1373 1374 index singles portfolio distinction 3:787–8, 880, 974 optimal portfolio under threat of crash 3:1062–70 singularities, phase plane 2:588 skews, implied volatility 3:824–5, 839 skills factor, traders 3:1180–6 skirt hemlines, economies 1:357 slippage 2:406 smiles, implied volatility 1:132; 3:756–7, 824–5, 826–7, 839 SMM see single monthly mortality smooth pasting condition 1:154; 3:1245 Sobol’ sequence 3:1288, 1292 soft barrier options 2:400 SOR method see successive over-relaxation method Soss, NM 3:1159–60 special utility functions 3:1007–8 spectral radius, SOR method 3:1239 speculation 1:40, 121–2, 181, 356; 3:953–68, 972 barrier options 2:401 Black-Scholes assumption 3:759 closure 3:962–5 definition 1:21–2 diffusive drift 3:959, 962 drift rates 3:954–7, 959–62 early closure 3:962–5 hedging 3:954, 966 jump drift 3:959–62, 962–5 models 3:954–62 overview 3:953–4 present value of expected payoff 3:954–5 standard deviation 3:955–7 speed, option 1:126–7 spot interest rates 1:241 credit risks 2:658, 668–9 drift structure 2:599–601, 603, 606–7 empirical behavior 2:595–608 forward-rate curves 2:604–6, 609–25 HJM comparison 2:615 implied modeling 2:597–8, 606–7 interest rate derivatives 2:533–52 multi-factor modeling 2:581–93 non-probabilistic model 3:1078, 1119–21 one-factor modeling 2:509–24, 525–32, 596–7, 603, 616–17 overview 2:595 popular models 2:596–7 risk-neutrality 2:513 volatility structure 2:598, 606–7, 616–17 yield curve slope 2:590–1, 606–7 spot prices forward contracts 1:22–3 storage costs and 1:144 spread options 2:542; 3:1150 spreads 1:44–5, 49–50; 3:792–3, 945, 1082–7 basis spreads 3:1149 bid-offer spreads 3:1126–7 CreditMetrics 2:705 long/short-term interest rates 2:584, 606–7 non-linear equations 3:973–6 static hedging 3:974–6 yield spread 2:671–2, 683 spreadsheets bootstrapping 1:338–9, 340 CrashMetrics 2:714–15 delta hedging 3:765 efficient frontiers 1:321, 323–4 exponentially-weighted volatility 2:704–5 fixed-income analysis 1:245, 246 forward-rates 2:617–20 jump diffusion 3:931, 932 Monte Carlo simulation 3:1264–6, 1273 random walk 1:67 returns 1:58–62 see also Excel stable singularities 2:588 Standard & Poor’s 1:17; 2:649, 663, 685, 728 standard deviation 3:822 of asset price change 1:276 concept 3:1325 profit hedging 1:208–11 state variables 2:418 static hedging 1:93; 3:828–9, 896–8, 969–87, 1096 calibration 3:978 crash modeling 3:946–7 definition 1:136; 3:880 delta hedging 3:969, 975–6 interest rate modeling 3:1078–97 non-probabilistic model 3:1082–116 non-linear models 3:972–4 overview 3:969 portfolio optimization 3:894 spreads 3:975–6 target-contract matching 3:970–1 value, definitions 3:898 see also optimal static hedging static replication 3:969–71 stationarity 1:328 steady-state distribution, random walk 1:173–4 step-up swaps 2:540 sticky delta 1:21617 sticky strike 1:216 stochastic calculus 1:69, 71–89 coin tossing 1: 71–5 index definitions 1:75–6 examples 1:84–90 overview 1:71 transition probability density functions 1:169–70 stochastic control 2:453–60 stochastic default risks 2:655–7 stochastic dividends 3:1040 stochastic interest rates 2:509–10, 564–7 stochastic variables, functions 1:78–80 stochastic volatility 3:827–8, 853–67 biases 3:864 Black-Scholes assumption 3:757, 758, 853–67 differential equation 3:854 empirical analysis 3:881–8 example 3:858–60 GARCH 3:860, 861 Heston model 3:861, 862 Hull & White model 2:586–7; 3:860–1 with jumps 3:862 market price of risk 3:758, 856–8 mean-variance analysis 3:758, 829, 889–99 models 3:860–3 overview 3:853 Ornstein-Uhlenbeck process 3:861–2 pricing 3:855–8, 886–7 program code 3:1303–4 REGARCH 3:862–4 Schăonbucher model 3:8656 3/2 model 3:861 time evolution 3:887 uncertain volatility 3:887 stochasticity, models 3:752 stock markets 1:7–14, 17 stocks borrowing 1:145–6 dividend dates 1:140–3; 3:1037–44 overview 1:7–14 splits 1:13–14 see also equities storage costs 1:144 straddles overview 1:46–8 perpetual American straddles 1:157–8 to skews and smiles 3:824 swaptions 2:742 volatility information 3:846 straight value, CBs 2:554 strangles, overview 1:46–8 stratified sampling 3:1293 strike options 2:428, 437–40, 445, 450–2 strike prices 1:39 delta alternative 3:848 implied volatility 3:839 overview 1:26–30 STRIPS 1:229 strong path dependency 1:371; 2:371, 381–2, 417–26 continuous sampling equation 2:420–1 discrete sampling equation 2:422–5 early exercise 2:426 expectations 2:425–6 higher dimensions 2:425 integral representations 2:418–19 jump conditions 2:423–4 sampling dates 2:421–2 updating rule 2:421–2 successive over-relaxation (SOR) method, Crank-Nicolson 3:1236–9, 1246 suicide 3:1159–60 supply, trading strategy 3:991–6 support concept 1:345, 346 surfaces, volatility 3:756–7, 826–7, 840–5, 889 swaps 2:445–6, 251–9, 733, 737–9 bonds 1:254–7 bootstrapping 1:257 comparative advantages 1:253–4 correlation 2:472 curve 1:254 index amortizing rate swaps 1:258; 2:542–6; 3:1110–13 inflation 3:1130 interest rates 1:19, 251–9; 3:1091–4 LIBOR-in-arrears swaps 2:551 non-probabilistic model 3:1108–13 overview 1:251 step-up swaps 2:540 types 1:251–2, 258–9 variance 2:471–2 yield curve swaps 2:590–1 swaptions 2:541, 551 inflation 3:1130 straddles 2:742 swing options, energy 3:1150 systematic risks 1:327 ‘tail event’ risks 3:940 tail index 2:696–7 target-contract matching, static hedging 3:970–1 taxation dividends 3:1038 stock prices 1:13 Taylor series 1:81–3; 3:1206, 1321–4 technical analysis 1:56, 343–53 telegraph equation 3:1047–9 1375 1376 index ‘Tequila effect’ 2:654, 659–61 termsheets 2:687–9 basket options 2:483–6 chooser range note 2:628 cliquet option 2:500–1; 3:916 double knock-out note 2:386, 487 equity and 2:481–505 fixed-income 2:627–35 index amortizing rate swaps 2:545, 632 instalment knockout 2:490 interest rate derivatives 2:545, 548–50 knocked-out options 2:397 lookback swaps 2:445–6, 497–8 multi-asset options 1:191 OTC 1:51–3 passport option 2:503, 504 perfect trader options 2:454–5 range notes 2:379–80, 494 yield curve swaps 2:590–1 term structure Asian options 2:440–1 dividends 3:1038–40 terms, Black-Scholes model 1:102–3 theta binomial model 1:288–9 CrashMetrics 2:726, 727 finite-difference methods 3:1202–3, 1206 formulae 1:126 Thorp & Kassouf 1:93 Thorp, E 2:741 three time-level methods 3:1242–3 3/2 model 3:861 three-for-one splits 1:13 time Black-Scholes assumption 1:96, 147–8; 3:756–7, 780 calculation 3:1276–7 CrashMetrics 2:724 dividends 1:147–8; 3:1035–44 evolution, stochastic volatility 3:887 limit, binomial model 1:291 theta 1:126 to expiry 1:38; 3:869 value 1:5–7, 31; 2:724 time-dependency 2:369 hazard rates 2:655, 690–1 interest rates 1:241–2 LU decomposition 3:1236 parameters, Black-Scholes assumptions 1:147–8 path dependency 2:421–2 trading strategy 3:996–1002 volatility 1:147–8; 3:835–8, 848 yield curve fitting 2:525–32 time-periodic behavior 1:217–19 time steps 3:1264–5 ADI 3:1259–60 binomial model 1:261–94 Crank-Nicolson method 3:1230 explicit finite-difference methods 3:1210–13, 1255–8 fully implicit finite-difference methods 3:1227–8 Hopscotch method 3:1260–1 Leland model 3:784–5 random numbers 3:1267–8 returns 1:62–5 three time-level methods 3:1242–3 time swaps 2:739 timescales, assets 1:62–5 total hedging error 3:765, 776–7 total rate of return swaps 2:679–80 total return swaps (TRS) 2:679–80 tranches 2:697 tracking, indexes 1:327–8 tractable models, bond pricing equation 2:513–16, 518–19 trade effects, underlying 3:760, 857, 989, 996 traders bonuses 3:1175–87 dismissal issues 3:1186–7 imitative actions 1:357 skill factor 3:1180–6 types 1:356 trades optimal trades 2:457–9 passport options 2:456–9; 3:1029–33 trading accounts 2:453 trading games 1:359–63 trading strategy 3:990–1001, 1175–87 American options 3:1016–27 incorporation 3:991–3 time-dependence 3:996–1002 writers 2:459 see also replication trading talent 1:339 traditional close-to-close measure 3:819 transaction costs 3:764, 783–811, 972 arbitrary cost structure 3:796–7 asset allocation 3:1058–60 asymptotic analysis 3:795–6 Black-Scholes assumption 1:96; 3:756, 783–811 brief look 3:1058–60 bull spreads 3:790–1 butterfly spreads 3:791–2 Davis, Panas & Zariphopoulou model 3:794, 795 index delta-tolerance strategy 3:802–6 discrete hedging and 3:807–8 economies of scale 3:783 effects 3:783–4, 792–3 empirical testing 3:800–6 Hodges & Neuberger model 3:794 Hoggard, Whalley & Wilmott model 3:785–90, 798, 809–11, 873, 879 Leland model 3:784–5, 788, 801–2, 806 marginal effects 3:792–3 market movement strategy 3:802–4 model interpretations 3:797–9 negative option prices 3:798 non-normal returns 3:800 nonlinearity 3:797 optimal rebalance point 3:796–7 overview 3:783, 1058–60 real data 3:806 utility-based models 3:794–7, 802–6 Whalley & Wilmott & Henrotte model 3:793–4 Whalley & Wilmott asymptotic analysis 3:795–6 transition matrices credit rating 2:665–7 CreditMetrics 2:706 transitivity utility theory 3:1006 Treasury-linked swaps 2:738 tree structures, HJM 2:614 trendlines 1:345, 347 Treynor ratio, reward to volatility 1:329–30 