Martingale methods in financial modelling

Stochastic Mechanics Random Media Signal Processing and Image Synthesis Mathematical Economics and Finance Stochastic Optimization Stochastic Control Stochastic Models in Life Sciences Stochastic Modelling and Applied Probability (Formerly: Applications of Mathematics) 36 Edited by B Rozovski˘ı G Grimmett Advisory Board D Dawson D Geman I Karatzas F Kelly Y Le Jan B Øksendal G Papanicolaou E Pardoux Marek Musiela Marek Rutkowski Martingale Methods in Financial Modelling Second Edition Authors Marek Musiela BNP Paribas 10 Harewood Avenue London NW1 6AA UK marek.musiela@bnpparibas.com Marek Rutkowski Technical University Warszawa Inst Mathematics Pl Politechniki 00-661 Warszawa Poland markrut@alpha.mini.pw.edu.pl Managing Editors B Rozovski˘ı Division of Applied Mathematics Brown University 182 George St Providence, RI 02912 USA rozovsky@dam.brown.edu G Grimmett Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB UK g.r.grimmett@statslab.cam.ac.uk Cover illustration: Cover pattern courtesy of Rick Durrett, Cornell University, Ithaca, New York ISBN 978-3-540-20966-9 DOI 10.1007/978-3-540-26653-2 e-ISBN 978-3-540-26653-2 Stochastic Modelling and Applied Probability ISSN 0172-4568 Library of Congress Control Number: 2004114482 Mathematics Subject Classification (2000): 60Hxx, 62P05, 90A09 2nd ed 2005 Corr 3rd printing 2009 © 2005, 1997 Springer-Verlag Berlin Heidelberg This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper springer.com Preface to the Second Edition During the seven years that elapsed between the first and second editions of the present book, considerable progress was achieved in the area of financial modelling and pricing of derivatives Needless to say, it was our intention to incorporate into the second edition at least the most relevant and commonly accepted of these developments Since at the same time we had the strong intention not to expand the book to an unbearable size, we decided to leave out from the first edition of this book some portions of material of lesser practical importance Let us stress that we have only taken out few sections that, in our opinion, were of marginal importance for the understanding of the fundamental principles of financial modelling of arbitrage valuation of derivatives In view of the abundance of new results in the area, it would be in any case unimaginable to cover all existing approaches to pricing and hedging financial derivatives (not to mention all important results) in a single book, no matter how voluminous it were Hence, several intensively studied areas, such as: mean-variance hedging, utility-based pricing, entropybased approach, financial models with frictions (e.g., short-selling constraints, bidask spreads, transaction costs, etc.) either remain unmentioned in this text, or are presented very succinctly Although the issue of market incompleteness is not totally neglected, it is examined primarily in the framework of models of stochastic (or uncertain) volatility Luckily enough, the afore-mentioned approaches and results are covered exhaustively in several excellent monographs written in recent years by our distinguished colleagues, and thus it is our pleasure to be able to refer the interested reader to these texts Let us comment briefly on the content of the second edition and the differences with respect to the first edition Part I was modified to a lesser extent and thus is not very dissimilar to Part I in the first edition However, since, as was mentioned already, some sections from the first edition were deliberately taken out, we decided for the sake of better readability to merge some chapters Also, we included in Part I a new chapter entirely devoted to volatility risk and related modelling issues As a consequence, the issues of hedging of plain-vanilla options and valuation of exotic options are no longer limited to the classical Black-Scholes framework with constant volatility The theme of stochastic volatility also reappears systematically in the second part of the book vi Preface to the Second Edition Part II has been substantially revised and thus its new version constitutes a major improvement of the present edition with respect to the first one We present there alternative interest rate models, and we provide the reader with an