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