tridiagonal matrices 3:1234–6 triggered derivatives, by default 2:680–3 triggered options 3:980–1 triggers 2:541 trinomial model, random walk 1:170–1, 291–2; 3:763–4 trinomial trees, crash modeling 3:941–2 triple tops and bottoms 1:349, 351 tulip curves 3:997, 1001 two-factor interest rate modeling CBs 2:564–7 implied modeling 2:606–7 two-factor models, finite-difference methods 3:1253–62 two-state drift model, speculation 3:959–62 UK see United Kingdom uncertain correlation 3:879 uncertain dividends 3:761, 877–8, 1035, 1040–3 uncertain interest rates 3:877 uncertain parameters 3:757, 828, 869–71, 881–2, 972, 972–4 Avellaneda, Levy & paras, Lyons model 3:871, 872–3 best/worst cases 3:871–8, 1040–3 correlation 3:879 dividends 3:877–8, 1035, 1040–3 interest rates 3:877 multi-asset options 3:879 overview 3:869–71 volatility 3:871–7 see also parameters uncertain volatility 3:872–7, 880, 919–23, 1304 stochastic volatility 3:887 uncertainty bands 3:1119–21 dividends 3:761, 877–8, 1035, 1040–3 uncorrelated assets, portfolio management 1:319 underlying assumptions 3:989 basket options 1:186–7 Black-Scholes model 1:95–6, 116–19; 3:758–9 CBs 2:553–70 correlation 1:92 definition 1:27, 29, 31 delta 1:92, 109, 121–2 dividends 1:139–40 gamma 1:124–5 measurability 3:869–71 perpetual options 1:152, 155 price determinant 1:38–9 trade effects 3:760, 857, 989, 996 United Kingdom (UK), bond market 1:230 United States of America (USA), bond market 1:229–30 untriggered options 3:980–1 up-and-in call options, formula 2:409 up-and-in options 2:389, 409, 415 up-and-in put options 2:409, 415 up-and-out call options 2:402, 408 410; 3:970–1 best/worst prices 3:873–7 formula 2:393, 395–6 mean-variance analysis 3:894–6 up-and-out put options 2:409, 414 up barrier options 2:386, 388, 391–6; 3:873–7, 894–6 updating rule lookback-Asian options 2:465–6 path dependency 2:421–2 upwind differencing 3:1224–6 US Treasuries 2:742 USA see United States of America utility-based models, transaction costs 3:794–7, 802–6 1377 1378 index utility functions concept 3:1007, 1016–33 portfolio management 1:323–4 utility theory 3:1005–12, 1016–18, 898 Black-Scholes assumption 3:761 certainty equivalent wealth 3:1008–10 event ranking 3:1005–7 functions 3:1006–12, 1016–27 maximization 3:1010–11, 1019–23, 1030–2, 1052–3 von Neumann-Morgenstern function 3:1011 wealth 3:1006–11, 1019–23 valuation MBSs 2:578–9 VaR models 1:338 value at risk (VaR) 1:331–42 bootstrapping 1:338–9 coherence 1:341–2 crash modeling 3:939–40, 946–7 definition 1:331–2; 2:701 delta approximations 1:335 delta-gamma approximations 1:336–7 Extreme Value Theory 1:339–41 fixed-income portfolios 1:227–8 Monte Carlo simulation 1:338 overview 1:331 performance measurement usage 1:339 portfolios 1:334–5 reduction 3:946–7 simulations 1:338 single assets 1:332–4 valuation models 1:337 see also RiskMetrics value, speculator, definition 3:954–5 vanilla options 1:30, 43; 2:373–4, 385 asymptotic analysis 3:908–10 chooser options 2:376–8 compound options 2:375–6 decomposition into 3:505 implied volatilities 3:910–13 local volatility surface 3:849, 971–2 optimal static hedging 3:978–81 shout 2:463–4 swaps 1:251–9 up-and-in options 2:389 variance swaps 471–2 volatility for 2:440–1 VaR see value at risk variables 1:78–80, 91–2; 3:1277–8, 1292–3 credit risks 2:642–3 dimensionality 2:372–3 measurability 3:869–71 parameters 1:38 state variables 2:418 variance analysis 3:890–1 concept 3:1324–5 hedging with implied volatility 1:213 interpretation 3:892–3 portfolio of options 1:224 randomness 1:58 single option: 222–4 see also mean-variance analysis Vasicek model 2:518–19, 526–7, 567, 591–3, 669; 3:751, 1114, 1115 vega 1:127–30, 136; 3:889, 897, 1035–7 basket options 1:195 binomial model 1:288–9 cliquet option 915 formulae 1:127–30 matching 3:971–2 uncertain parameters 3:874 up-and-out call options 2:402 variance swaps 2:471 viaticals 3:1161–74 Visual Basic code see programs volatility 1:184, 185; 3:773, 774 actual volatility 1:197, 198–9, 200–2, 215; 3:814 asymptotic analysis 3:758, 829, 901–13 at-the-money straddles 3:846 Avellaneda, Levy & Paras, Lyons model 3:872, 873–7 barrier options 2:389–93, 401–5 Black-Scholes model 1:131–3; 3:756–7, 833–52 calculation 1:65–7 cliquet option 915–26 constancy 3:756, 757, 815, 833–52, 854, 857, 917–18, 919 crash effects 2:727 crash modeling 3:951 curve fitting 3:850–2 definition 1:39 dividend-yield sensitivity 3:1035–7 drift 3:884–5 empirical analysis 3:758, 881–8 energy derivatives 3:1145–8 estimation 1:65–7; 3:815–20 fast mean reversion 3:901–3 forward 3:814 gamma 1:124–5 historical (realized) volatility 2:467–9; 3:814, 834, 854, 869–70 index HJM 2:613, 616–17 jump processes 3:935–6 local volatility surface 3:840–5, 889, 971–2 major news effects 3:833–4 measurability 3:869–71 modeling 3:813–31 principal component analysis 2:617–20 range-based estimation 3:818–20 risk-neutrality 3:857, 858, 866, 889–99, 1303–4 risk-reversals 3:847 RiskMetrics 2:702–3 spot interest rates 2:595–608 straddles/strangles 1:46–8; 3:846 surfaces 3:756–7, 840–5, 889 time-dependence 1:147–8; 3:835–8, 848 Treynor ratio 1:329–30 types 3:813–14 uncertainty 3:872–7, 880, 887, 919–23, 1304 value at risk 1:335 vanilla option 2:440–1 vega 1:127–30, 131–4 volatility of 3:901–3 see also implied ; stochastic volume, charting 1:356 von Neumann-Morgenstern utility function 3:1011 WAC see weight-averaged Wall Street Journal Europe 1:9–10, 15, 16, 27–9, 48 warrants overview 1:51 real expected value 3:967 wave theory 1:353–5 weak path dependency 1:371–2 wealth continuous-time investments 3:1051–60 utility 3:1007, 1019–23 Weibull distribution 1:341 weight-averaged coupon (WAC) 2:575, 577 Whalley & Wilmott, asymptotic analysis 3:795–6 Whalley & Wilmott & Henrotte model 3:793–4 Wiener process, overview 1:67–9 worst-case scenarios crash modeling 3:940–5, 948–9 interest rate modeling 3:1077–97 nonlinear equations 3:1079–81 uncertain parameters 3:872–8 see also CrashMetrics writers American options 1:156; 3:1014–27 non-optimal trading 2:459 optimal 2:639–31 overview 1:37 risks 1:40 Yankee bonds 1:230 yield convenience yield 3:1146 duration relationship 1:237 energy 3:1146 envelope 3:1087–91 measures 1:231–2 price relationship 1:233 risky bonds 2:649–50, 654–5; 3:1307–9 spread 2:671–2, 683 yield curves 2:518–19, 548; 3:1087–103, 111–13 arguments 2:527–30 concept 1:233, 234 credit risks 2:658–60, 670 CreditMetrics 2:705 multi-factor interest rate modeling 2:582–3 named models 2:525–7, 530–1 one-factor interest rate modeling 2:525–32 overview 2:525 parallel shifts 1:241 spot rate slope 2:601–3, 606–7 swaps 2:590–1 yield to maturity (YTM) 1:232, 242–8; 2:536 zero-coupon bonds 1:225–6, 235–6, 246 bills 1:229 non-probabilistic model 3:1103–5 STRIPS 1:229 zetas, formulae 1:127–30 Index compiled by Annette Musker 1379 WILEY COPYRIGHT INFORMATION AND TERMS OF USE CD supplement to Paul Wilmott on Quantitative Finance, Second Edition by Paul Wilmott ISBN-13: 978-0-470-01870-5 (HB) ISBN-10: 0-470-01870-4 (HB) Copyright  2006 Paul Wilmott Published by John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England All rights-reserved All material contained herein is protected by copyright, whether or not a copyright notice appears on the particular screen where the material is displayed No part of the material may be reproduced or transmitted in any form or by any means, or stored in a computer for retrieval purposes or otherwise, without written permission from Wiley, unless this is expressly permitted in a copyright notice or usage statement accompanying the materials Requests for permission to store or reproduce material for any purpose, or to distribute it on a network, should be addressed to the Permissions Department, John Wiley & Sons, Ltd., The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK; fax +44 (0) 1243 770571; Email permreq@wiley.co.uk Neither the author nor John Wiley & Sons, Ltd accept any responsibility or liability for loss or damage occasioned to any person or property through using materials, instructions, methods or ideas contained herein, or acting or refraining from acting as a result of such use The author and Publisher expressly disclaim all implied warranties, including merchantability or fitness for any particular purpose There will be no duty on the author or Publisher to correct any errors or defects in the software ... option, explicit finite difference Passport option, explicit finite difference 13 0 13 1 286 290 490 493 497 5 01 634 923 983 12 12 12 13 12 15 12 19 12 21 1225 12 34 12 35 12 38 12 46 12 48 12 49 12 57 12 57 12 69... =D6*IF(RAND() >1- $B$4,$B$2,$B$3) 99.960 01 98.960 41 99.950 01 100.9495 99.940 01 98.940 61 97.9 512 1 98.93072 97.9 414 1 98.92083 99. 910 04 98. 910 94 97.9 218 3 98.9 010 4 97. 912 03 98.8 911 5 99.88007 10 0.8789 10 1.8877 10 0.8688... Extensions to the Non-probabilistic Interest-rate Model 11 17 71 Modeling Inflation 11 29 72 Energy Derivatives 11 41 73 Real Options 11 51 74 Life Settlements and Viaticals 11 61 75 Bonus Time 11 75 PART SIX