analysis of each of them, which is very much more detailed than in the first edition Although we did not even try to appraise the efficiency of real-life implementations of each approach, we have stressed on each occasion that, when dealing with derivatives pricing models, one should always have in mind a specific practical perspective Put another way, we advocate the opinion, put forward by many researchers, that the choice of model should be tied to observed real features of a particular sector of the financial market or even a product class Consequently, a necessary first step in modelling is a detailed study of functioning of a given market we wish to model The goal of this preliminary stage is to become familiar with existing liquid primary and derivative assets (together with their sometimes complex specifications), and to identify sources of risks associated with trading in these instruments It was our hope that by concentrating on the most pertinent and widely accepted modelling approaches, we will be able to provide the reader with a text focused on practical aspects of financial modelling, rather than theoretical ones We leave it, of course, to the reader to assess whether we have succeeded achieving this goal to a satisfactory level Marek Rutkowski expresses his gratitude to Marek Musiela and the members of the Fixed Income Research and Strategies Team at BNP Paribas for their hospitality during his numerous visits to London Marek Rutkowski gratefully acknowledges partial support received from the Polish State Committee for Scientific Research under grant PBZ-KBN-016/P03/1999 We would like to express our gratitude to the staff of Springer-Verlag We thank Catriona Byrne for her encouragement and invaluable editorial supervision, as well as Susanne Denskus for her invaluable technical assistance London and Sydney September 2004 Marek Musiela Marek Rutkowski Note on the Second Printing The second printing of the second edition of this book expands and clarifies further its contents exposition Several proofs previously left to the reader are now included The presentation of LIBOR and swap market models is expanded to include the joint dynamics of the underlying processes under the relevant probability measures The appendix in completed with several frequently used theoretical results making the book even more self-contained The bibliographical references are brought up to date as far as possible This printing corrects also numerous typographical errors and mistakes We would like the express our gratitude to Alan Bain and Imanuel Costigan who uncovered many of them London and Sydney August 2006 Marek Musiela Marek Rutkowski Preface to the First Edition The origin of this book can be traced to courses on financial mathematics taught by us at the University of New South Wales in Sydney, Warsaw University of Technology (Politechnika Warszawska) and Institut National Polytechnique de Grenoble Our initial aim was to write a short text around the material used in two one-semester graduate courses attended by students with diverse disciplinary backgrounds (mathematics, physics, computer science, engineering, economics and commerce) The anticipated diversity of potential readers explains the somewhat unusual way in which the book is written It starts at a very elementary mathematical level and does not assume any prior knowledge of financial markets Later, it develops into a text which requires some familiarity with concepts of stochastic calculus (the basic relevant notions and results are collected in the appendix) Over time, what was meant to be a short text acquired a life of its own and started to grow The final version can be used as a textbook for three one-semester courses – one at undergraduate level, the other two as graduate courses The first part of the book deals with the more classical concepts and results of arbitrage pricing theory, developed over the last thirty years and currently widely applied in financial markets The second part, devoted to interest rate modelling is more subjective and thus less standard A concise survey of short-term interest rate models is presented However, the special emphasis is put on recently developed models built upon market interest rates We are grateful to the Australian Research Council for providing