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  • Paul Wilmott On Quantitative Finance

    • contents of volume one

    • contents of volume two

    • contents of volume three

    • Visual Basic Code

    • Prolog to the Second Edition

    • PART ONE MATHEMATICAL AND FINANCIAL FOUNDATIONS; BASIC THEORY OF DERIVATIVES; RISK AND RETURN

      • 1 Products and Markets

        • 1.1 Introduction

        • 1.2 The time value of money

        • 1.3 Equities

          • 1.3.1 Dividends

          • 1.3.2 Stock splits

        • 1.4 Commodities

        • 1.5 Currencies

        • 1.6 Indices

        • 1.7 Fixed-income securities

        • 1.8 Inflation-proof bonds

        • 1.9 Forwards and futures

          • 1.9.1 A first example of no arbitrage

        • 1.10 Summary

      • 2 Derivatives

        • 2.1 Introduction

        • 2.2 Options

        • 2.3 Definition of common terms

        • 2.4 Payoff diagrams

          • 2.4.1 Other representations of value

        • 2.5 Writing options

        • 2.6 Margin

        • 2.7 Market conventions

        • 2.8 The value of the option before expiry

        • 2.9 Factors affecting derivatives prices

        • 2.10 Speculation and gearing

        • 2.11 Early exercise

        • 2.12 Put-call parity

        • 2.13 Binaries or digitals

        • 2.14 Bull and bear spreads

        • 2.15 Straddles and strangles

        • 2.16 Risk reversal

        • 2.17 Butterflies and condors

        • 2.18 Calendar spreads

        • 2.19 LEAPS and FLEX

        • 2.20 Warrants

        • 2.21 Convertible bonds

        • 2.22 Over the counter options

        • 2.23 Summary

      • 3 The Random Behavior of Assets

        • 3.1 Introduction

        • 3.2 The popular forms of ‘analysis’

        • 3.3 Why we need a model for randomness: Jensen’s Inequality

        • 3.4 Similarities between equities, currencies, commodities and indices

        • 3.5 Examining returns

        • 3.6 Timescales

          • 3.6.1 The drift

          • 3.6.2 The volatility

        • 3.7 Estimating volatility

        • 3.8 The random walk on a spreadsheet

        • 3.9 The Wiener process

        • 3.10 The widely accepted model for equities, currencies, commodities and indices

        • 3.11 Summary

      • 4 Elementary Stochastic Calculus

        • 4.1 Introduction

        • 4.2 A motivating example

        • 4.3 The Markov property

        • 4.4 The martingale property

        • 4.5 Quadratic variation

        • 4.6 Brownian motion

        • 4.7 Stochastic integration

        • 4.8 Stochastic differential equations

        • 4.9 The mean square limit

        • 4.10 Functions of stochastic variables and Itˆo’s lemma

        • 4.11 Interpretation of Itˆo’s lemma

        • 4.12 Itˆo and Taylor

          • 4.12.1 The intuition

          • 4.12.2 Simple generalization

        • 4.13 Itˆo in higher dimensions

        • 4.14 Some pertinent examples

          • 4.14.1 Brownian motion with drift

          • 4.14.2 The lognormal random walk

          • 4.14.3 A mean-reverting random walk

          • 4.14.4 And another mean-reverting random walk

        • 4.15 Summary

      • 5 The Black–Scholes Model

        • 5.1 Introduction

        • 5.2 A very special portfolio

        • 5.3 Elimination of risk: delta hedging

        • 5.4 No arbitrage

        • 5.5 The Black–Scholes equation

        • 5.6 The Black–Scholes assumptions

        • 5.7 Final conditions

        • 5.8 Options on dividend-paying equities

        • 5.9 Currency options

        • 5.10 Commodity options

        • 5.11 Options on futures

        • 5.12 Some other ways of deriving the Black–Scholes equation

          • 5.12.1 The martingale approach

          • 5.12.2 The binomial model

          • 5.12.3 CAPM/utility

        • 5.13 Summary

      • 6 Partial Differential Equations

        • 6.1 Introduction

        • 6.2 Putting the Black–Scholes equation into historical perspective

        • 6.3 The meaning of the terms in the Black–Scholes equation

        • 6.4 Boundary and initial/final conditions

        • 6.5 Some solution methods

          • 6.5.1 Transformation to constant coefficient diffusion equation

          • 6.5.2 Green’s functions

          • 6.5.3 Series solution

        • 6.6 Similarity reductions

        • 6.7 Other analytical techniques

        • 6.8 Numerical solution

        • 6.9 Summary

      • 7 The Black–Scholes Formulae and the ‘Greeks’

        • 7.1 Introduction

        • 7.2 Derivation of the formulae for calls, puts and simple digitals

          • 7.2.1 Formula for a call

          • 7.2.2 Formula for a put

          • 7.2.3 Formula for a binary call

          • 7.2.4 Formula for a binary put

        • 7.3 Delta

        • 7.4 Gamma

        • 7.5 Theta

        • 7.6 Speed

        • 7.7 Vega

        • 7.8 Rho

        • 7.9 Implied volatility

        • 7.10 A classification of hedging types

          • 7.10.1 Why hedge?

          • 7.10.2 The two main classifications

          • 7.10.3 Delta hedging

          • 7.10.4 Gamma hedging

          • 7.10.5 Vega hedging

          • 7.10.6 Static hedging

          • 7.10.7 Margin hedging

          • 7.10.8 Crash (platinum) hedging

        • 7.11 Summary

      • 8 Simple Generalizations of the Black–Scholes World

        • 8.1 Introduction

        • 8.2 Dividends, foreign interest and cost of carry

        • 8.3 Dividend structures

        • 8.4 Dividend payments and no arbitrage

        • 8.5 The behavior of an option value across a dividend date

        • 8.6 Commodities

          • 8.6.1 Futures prices and arbitrage

          • 8.6.2 Storage costs

          • 8.6.3 Convenience yield

          • 8.6.4 Cost of carry

          • 8.6.5 Effect on options

        • 8.7 Stock borrowing and repo

        • 8.8 Time-dependent parameters

        • 8.9 Formulae for power options

        • 8.10 The log contract

        • 8.11 Summary

      • 9 Early Exercise and American Options

        • 9.1 Introduction

        • 9.2 The perpetual American put

        • 9.3 Perpetual American call with dividends

        • 9.4 Mathematical formulation for general payoff

        • 9.5 Local solution for call with constant dividend yield

        • 9.6 Other dividend structures

        • 9.7 One-touch options

        • 9.8 Other features in American-style contracts

          • 9.8.1 Bermudan options

          • 9.8.2 Make your mind up

        • 9.9 Other issues

          • 9.9.1 Non-linearity

          • 9.9.2 Free-boundary problems

          • 9.9.3 Numerical solution

        • 9.10 Summary

      • 10 Probability Density Functions and First-exit Times

        • 10.1 Introduction

        • 10.2 The transition probability density function

        • 10.3 A trinomial model for the random walk

        • 10.4 The forward equation

        • 10.5 The steady-state distribution

        • 10.6 The backward equation

        • 10.7 First-exit times

        • 10.8 Cumulative distribution functions for first-exit times

        • 10.9 Expected first-exit times

        • 10.10 Another example of optimal stopping

        • 10.11 Expectations and Black–Scholes

        • 10.12 A common misconception

        • 10.13 Summary

      • 11 Multi-asset Options

        • 11.1 Introduction

        • 11.2 Multi-dimensional lognormal random walks

        • 11.3 Measuring correlations

        • 11.4 Options on many underlyings

        • 11.5 The pricing formula for European non-path-dependent options on dividend-paying assets

        • 11.6 Exchanging one asset for another: A similarity solution

        • 11.7 Quantos

        • 11.8 Two examples

        • 11.9 Other features

        • 11.10 Realities of pricing basket options

          • 11.10.1 Easy problems

          • 11.10.2 Medium problems

          • 11.10.3 Hard problems

        • 11.11 Realities of hedging basket options

        • 11.12 Correlation versus cointegration

        • 11.13 Summary

      • 12 How to Delta Hedge

        • 12.1 Introduction

        • 12.2 What if implied and actual volatilities are different?

        • 12.3 Implied versus actual; delta hedging but using which volatility?

        • 12.4 Case 1: Hedge with actual volatility, σ

        • 12.5 Case 2: Hedge with implied volatility, σ

          • 12.5.1 The expected profit after hedging using implied volatility

          • 12.5.2 The variance of profit after hedging using implied volatility

          • 12.5.3 Hedging with different volatilities

        • 12.6 Portfolios when hedging with implied volatility

          • 12.6.1 Expectation

          • 12.6.2 Variance

          • 12.6.3 Portfolio optimization possibilities

        • 12.7 Hedging when implied volatility is stochastic

          • 12.7.1 Case 1: Hedge with actual volatility, σ

          • 12.7.2 Case 2: Hedge with implied volatility, σ?

        • 12.8 How does implied volatility behave?

          • 12.8.1 Sticky strike

          • 12.8.2 Sticky delta

          • 12.8.3 Time-periodic behavior

        • 12.9 Summary

      • 13 Fixed-income Products and Analysis: Yield, Duration and Convexity

        • 13.1 Introduction

        • 13.2 Simple fixed-income contracts and features

          • 13.2.1 The zero-coupon bond

          • 13.2.2 The coupon-bearing bond

          • 13.2.3 The money market account

          • 13.2.4 Floating rate bonds

          • 13.2.5 Forward rate agreements

          • 13.2.6 Repos

          • 13.2.7 STRIPS

          • 13.2.8 Amortization

          • 13.2.9 Call provision

        • 13.3 International bond markets

          • 13.3.1 United States of America

          • 13.3.2 United Kingdom

          • 13.3.3 Japan

        • 13.4 Accrued interest

        • 13.5 Day-count conventions

        • 13.6 Continuously and discretely compounded interest

        • 13.7 Measures of yield

          • 13.7.1 Current yield

          • 13.7.2 The yield to maturity (YTM) or internal rate of return (IRR)

        • 13.8 The yield curve

        • 13.9 Price/yield relationship

        • 13.10 Duration

        • 13.11 Convexity

        • 13.12 An example

        • 13.13 Hedging

        • 13.14 Time-dependent interest rate

        • 13.15 Discretely paid coupons

        • 13.16 Forward rates and bootstrapping

          • 13.16.1 Discrete data

          • 13.16.2 On a spreadsheet

        • 13.17 Interpolation

        • 13.18 Summary

      • 14 Swaps

        • 14.1 Introduction

        • 14.2 The vanilla interest rate swap

        • 14.3 Comparative advantage

        • 14.4 The swap curve

        • 14.5 Relationship between swaps and bonds

        • 14.6 Bootstrapping

        • 14.7 Other features of swaps contracts

        • 14.8 Other types of swap

          • 14.8.1 Basis rate swap

          • 14.8.2 Equity swaps

          • 14.8.3 Currency swaps

        • 14.9 Summary

      • 15 The Binomial Model

        • 15.1 Introduction

        • 15.2 Equities can go down as well as up

        • 15.3 The option value

        • 15.4 Which part of our ‘model’ didn’t we need?

        • 15.5 Why should this ‘theoretical price’ be the ‘market price’?

          • 15.5.1 The role of expectations

        • 15.6 How did I know to sell 12 of the stock for hedging?

          • 15.6.1 The general formula for

        • 15.7 How does this change if interest rates are non-zero?

        • 15.8 Is the stock itself correctly priced?

        • 15.9 Complete markets

        • 15.10 The real and risk-neutral worlds

          • 15.10.1 Non-zero interest rates

        • 15.11 And now using symbols

          • 15.11.1 Average asset change

          • 15.11.2 Standard deviation of asset price change

        • 15.12 An equation for the value of an option

          • 15.12.1 Hedging

          • 15.12.2 No arbitrage

        • 15.13 Where did the probability p go?

        • 15.14 Counterintuitive?

        • 15.15 The binomial tree

        • 15.16 The asset price distribution

        • 15.17 Valuing back down the tree

        • 15.18 Programming the binomial method

        • 15.19 The greeks

        • 15.20 Early exercise

        • 15.21 The continuous-time limit

        • 15.22 No arbitrage in the binomial, Black–Scholes and ‘other’ worlds

        • 15.23 Summary

      • 16 How Accurate is the Normal Approximation?

        • 16.1 Introduction

        • 16.2 Why we like the Normal distribution: the Central Limit Theorem

        • 16.3 Normal versus lognormal

        • 16.4 Does my tail look fat in this?

          • 16.4.1 Probability of a 20% SPX fall: empirical

          • 16.4.2 Probability of a 20% SPX fall: theoretical

        • 16.5 Use a different distribution, perhaps

        • 16.6 Serial Autocorrelation

        • 16.7 Summary

      • 17 Investment Lessons from Blackjack and Gambling

        • 17.1 Introduction

        • 17.2 The rules of blackjack

        • 17.3 Beating the dealer

          • 17.3.1 Summary of winning at blackjack

        • 17.4 The distribution of profit in blackjack

        • 17.5 The Kelly criterion

        • 17.6 Can you win at roulette?

        • 17.7 Horse race betting and no arbitrage

          • 17.7.1 Setting the odds in a sporting game

          • 17.7.2 The mathematics

        • 17.8 Arbitrage

          • 17.8.1 How best to profit from the opportunity?

        • 17.9 How to bet

        • 17.10 Summary

      • 18 Portfolio Management

        • 18.1 Introduction

        • 18.2 Diversification

          • 18.2.1 Uncorrelated assets

        • 18.3 Modern Portfolio Theory

          • 18.3.1 Including a risk-free investment

        • 18.4 Where do I want to be on the efficient frontier?

        • 18.5 Markowitz in practice

        • 18.6 Capital Asset Pricing Model

          • 18.6.1 The single-index model

          • 18.6.2 Choosing the optimal portfolio

        • 18.7 The multi-index model

        • 18.8 Cointegration

        • 18.9 Performance measurement

        • 18.10 Summary

      • 19 Value at Risk

        • 19.1 Introduction

        • 19.2 Definition of value at risk

        • 19.3 VaR for a single asset

        • 19.4 VaR for a portfolio

        • 19.5 VaR for derivatives

          • 19.5.1 The delta approximation

          • 19.5.2 Which volatility do I use?

          • 19.5.3 The delta-gamma approximation

          • 19.5.4 Use of valuation models

          • 19.5.5 Fixed-income portfolios

        • 19.6 Simulations

          • 19.6.1 Monte Carlo

          • 19.6.2 Bootstrapping

        • 19.7 Use of VaR as a performance measure

        • 19.8 Introductory extreme value theory

          • 19.8.1 Some EVT results

        • 19.9 Coherence

        • 19.10 Summary

      • 20 Forecasting the Markets?

        • 20.1 Introduction

        • 20.2 Technical analysis

          • 20.2.1 Plotting

          • 20.2.2 Support and resistance

          • 20.2.3 Trendlines

          • 20.2.4 Moving averages

          • 20.2.5 Relative strength

          • 20.2.6 Oscillators

          • 20.2.7 Bollinger bands

          • 20.2.8 Miscellaneous patterns

          • 20.2.9 Japanese candlesticks

          • 20.2.10 Point and figure charts

        • 20.3 Wave theory

          • 20.3.1 Elliott waves and Fibonacci numbers

          • 20.3.2 Gann charts

        • 20.4 Other analytics

        • 20.5 Market microstructure modeling

          • 20.5.1 Effect of demand on price

          • 20.5.2 Combining market microstructure and option theory

          • 20.5.3 Imitation

        • 20.6 Crisis prediction

        • 20.7 Summary

      • 21 A Trading Game

        • 21.1 Introduction

        • 21.2 Aims

        • 21.3 Object of the game

        • 21.4 Rules of the game

        • 21.5 Notes

        • 21.6 How to fill in your trading sheet

          • 21.6.1 During a trading round

          • 21.6.2 At the end of the game

    • PART TWO EXOTIC CONTRACTS AND PATH DEPENDENCY

      • 22 An Introduction to Exotic and Path-dependent Derivatives

        • 22.1 Introduction

        • 22.2 Option classification

        • 22.3 Time dependence

        • 22.4 Cashflows

        • 22.5 Path dependence

          • 22.5.1 Strong path dependence

          • 22.5.2 Weak path dependence

        • 22.6 Dimensionality

        • 22.7 The order of an option

        • 22.8 Embedded decisions

        • 22.9 Classification tables

        • 22.10 Examples of exotic options

          • 22.10.1 Compounds and choosers

          • 22.10.2 Range notes

          • 22.10.3 Barrier options

          • 22.10.4 Asian options

          • 22.10.5 Lookback options

        • 22.11 Summary of math/coding consequences

        • 22.12 Summary

      • 23 Barrier Options

        • 23.1 Introduction

        • 23.2 Different types of barrier options

        • 23.3 Pricing methodologies

          • 23.3.1 Monte Carlo simulation

          • 23.3.2 Partial differential equations

        • 23.4 Pricing barriers in the partial differential equation framework

          • 23.4.1 ‘Out’ barriers

          • 23.4.2 ‘In’ barriers

          • 23.4.3 Some formulae when volatility is constant

          • 23.4.4 Some more examples

        • 23.5 Other features in barrier-style options

          • 23.5.1 Early exercise

          • 23.5.2 The intermittent barrier

          • 23.5.3 Repeated hitting of the barrier

          • 23.5.4 Resetting of barrier

          • 23.5.5 Outside barrier options

          • 23.5.6 Soft barriers

          • 23.5.7 Parisian options

          • 23.5.8 The emergency exit

        • 23.6 First-exit time

        • 23.7 Market practice: What volatility should I use?