partial financial support throughout the development of this book We would like to thank Alan Brace, Ben Goldys, Dieter Sondermann, Erik Schlögl, Lutz Schlögl, Alexander Mürmann, and Alexander Zilberman, who offered useful comments on the first draft, and Barry Gordon, who helped with editing Our hope is that this book will help to bring the mathematical and financial communities closer together, by introducing mathematicians to some important problems arising 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Theor Appl Finance 1, 289–310 Index Accrual period of a caplet, 477 of a swap, 518 Actual probability, 10 Adapted process, 35, 42 Affine term structure, 390 American claim arbitrage price, 88 buyer’s price, 100 definition, 88 payoff process, 88 rational exercise time, 98 seller’s price, 97 super-hedging, 97 American option continuation region, 216 Cox-Ross-Rubinstein model, 58–65 critical stock price, 217 discrete time, 89, 101 dividend-paying stock, 224 early exercise premium, 217 free boundary problem, 219 Geske-Johnson method, 222 hedging, 212 method of lines, 224 one-period case, 27–31 perpetual put, 220 price (premium), rational exercise time, 215 reward function, 209 smooth fit principle, 220 stopping region, 216 variational inequality, 221 Appreciation rate, 114, 131 Arbitrage free, 14, 25, 69, 82, 210, 318 long or short, 210 opportunity, 14, 69, 82, 317 price, 15, 25, 70, 124, 320, 377, 445 Asian option, see exotic option Asset, see security Average option, see Asian option B-cap, see cap, bounded Bachelier equation, 158 formula, 156 model, 155 sensitivity, 158 Backward SDE, 345 Barrier option, see exotic option Basis, 359 Bayes formula, 615 BDT, see Black-Derman-Toy Black-Derman-Toy model, 398 Black-Karasinski model, 398 Black-Scholes equation, 127 formula, 54, 126 model completeness, 335 mean-variance hedging, 337 multidimensional, 333 one-dimensional, 114 risk-minimizing hedging, 338–345 robustness, 168 708 Index transaction costs, 347 sensitivity, 141 Black’s equation, 164 formula, 161 model, 161 Bond callable, 444 convertible, 102, 111 coupon-bearing, 354 defaultable, 607 discount, see zero-coupon equivalent yield, 357 face value, see principal maturity, 351 principal, 351 zero-coupon, 351 Brace-G¸atarek-Musiela model, see model of LIBOR Brownian motion definition, 617 forward, 376 functionals of, 650 geometric, 114 Lévy’s characterization of, 633 local time, 632 maximum of, 650 multidimensional, 631 natural filtration, 117, 644 one-dimensional, 617 quadratic variation, 618 reflection principle, 650 BSDE, see backward SDE Buy-and-hold strategy, 209 Cap, 477 basket, 602 bounded, 481 caplet, 477 cumulative, 481 dual strike, 481 quanto, 604 valuation formula Flesaker-Hughston model, 570 Gaussian HJM model, 480 lognormal LIBOR model, 506 market, 484 Caplet, see cap Caption, 483 CBOT, Ceiling rate agreement, see cap CEV, see model of stock price Change of measure, see Girsanov’s theorem Change of variable, see Itô’s formula Cheapest-to-deliver bond, 359 Chooser option, see exotic option CIR, see Cox-Ingersoll-Ross CME, CMS, see swap, constant maturity Combined option, see exotic option Completeness CRR model, 42 finite market, 75 Compound option, see exotic option Conditional expectation, 612 probability, 612 Consol, 404 Contingent claim American, 30, 62, 88, 208 attainable, 14, 42, 69 European, 14, 68 game, 101, 226 path-dependent, 39, 135 path-independent, 39, 62, 135 Convexity, 357 Coupon rate, 355 Cox-Ingersoll-Ross model, 392, 411 Cox-Ross-Rubinstein formula, 43 model, 36 CRA, 477 Credit risk, 607 Critical stock price, 217 Cross-currency rate, see exchange rate swap, see swap swaption, see swaption Cross-variation, 631 CRR, see Cox-Ross-Rubinstein Daily settlement, see marking to market Date reset, 518 settlement, 518 Decomposition of Ω, 47, 611 Default risk, 607 Index Defaultable bonds, 607 Derivative security, Differential swap, see swap, cross-currency Digital option, see exotic option Discount function, 352 rate, 360 Dividend, 63, 147, 148, 224 Doléans exponential Brownian case, 639 general case, 330 Doob-Meyer decomposition, 98, 211 Dothan’s model, 392 DSR, see Longstaff’s model Dynamic portfolio, see trading strategy Dynkin game definition, 102 Nash equilibrium, 104 optimal stopping times, 104 saddle point, 104 Stackelberg equilibrium, 