        • 23.8 Hedging barrier options

        • 23.9 Slippage costs

        • 23.10 Summary

      • 24 Strongly Path-Dependent Derivatives

        • 24.1 Introduction

        • 24.2 Path-dependent quantities represented by an integral

          • 24.2.1 Examples

        • 24.3 Continuous sampling: The pricing equation

          • 24.3.1 Example

        • 24.4 Path-dependent quantities represented by an updating rule

          • 24.4.1 Examples

        • 24.5 Discrete sampling: The pricing equation

          • 24.5.1 Examples

          • 24.5.2 The algorithm for discrete sampling

        • 24.6 Higher dimensions

        • 24.7 Pricing via expectations

        • 24.8 Early exercise

        • 24.9 Summary

      • 25 Asian Options

        • 25.1 Introduction

        • 25.2 Payoff types

        • 25.3 Types of averaging

          • 25.3.1 Arithmetic or geometric

          • 25.3.2 Discrete or continuous

        • 25.4 Solution methods

          • 25.4.1 Monte Carlo simulation

        • 25.5 Extending the Black–Scholes equation

          • 25.5.1 Continuously sampled averages

          • 25.5.2 Discretely sampled averages

          • 25.5.3 Exponentially weighted and other averages

          • 25.5.4 The Asian tail

        • 25.6 Early exercise

        • 25.7 Asian options in higher dimensions

        • 25.8 Similarity reductions

          • 25.8.1 Put-call parity for the European average strike

        • 25.9 Closed-form solutions and approximations

          • 25.9.1 Kemna and Vorst (1990)

          • 25.9.2 Turnbull and Wakeman (1991)

          • 25.9.3 Curran (1992)

          • 25.9.4 Thompson (2000)

        • 25.10 Term-structure effects

          • 25.10.1 Some results

        • 25.11 Some formulae

        • 25.12 Summary

      • 26 Lookback Options

        • 26.1 Introduction

        • 26.2 Types of payoff

        • 26.3 Continuous measurement of the maximum

        • 26.4 Discrete measurement of the maximum

        • 26.5 Similarity reduction

        • 26.6 Some formulae

        • 26.7 Summary

      • 27 Derivatives and Stochastic Control

        • 27.1 Introduction

        • 27.2 Perfect trader and passport options

          • 27.2.1 Similarity solution

        • 27.3 Limiting the number of trades

        • 27.4 Limiting the time between trades

        • 27.5 Non-optimal trading and the bene.ts to the writer

        • 27.6 Summary

      • 28 Miscellaneous Exotics

        • 28.1 Introduction

        • 28.2 Forward-start options

        • 28.3 Shout options

        • 28.4 Capped lookbacks and Asians

        • 28.5 Combining path-dependent quantities: The lookback-Asian etc.

          • 28.5.1 The maximum of the asset and the average of the asset

          • 28.5.2 The average of the asset and the maximum of the average

          • 28.5.3 The maximum of the asset and the average of the maximum

        • 28.6 The volatility option

          • 28.6.1 The continuous-time limit

          • 28.6.2 Hedging variance swaps with vanilla options

        • 28.7 Correlation swap

          • 28.7.1 Dispersion trading

        • 28.8 Ladders

        • 28.9 Parisian options

          • 28.9.1 Examples

        • 28.10 Yet more exotics

        • 28.11 Summary

      • 29 Equity and FX Term Sheets

        • 29.1 Introduction

        • 29.2 Contingent premium put

        • 29.3 Basket options

          • 29.3.1 Simple basket option

          • 29.3.2 Basket option with averaging over time

        • 29.4 Knockout options

          • 29.4.1 Double knockout

          • 29.4.2 Instalment knockout

        • 29.5 Range notes

          • 29.5.1 A really simple range note

        • 29.6 Lookbacks

        • 29.7 Cliquet option

          • 29.7.1 Path dependency, constant volatility

        • 29.8 Passport options

        • 29.9 Decomposition of exotics into vanillas

    • PART THREE FIXED-INCOME MODELING AND DERIVATIVES

      • 30 One-factor Interest Rate Modeling

        • 30.1 Introduction

        • 30.2 Stochastic interest rates

        • 30.3 The bond pricing equation for the general model

        • 30.4 What is the market price of risk?

        • 30.5 Interpreting the market price of risk, and risk neutrality

        • 30.6 Tractable models and solutions of the bond pricing equation

        • 30.7 Solution for constant parameters

        • 30.8 Named models

          • 30.8.1 Vasicek

          • 30.8.2 Cox, Ingersoll & Ross

          • 30.8.3 Ho & Lee

          • 30.8.4 Hull & White

        • 30.9 Equity and FX forwards and futures when rates are stochastic

          • 30.9.1 Forward contracts

          • 30.9.2 Futures contracts

          • 30.9.3 The convexity adjustment

        • 30.10 Summary

      • 31 Yield Curve Fitting

        • 31.1 Introduction

        • 31.2 Ho & Lee

        • 31.3 The extended Vasicek model of Hull & White

        • 31.4 Yield-curve fitting: For and against

          • 31.4.1 For

          • 31.4.2 Against

        • 31.5 Other models

        • 31.6 Summary

      • 32 Interest Rate Derivatives

        • 32.1 Introduction

        • 32.2 Callable bonds

        • 32.3 Bond options

          • 32.3.1 Market practice

        • 32.4 Caps and floors

          • 32.4.1 Cap/floor parity

          • 32.4.2 The relationship between a caplet and a bond option

          • 32.4.3 Market practice

          • 32.4.4 Collars

          • 32.4.5 Step-up swaps, caps and .oors

        • 32.5 Range notes

        • 32.6 Swaptions, captions and floortions

          • 32.6.1 Market practice

        • 32.7 Spread options

        • 32.8 Index amortizing rate swaps

          • 32.8.1 Other features in the index amortizing rate swap

        • 32.9 Contracts with embedded decisions

        • 32.10 When the interest rate is not the spot rate

          • 32.10.1 The relationship between the spot interest rate and other rates

        • 32.11 Some examples

        • 32.12 More interest rate derivatives

        • 32.13 Summary

      • 33 Convertible Bonds

        • 33.1 Introduction

        • 33.2 Convertible bond basics

          • 33.2.1 What are CBs for?

          • 33.2.2 The issuers of CBs

          • 33.2.3 Why issue a convertible?

          • 33.2.4 Why buy a convertible?

          • 33.2.5 Some statistics

        • 33.3 Market practice

        • 33.4 Converts as options

        • 33.5 Pricing CBs with known interest rate

          • 33.5.1 Call and put features

        • 33.6 Two-factor modeling: Convertible bonds with stochastic interest rate

        • 33.7 A special model

        • 33.8 Path dependence in convertible bonds

        • 33.9 Dilution

        • 33.10 Credit risk issues

        • 33.11 Summary

      • 34 Mortgage-backed Securities

        • 34.1 Introduction

        • 34.2 Individual mortgages

          • 34.2.1 Monthly payments in the fixed rate mortgage

          • 34.2.2 Prepayment

        • 34.3 Mortgage-backed securities

          • 34.3.1 The issuers

        • 34.4 Modeling prepayment

          • 34.4.1 The statistics of repayment

          • 34.4.2 The PSA model

          • 34.4.3 More realistic models

        • 34.5 Valuing MBSs

        • 34.6 Summary

      • 35 Multi-factor Interest Rate Modeling

        • 35.1 Introduction

        • 35.2 Theoretical framework for two factors

          • 35.2.1 Special case: Modeling a long-term rate

          • 35.2.2 Special case: Modeling the spread between the long and the short rate

        • 35.3 Popular models

        • 35.4 The market price of risk as a random factor

        • 35.5 The phase plane in the absence of randomness

        • 35.6 The yield curve swap

        • 35.7 General multi-factor theory

          • 35.7.1 Tractable affine models

        • 35.8 Summary

      • 36 Empirical Behavior of the Spot Interest Rate

        • 36.1 Introduction

        • 36.2 Popular one-factor spot-rate models

        • 36.3 Implied modeling: One factor

        • 36.4 The volatility structure

        • 36.5 The drift structure

        • 36.6 The slope of the yield curve and the market price of risk

        • 36.7 What the slope of the yield curve tells us

        • 36.8 Properties of the forward rate curve ‘on average’

        • 36.9 Implied modeling: Two factor

        • 36.10 Summary

      • 37 The Heath, Jarrow & Morton and Brace, Gatarek & Musiela Models

        • 37.1 Introduction

        • 37.2 The forward rate equation

        • 37.3 The spot rate process

          • 37.3.1 The non-Markov nature of HJM

        • 37.4 The market price of risk

        • 37.5 Real and risk neutral

          • 37.5.1 The relationship between the risk-neutral forward rate drift and volatility

        • 37.6 Pricing derivatives

        • 37.7 Simulations

        • 37.8 Trees

        • 37.9 The Musiela parameterization

        • 37.10 Multi-factor HJM

        • 37.11 Spreadsheet implementation

        • 37.12 A simple one-factor example: Ho & Lee

        • 37.13 Principal Component Analysis

          • 37.13.1 The power method

        • 37.14 Options on equities etc.

        • 37.15 Non-in.nitesimal short rate

        • 37.16 The Brace, Gatarek and Musiela model

        • 37.17 Simulations

        • 37.18 PVing the cashflows

        • 37.19 Summary

      • 38 Fixed-income Term Sheets

        • 38.1 Introduction

        • 38.2 Chooser range note

          • 38.2.1 Optimal choice of ranges?

          • 38.2.2 Pricing

          • 38.2.3 Differences between optimal for the writer and the buyer

        • 38.3 Index amortizing rate swap

          • 38.3.1 Similarity solution

          • 38.3.2 The code

    • PART FOUR CREDIT RISK

      • 39 Value of the Firm and the Risk of Default

        • 39.1 Introduction

        • 39.2 The Merton model: Equity as an option on a company’s assets

          • 39.2.1 Default before maturity

          • 39.2.2 Probability of default

          • 39.2.3 Stochastic interest rates

        • 39.3 Modeling with measurable parameters and variables

        • 39.4 Calculating the value of the firm

        • 39.5 Summary

      • 40 Credit Risk

        • 40.1 Introduction

        • 40.2 Risky bonds

        • 40.3 Modeling the risk of default

        • 40.4 The Poisson process and the instantaneous risk of default

          • 40.4.1 A note on hedging

        • 40.5 Time-dependent intensity and the term structure of default

        • 40.6 Stochastic risk of default

        • 40.7 Positive recovery

        • 40.8 Special cases and yield curve .tting

        • 40.9 A case study: The Argentine Par bond

        • 40.10 Hedging the default

        • 40.11 Is there any information content in the market price?

          • 40.11.1 Implied hazard rate and duration

        • 40.12 Credit rating

        • 40.13 A model for change of credit rating

          • 40.13.1 The forward equation

          • 40.13.2 The backward equation

        • 40.14 The pricing equation

          • 40.14.1 Constant interest rates

          • 40.14.2 Stochastic interest rates

        • 40.15 Credit risk in CBs

          • 40.15.1 Bankruptcy when stock reaches a critical level

          • 40.15.2 Incorporating the instantaneous risk of default

        • 40.16 Modeling liquidity risk

        • 40.17 Summary

      • 41 Credit Derivatives

        • 41.1 Introduction

        • 41.2 What are Credit Derivatives?

          • 41.2.1 Uses of credit derivatives: Banks

          • 41.2.2 Uses of credit derivatives: Investors

          • 41.2.3 Uses of credit derivatives: Corporates

        • 41.3 Popular credit derivatives

          • 41.3.1 Asset swap

          • 41.3.2 Total return swaps or total rate of return swaps

        • 41.4 Derivatives triggered by default

          • 41.4.1 Basic definitions

          • 41.4.2 What defines a ‘credit event’?

          • 41.4.3 Credit default swap

          • 41.4.4 Limited recourse note

          • 41.4.5 First to default

          • 41.4.6 nth to default

        • 41.5 Derivatives of the yield spread

          • 41.5.1 Default calls and puts

          • 41.5.2 Exchange options

          • 41.5.3 Credit spread options

        • 41.6 Payment on change of rating

        • 41.7 Using default swaps in CB arbitrage

          • 41.7.1 Exploiting your credit risk view

        • 41.8 Term sheets

          • 41.8.1 Put on credit spread on XYZ bond

          • 41.8.2 Binary payoff bond

          • 41.8.3 Digital spread option, one-year note linked to Venezuelan bond

          • 41.8.4 Basket credit-linked note

          • 41.8.5 Asset swap put option on one year ABC 6.65% bonds 10/2006 asset swap

        • 41.9 Pricing credit derivatives

        • 41.10 An exchange option

        • 41.11 Default only when payment is due

          • 41.11.1 The market’s estimate of default risk

          • 41.11.2 Hedging

        • 41.12 Payoff on change of rating

        • 41.13 Multi-factor derivatives

        • 41.14 Copulas: Pricing credit derivatives with many underlyings

          • 41.14.1 The copula function

          • 41.14.2 The mathematical definition

          • 41.14.3 Examples of copulas

        • 41.15 Collateralized Debt Obligations

        • 41.16 Summary

      • 42 RiskMetrics and CreditMetrics

        • 42.1 Introduction

        • 42.2 The RiskMetrics datasets

        • 42.3 Calculating the parameters the RiskMetrics way

          • 42.3.1 Estimating volatility

          • 42.3.2 Correlation

        • 42.4 The CreditMetrics dataset

          • 42.4.1 Yield curves

          • 42.4.2 Spreads

          • 42.4.3 Transition matrices

          • 42.4.4 Correlations

        • 42.5 The CreditMetrics methodology

        • 42.6 A portfolio of risky bonds

        • 42.7 CreditMetrics model outputs

        • 42.8 Summary

      • 43 CrashMetrics

        • 43.1 Introduction

        • 43.2 Why do banks go broke?

        • 43.3 Market crashes

        • 43.4 CrashMetrics

        • 43.5 CrashMetrics for one stock

        • 43.6 Portfolio optimization and the Platinum Hedge

          • 43.6.1 Other ‘cost’ functions

        • 43.7 The multi-asset/single-index model

          • 43.7.1 Assuming Taylor series for the moment

        • 43.8 Portfolio optimization and the Platinum Hedge in the multi-asset model

          • 43.8.1 The marginal effect of an asset

        • 43.9 The multi-index model

        • 43.10 Incorporating time value

        • 43.11 Margin calls and margin hedging

          • 43.11.1 What is margin?

          • 43.11.2 Modeling margin

          • 43.11.3 The single-index model

        • 43.12 Counterparty risk

        • 43.13 Simple extensions to CrashMetrics

        • 43.14 The CrashMetrics Index (CMI)

        • 43.15 Summary

      • 44 Derivatives **** Ups

        • 44.1 Introduction

        • 44.2 Orange County

        • 44.3 Proctor and Gamble

        • 44.4 Metallgesellschaft

          • 44.4.1 Basis risk

        • 44.5 Gibson Greetings

        • 44.6 Barings

        • 44.7 Long-Term Capital Management

        • 44.8 Summary

    • PART FIVE ADVANCED TOPICS

      • 45 Financial Modeling

        • 45.1 Introduction

        • 45.2 Warning: Modeling as it is currently practiced

          • 45.2.1 Models, a personal view

          • 45.2.2 The find-and-replace school of mathematical modeling

        • 45.3 Summary

      • 46 Defects in the Black–Scholes Model

        • 46.1 Introduction

        • 46.2 Discrete hedging

        • 46.3 Transaction costs

        • 46.4 Overview of volatility modeling

        • 46.5 Deterministic volatility surfaces

        • 46.6 Stochastic volatility

        • 46.7 Uncertain parameters

        • 46.8 Empirical analysis of volatility

        • 46.9 Stochastic volatility and mean-variance analysis

        • 46.10 Asymptotic analysis of volatility

        • 46.11 Jump diffusion

        • 46.12 Crash modeling

        • 46.13 Speculating with options

        • 46.14 Optimal static hedging

        • 46.15 The feedback effect of hedging in illiquid markets

        • 46.16 Utility theory

        • 46.17 More about American options and related matters

        • 46.18 Advanced dividend modeling

        • 46.19 Serial autocorrelation in returns

        • 46.20 Summary

      • 47 Discrete Hedging

        • 47.1 Introduction

        • 47.2 Motivating example: The trinomial model

        • 47.3 A model for a discretely hedged position

        • 47.4 A higher-order analysis

          • 47.4.1 Choosing the best

          • 47.4.2 The hedging error

          • 47.4.3 Observations about the hedging error

          • 47.4.4 Pricing the option

          • 47.4.5 The adjusted and option value

        • 47.5 The real distribution of returns and the hedging error

        • 47.6 Total hedging error for the real distribution of returns

        • 47.7 Which models allow perfect delta hedging

        • 47.8 Summary

      • 48 Transaction Costs

        • 48.1 Introduction

        • 48.2 The effect of costs

        • 48.3 The model of Leland (1985)

        • 48.4 The model of Hoggard, Whalley & Wilmott (1992)

        • 48.5 Non-single-signed gamma

        • 48.6 The marginal effect of transaction costs

        • 48.7 Other cost structures

        • 48.8 Hedging to a bandwidth: The model of Whalley & Wilmott (1993) and Henrotte (1993)

        • 48.9 Utility-based models

          • 48.9.1 The model of Hodges & Neuberger (1989)

          • 48.9.2 The model of Davis, Panas & Zariphopoulou (1993)

          • 48.9.3 The asymptotic analysis of Whalley & Wilmott (1993)

          • 48.9.4 Arbitrary cost structure

        • 48.10 Interpretation of the models

          • 48.10.1 Nonlinearity

          • 48.10.2 Negative option prices

          • 48.10.3 Existence of solutions

        • 48.11 Non-normal returns

        • 48.12 Empirical testing

          • 48.12.1 Black–Scholes and Leland hedging

          • 48.12.2 Market movement or delta-tolerance strategy

          • 48.12.3 The utility strategy

          • 48.12.4 Using the real data

          • 48.12.5 And the winner is. . .

        • 48.13 Transaction costs and discrete hedging put together

        • 48.14 Summary

      • 49 Overview of Volatility Modeling

        • 49.1 Introduction

        • 49.2 The different types of volatility

          • 49.2.1 Actual volatility

          • 49.2.2 Historical or realized volatility

          • 49.2.3 Implied volatility

          • 49.2.4 Forward volatility

        • 49.3 Volatility estimation by statistical means

          • 49.3.1 The simplest volatility estimate: Constant volatility/moving window

          • 49.3.2 Incorporating mean reversion

          • 49.3.3 Exponentially weighted moving average

          • 49.3.4 A simple GARCH model

          • 49.3.5 Expected future volatility

          • 49.3.6 Beyond close-close estimators: Range-based estimation of volatility

        • 49.4 Maximum likelihood estimation

          • 49.4.1 A simple motivating example: Taxi numbers

          • 49.4.2 Three hats

          • 49.4.3 The maths behind this: Find the standard deviation

          • 49.4.4 Quants’ salaries

        • 49.5 Skews and smiles

          • 49.5.1 Sensitivity of the straddle to skews and smiles

          • 49.5.2 Sensitivity of the risk reversal to skews and smiles

        • 49.6 Different approaches to modeling volatility

          • 49.6.1 To calibrate or not?

          • 49.6.2 Deterministic volatility surfaces

          • 49.6.3 Stochastic volatility

          • 49.6.4 Uncertain parameters

          • 49.6.5 Empirical analysis of volatility

          • 49.6.6 Static hedging

          • 49.6.7 Stochastic volatility and mean-variance analysis

          • 49.6.8 Asymptotic analysis of volatility

          • 49.6.9 Volatility case study: The cliquet option

        • 49.7 The choices of volatility models

        • 49.8 Summary

      • 50 Deterministic Volatility Surfaces

        • 50.1 Introduction

        • 50.2 Implied volatility

        • 50.3 Time-dependent volatility

        • 50.4 Volatility smiles and skews

        • 50.5 Volatility surfaces

        • 50.6 Backing out the local volatility surface from European call option prices

        • 50.7 A simple volatility surface parameterization

        • 50.8 An approximate solution

        • 50.9 Volatility information contained in an at-the-money straddle

        • 50.10 Volatility information contained in a risk-reversal

        • 50.11 Time dependence

        • 50.12 A market convention

        • 50.13 How do I use the local volatility surface?

        • 50.14 Summary

      • 51 Stochastic Volatility

        • 51.1 Introduction

        • 51.2 Random volatility

        • 51.3 A stochastic differential equation for volatility

        • 51.4 The pricing equation

        • 51.5 The market price of volatility risk

          • 51.5.1 Aside: The market price of risk for traded assets

        • 51.6 The value as an expectation

        • 51.7 An example

        • 51.8 Choosing the model

        • 51.9 Named/popular models

          • 51.9.1 The Heston model

          • 51.9.2 The REGARCH model and its diffusion limit

        • 51.10 A note on biases

        • 51.11 Stochastic implied volatility: The model of Sch¨onbucher

        • 51.12 Summary

      • 52 Uncertain Parameters

        • 52.1 Introduction

        • 52.2 Best and worst cases

          • 52.2.1 Uncertain volatility: The model of Avellaneda, Levy & Par´as (1995) and Lyons (1995)

          • 52.2.2 Example: An up-and-out call

          • 52.2.3 Uncertain interest rate

          • 52.2.4 Uncertain dividends

        • 52.3 Uncertain correlation

        • 52.4 Nonlinearity

        • 52.5 Summary

      • 53 Empirical Analysis of Volatility

        • 53.1 Introduction

        • 53.2 Stochastic volatility and uncertain parameters revisited

        • 53.3 Deriving an empirical stochastic volatility model

        • 53.4 Estimating the volatility of volatility

        • 53.5 Estimating the drift of volatility

        • 53.6 Out-of-sample results

        • 53.7 How to use the model

          • 53.7.1 Option pricing with stochastic volatility

          • 53.7.2 The time evolution of stochastic volatility

          • 53.7.3 Stochastic volatility, certainty bands and confidence limits

        • 53.8 Summary

      • 54 Stochastic Volatility and Mean-variance Analysis

        • 54.1 Introduction

        • 54.2 The model for the asset and its volatility

        • 54.3 Analysis of the mean

        • 54.4 Analysis of the variance

        • 54.5 Choosing to minimize the variance

        • 54.6 The mean and variance equations

        • 54.7 How to interpret and use the mean and variance

        • 54.8 Static hedging and portfolio optimization

        • 54.9 Example: Valuing and hedging an up-and-out call

        • 54.10 Static hedging

        • 54.11 Other definitions of ‘value’

        • 54.12 Summary

      • 55 Asymptotic Analysis of Volatility

        • 55.1 Introduction

        • 55.2 Fast mean reversion and high volatility of volatility

        • 55.3 Conditions on the models

        • 55.4 Examples of models

          • 55.4.1 Scott’s model

          • 55.4.2 The Heston/Ball–Roma model

        • 55.5 Notation

        • 55.6 Asymptotic analysis

        • 55.7 Vanilla options: Asymptotics for values

        • 55.8 Vanilla options: Implied volatilities

        • 55.9 Summary

      • 56 Volatility Case Study: The Cliquet Option

        • 56.1 Introduction

        • 56.2 The subtle nature of the cliquet option

        • 56.3 Path dependency, constant volatility

        • 56.4 Results

          • 56.4.1 Constant volatility

          • 56.4.2 Uncertain volatility

        • 56.5 Code: Cliquet with uncertain volatility, in similarity variables

        • 56.6 Summary

      • 57 Jump Diffusion

        • 57.1 Introduction

        • 57.2 Evidence for jumps

        • 57.3 Poisson processes

        • 57.4 Hedging when there are jumps

        • 57.5 Hedging the diffusion

        • 57.6 Hedging the jumps

        • 57.7 Hedging the jumps and risk neutrality

        • 57.8 The downside of jump-diffusion models

        • 57.9 Jump volatility

        • 57.10 Jump volatility with deterministic decay

        • 57.11 Summary

      • 58 Crash Modeling

        • 58.1 Introduction

        • 58.2 Value at risk

        • 58.3 A simple example: The hedged call

        • 58.4 A mathematical model for a crash

          • 58.4.1 Case I: Black–Scholes Hedging

          • 58.4.2 Case II: Crash Hedging

        • 58.5 An example

        • 58.6 Optimal static hedging: VaR reduction

        • 58.7 Continuous-time limit

        • 58.8 A range for the crash

        • 58.9 Multiple crashes

          • 58.9.1 Limiting the total number of crashes

          • 58.9.2 Limiting the frequency of crashes

        • 58.10 Crashes in a multi-asset world

        • 58.11 Fixed and floating exchange rates

        • 58.12 Summary

      • 59 Speculating with Options

        • 59.1 Introduction

        • 59.2 A simple model for the value of an option to a speculator

          • 59.2.1 The present value of expected payoff

          • 59.2.2 Standard deviation

        • 59.3 More sophisticated models for the return on an asset

          • 59.3.1 Diffusive drift

          • 59.3.2 Jump drift

        • 59.4 Early closing

        • 59.5 To hedge or not to hedge?

        • 59.6 Other issues

        • 59.7 Summary

      • 60 Static Hedging

        • 60.1 Introduction

        • 60.2 Static replicating portfolio

        • 60.3 Matching a ‘target’ contract

        • 60.4 Vega matching

        • 60.5 Static hedging: Non-linear governing equation

        • 60.6 Non-linear equations

        • 60.7 Pricing with a non-linear equation

          • 60.7.1 Example: Non-linear model, unhedged

          • 60.7.2 Static hedging: A first attempt

          • 60.7.3 Static hedging: The best hedge

        • 60.8 Optimal static hedging: The theory

        • 60.9 Calibration?

        • 60.10 Hedging path-dependent options with vanilla options, non-linear models

          • 60.10.1 American options

          • 60.10.2 Barrier options

          • 60.10.3 Pricing and optimally hedging a portfolio of barrier options

        • 60.11 The mathematics of optimization

          • 60.11.1 Downhill simplex method

          • 60.11.2 Simulated annealing

        • 60.12 Summary

      • 61 The Feedback Effect of Hedging in Illiquid Markets

        • 61.1 Introduction

        • 61.2 The trading strategy for option replication

        • 61.3 The excess demand function

        • 61.4 Incorporating the trading strategy

        • 61.5 The influence of replication

        • 61.6 The forward equation

          • 61.6.1 The boundaries

        • 61.7 Numerical results

          • 61.7.1 Time-independent trading strategy

          • 61.7.2 Put replication trading

        • 61.8 Attraction and repulsion

        • 61.9 Summary

      • 62 Utility Theory

        • 62.1 Introduction

        • 62.2 Ranking events

        • 62.3 The utility function

        • 62.4 Risk aversion

        • 62.5 Special utility functions

        • 62.6 Certainty equivalent wealth

        • 62.7 Maximization of expected utility

          • 62.7.1 Ordinal and cardinal utility

        • 62.8 Summary

      • 63 More About American Options and Related Matters

        • 63.1 Introduction

        • 63.2 What Derivatives Week published

        • 63.3 Hold these thoughts

        • 63.4 Change of notation

        • 63.5 And finally, the paper . . .

        • 63.6 Introduction

        • 63.7 Preliminary: Pricing and Hedging

        • 63.8 Utility-Maximizing Exercise Time

          • 63.8.1 Constant absolute risk aversion

          • 63.8.2 Hyperbolic absolute risk aversion

          • 63.8.3 The expected return

        • 63.9 Profit from Selling American Options

        • 63.10 Concluding Remarks

        • 63.11 Who wins and who loses?

        • 63.12 FAQ

        • 63.13 Another situation where the same idea applies: Passport options

          • 63.13.1 Recap

          • 63.13.2 Utility maximization in the passport option

        • 63.14 Summary

      • 64 Advanced Dividend Modeling

        • 64.1 Introduction

        • 64.2 Why do we need dividend models?

        • 64.3 Effects of dividends on asset prices

          • 64.3.1 Market frictions

          • 64.3.2 Term structure of dividends

        • 64.4 Stochastic dividends

        • 64.5 Poisson jumps

        • 64.6 Uncertainty in dividend amount and timing

        • 64.7 Summary

      • 65 Serial Autocorrelation in Returns

        • 65.1 Introduction

        • 65.2 Evidence

        • 65.3 The Telegraph equation

        • 65.4 Pricing and hedging derivatives

        • 65.5 Summary

      • 66 Asset Allocation in Continuous Time

        • 66.1 Introduction

        • 66.2 One risk-free and one risky asset

          • 66.2.1 The wealth process

          • 66.2.2 Maximizing expected utility

          • 66.2.3 Stochastic control and the Bellman equation

          • 66.2.4 Constant relative risk aversion

          • 66.2.5 Constant absolute risk aversion

        • 66.3 Many assets

        • 66.4 Maximizing long-term growth

        • 66.5 A brief look at transaction costs

        • 66.6 Summary

      • 67 Asset Allocation under Threat of a Crash

        • 67.1 Introduction

        • 67.2 Optimal Portfolios under the Threat of a Crash: The single stock case

        • 67.3 Maximizing Growth Rate under the Threat of a Crash: n stocks

        • 67.4 Maximizing Growth Rate under the Threat of a Crash: An arbitrary number of crashes and other refinements

          • 67.4.1 Arbitrary upper bound for the number of crashes

          • 67.4.2 Changing volatility after a crash

          • 67.4.3 Further possible refinements

        • 67.5 Summary

      • 68 Interest-rate Modeling Without Probabilities

        • 68.1 Introduction

        • 68.2 What do I want from an interest rate model?

        • 68.3 A non-probabilistic model for the behavior of the short-term interest rate

        • 68.4 Worst-case scenarios and a non-linear equation

          • 68.4.1 Let’s see that again in slow motion

        • 68.5 Examples of hedging: Spreads for prices

          • 68.5.1 Hedging with one instrument

          • 68.5.2 Hedging with multiple instruments

        • 68.6 Generating the ‘Yield Envelope’

        • 68.7 Swaps

        • 68.8 Caps and floors

        • 68.9 Applications of the model

          • 68.9.1 Identifying arbitrage opportunities

          • 68.9.2 Establishing prices for the market maker

          • 68.9.3 Static hedging to reduce interest rate risk

          • 68.9.4 Risk management: A measure of absolute loss

          • 68.9.5 A remark on the validity of the model

        • 68.10 Summary

      • 69 Pricing and Optimal Hedging of Derivatives, the Non-probabilistic Model Cont’d

        • 69.1 Introduction

        • 69.2 A real portfolio

        • 69.3 Bond options

          • 69.3.1 Pricing the European option on a zero-coupon bond

          • 69.3.2 Pricing and hedging American options

        • 69.4 Contracts with embedded decisions

        • 69.5 The index amortizing rate swap

        • 69.6 Convertible bonds

        • 69.7 Summary

      • 70 Extensions to the Non-probabilistic Interest-rate Model

        • 70.1 Introduction

        • 70.2 Fitting forward rates

        • 70.3 Economic cycles

        • 70.4 Interest rate bands

          • 70.4.1 Estimating from past data

        • 70.5 Crash modeling

          • 70.5.1 A maximum number of crashes

          • 70.5.2 A maximum frequency of crashes

          • 70.5.3 Estimating from past data

        • 70.6 Liquidity

        • 70.7 Summary

      • 71 Modeling Inflation

        • 71.1 Introduction

        • 71.2 Inflation-linked products

          • 71.2.1 Bonds

          • 71.2.2 Inflation caps and floors

          • 71.2.3 Swaps

          • 71.2.4 Barriers

        • 71.3 Pricing, first thoughts

        • 71.4 What the data tell us

        • 71.5 Pricing, second thoughts

        • 71.6 Analyzing the data

        • 71.7 Can we model inflation independently of interest rates?

        • 71.8 Calibration and market price of risk

        • 71.9 Non-linear pricing methods

        • 71.10 Summary

      • 72 Energy Derivatives

        • 72.1 Introduction

        • 72.2 The energy market

        • 72.3 What’s so special about the energy markets?

        • 72.4 Why can’t we apply Black–Scholes theory to energy derivatives?

        • 72.5 The convenience yield

        • 72.6 The Pilopovi´c two-factor model

          • 72.6.1 Fitting

        • 72.7 Energy derivatives

          • 72.7.1 One-day options

          • 72.7.2 Asian options

          • 72.7.3 Caps and floors

          • 72.7.4 Cheapest to deliver

          • 72.7.5 Basis spreads

          • 72.7.6 Swing options

          • 72.7.7 Spread options

        • 72.8 Summary

      • 73 Real Options

        • 73.1 Introduction

        • 73.2 Financial options and Real options

        • 73.3 An introductory example: Abandonment of a machine

        • 73.4 Optimal investment: Simple example #2

        • 73.5 Temporary suspension of the project, costless

        • 73.6 Temporary suspension of the project, costly

        • 73.7 Sequential and incremental investment

        • 73.8 Ashanti: Gold mine case study

        • 73.9 Summary

      • 74 Life Settlements and Viaticals

        • 74.1 Introduction

        • 74.2 Life expectancy

          • 74.2.1 Sex

          • 74.2.2 Health

        • 74.3 Actuarial tables

        • 74.4 Death seen as default

        • 74.5 Pricing a single policy

          • 74.5.1 Internal rate of return

        • 74.6 Pricing portfolios

          • 74.6.1 Extension risk

        • 74.7 Summary

      • 75 Bonus Time

        • 75.1 Introduction

        • 75.2 One bonus period

          • 75.2.1 Bonus depending on the Sharpe ratio

          • 75.2.2 Numerical results

        • 75.3 The skill factor

        • 75.4 Putting skill into the equation

          • 75.4.1 Example

        • 75.5 Summary

    • PART SIX NUMERICAL METHODS AND PROGRAMS

      • 76 Overview of Numerical Methods

        • 76.1 Introduction

        • 76.2 Finite-difference methods

          • 76.2.1 Efficiency

          • 76.2.2 Program of study

        • 76.3 Monte Carlo methods

          • 76.3.1 Efficiency

          • 76.3.2 Program of study

        • 76.4 Numerical integration

          • 76.4.1 Efficiency

          • 76.4.2 Program of study

        • 76.5 Summary

      • 77 Finite-difference Methods for One-factor Models

        • 77.1 Introduction

        • 77.2 Overview

        • 77.3 Grids

        • 77.4 Differentiation using the grid

        • 77.5 Approximating θ

        • 77.6 Approximating

          • 77.6.1 One-sided differences

        • 77.7 Approximating

        • 77.8 Example

        • 77.9 Final conditions and payoffs

        • 77.10 Boundary conditions

          • 77.10.1 Other boundary conditions

        • 77.11 The explicit finite-difference method

          • 77.11.1 The Black–Scholes equation

        • 77.12 Convergence of the explicit method

        • 77.13 The Code # 1: European option

        • 77.14 The Code # 2: American exercise

        • 77.15 The Code # 3: 2-D output

        • 77.16 Bilinear interpolation

        • 77.17 Upwind differencing

        • 77.18 Summary

      • 78 Further Finite-difference Methods for One-factor Models

        • 78.1 Introduction

        • 78.2 Implicit finite-difference methods

        • 78.3 The Crank–Nicolson method

          • 78.3.1 Boundary condition type I: V k+1 0 given

          • 78.3.2 Boundary condition type II: Relationship between V k+1 0 and V k+ 1

          • 78.3.3 Boundary condition type III: ∂2V/∂S2 = 0

          • 78.3.4 The matrix equation

          • 78.3.5 LU decomposition

          • 78.3.6 Successive over-relaxation, SOR

          • 78.3.7 Optimal choice of ω

        • 78.4 Comparison of finite-difference methods

        • 78.5 Other methods

        • 78.6 Douglas schemes

        • 78.7 Three time-level methods

        • 78.8 Richardson extrapolation

        • 78.9 Free boundary problems and American options

          • 78.9.1 Early exercise and the explicit method

          • 78.9.2 Early exercise and Crank–Nicolson

        • 78.10 Jump conditions

          • 78.10.1 A discrete cashflow

          • 78.10.2 Discretely paid dividend

        • 78.11 Path-dependent options

          • 78.11.1 Discretely sampled quantities

          • 78.11.2 Continuously sampled quantities

        • 78.12 Summary

      • 79 Finite-difference Methods for Two-factor Models

        • 79.1 Introduction

        • 79.2 Two-factor models

        • 79.3 The explicit method

          • 79.3.1 Stability of the explicit method

        • 79.4 Calculation time

        • 79.5 Alternating Direction Implicit

        • 79.6 The Hopscotch method

        • 79.7 Summary

      • 80 Monte Carlo Simulation

        • 80.1 Introduction

        • 80.2 Relationship between derivative values and simulations: Equities, indices, currencies, commodities

        • 80.3 Generating paths

        • 80.4 Lognormal underlying, no path dependency

        • 80.5 Advantages of Monte Carlo simulation

        • 80.6 Using random numbers

        • 80.7 Generating normal variables

          • 80.7.1 Box–Muller

        • 80.8 Real versus risk neutral, speculation versus hedging

        • 80.9 Interest rate products

        • 80.10 Calculating the greeks

        • 80.11 Higher dimensions: Cholesky factorization

        • 80.12 Calculation time

        • 80.13 Speeding up convergence

          • 80.13.1 Antithetic variables

          • 80.13.2 Control variate technique

        • 80.14 Pros and cons of Monte Carlo simulations

        • 80.15 American options

        • 80.16 Longstaff & Schwartz regression approach for American options

        • 80.17 Basis Functions

        • 80.18 Summary

      • 81 Numerical Integration

        • 81.1 Introduction

        • 81.2 Regular grid

        • 81.3 Basic Monte Carlo integration

        • 81.4 Low-discrepancy sequences

        • 81.5 Advanced techniques

        • 81.6 Summary

      • 82 Finite-difference Programs

        • 82.1 Introduction

        • 82.2 Kolmogorov equation

        • 82.3 Explicit one-factor model for a convertible bond

        • 82.4 American call, implicit

        • 82.5 Explicit Parisian option

        • 82.6 Passport options

        • 82.7 Chooser passport option

        • 82.8 Explicit stochastic volatility

        • 82.9 Uncertain volatility

        • 82.10 Crash modeling

        • 82.11 Explicit Epstein–Wilmott solution

        • 82.12 Risky-bond calculator

      • 83 Monte Carlo Programs

        • 83.1 Introduction

        • 83.2 Monte Carlo pricing of a basket

        • 83.3 Quasi Monte Carlo pricing of a basket

        • 83.4 Monte Carlo for American options

    • Appendix A All the Math You Need. . . and No More (An Executive Summary)

      • A.1 Introduction

      • A.2 The different types of mathematics found in finance

        • A.2.1 Modeling approaches

        • A.2.2 The tools

      • A.3 e

      • A.4 log

      • A.5 Differentiation and Taylor series

      • A.6 Expectation and variance

      • A.7 Another look at Black–Scholes

      • A.8 Summary

    • Bibliography

    • Index

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