104 value process, 104 Early exercise premium, 217 Effective interest rate, see interest rate, actuarial Equilibrium general, 13, 131, 368, 371, 392 Nash, 104, 227 partial, 13, 46 rational, 14 Stackelberg, 104 Eurodollar futures market conventions, 362 price, 474 European option, see option Exchange rate cross-currency, 581 forward, 187 process, 182, 574 Exercise price, see strike price Exotic option Asian, 242 barrier, 235 basket, 246 break forward, 230 chooser, 232 collar, 230 combined, 252 compound, 233 digital, 234 Elf-X, 585 forward-start, 231, 269 gap, 234 knock-out, 235 lookback, 238 multi-asset, 252 package, 230 passport, 252 quantile, 249 quanto, 200, 585 range forward, 231 Russian, 252 Expectations hypothesis, 367 Expiry date, Fatou’s lemma, 614 Feynman-Kac formula, 135, 400 Filtration, 14, 42, 47, 616 augmentation, 635 Brownian, 635 complete, 635 natural, 47 usual conditions, 316 Fixed-for-floating cross-currency swap, 595–598 cross-currency swaption, 599–601 swap, 364, 518–521 Fixed-income market, 351 Flesaker-Hughston approach, 566 Floating-for-fixed swap, 364 Floating-for-floating cross-currency swap, 588–594, 606 cross-currency swaption, 601, 602 swap, 364 Floor, 477 Föllmer-Schweizer decomposition, 339 Foreign market forward contract, 191 futures contract, 194 option currency, 188, 582 Elf-X, 202, 585 equity, 198, 199, 583 quanto, 200, 585 swap, 588 swaption, 599 Formula 709 710 Index Bachelier, 156 Bayes, 615 Black, 161 Black-Scholes, 126 Cox-Ross-Rubinstein, 43 Feynman-Kac, 135, 400 integration by parts, 632 Itô, 630 Itô-Tanaka-Meyer, 214 Merton, 133 Forward Brownian motion, 376 contract, 6, 19, 374 interest rate, 353, 379, 424 martingale measure, 375, 424 price, 20, 85, 374, 376, 379, 576 rate agreement, 364 strategy, 328 swap measure co-sliding, 547 co-terminal, 531 swap rate, 520 wealth, 328 Forward neutral probability, see martingale measure Forward swap rate, 364 Forward-start option, see exotic option FRA, see forward rate agreement Free boundary problem, 219 Free lunch with bounded risk, 333 with vanishing risk, 333 Fundamental Theorem of Asset Pricing First, 74, 332 Second, 75 Futures contract, 21 Eurodollar, 362, 474 margin account, 22 options, 455 price, 24, 159, 452 Treasury bill, 360 Treasury bond, 359 hedging, 109 valuation, 109 Game option, 102 General equilibrium approach, 13 Geske-Johnson method, 222 Girsanov’s theorem, 639 Game contingent claim cancellation time, 101, 226 definition, 101, 226 Dynkin game, 108 exercise time, 101, 226 Level measure, 531 process co-terminal, 531 process (sliding), 547 Heath-Jarrow-Morton (HJM) empirical study, 449 Gaussian, 428, 475 Ho and Lee, 432 methodology, 420–427 Vasicek, 432 Hedge ratio, 11, 141, 158 Hedging mean-variance, 337 perfect, 212 risk-minimizing, 338, 339 HJM, see Heath-Jarrow-Morton Ho and Lee model, 419, 432 Hull and White model, 395 Indistinguishable processes, 318 Infinitesimal generator, 219 Inflation-based derivative, 607 Interest rate actuarial, 353 consol, 404 continuously compounded, 353 forward, 353, 379, 424 forward swap, 363 instantaneous, 353 LIBOR, 362, 471 short-term, 354, 424 swap, 363, 520 Israeli option, 102 Itô formula, 630 integral, 623 process, 629 Jamshidian’s model, see model of swap rates Jensen’s inequality, 28, 614 Jump-diffusion models, 315 Index LIBOR, 358 dynamics, 475, 476, 500–503 forward, 471, 474 futures, 474 lognormal model, 486–497 spot measure, 495 Local time, 214, 632 Localization, 622 Longstaff’s model, 394 Lookback option, see exotic option Manufacturing cost, 11, 14 Market Black-Scholes, 123, 333 cash, see spot complete, 68, 335 domestic, 184 finite, 35, 65, 66 fixed-income, 351 foreign, 185, 574 frictionless, futures, 22, 325 imperfections, 345 incomplete, 336 maker system, spot, 8, standard model, 321 Market price for risk, 371 Marking to market, Martingale, 47 localization, 626 measure, 12 probability, 12 quadratic variation, 626 representation property, 635 Martingale measure, 47, 70, 120, 319 domestic, 184 foreign, 185 forward, 375, 424 futures, 160 generalized, 70 minimal, 340 spot, 120, 424 strict, 332 Maturity, see expiry date Mean reversion, 386 Merton’s formula, 133 model, 384 711 Method domestic market, 580 foreign market, 580 martingale, 12 Miltersen-Sandmann-Sondermann model, see model of LIBOR Model completeness, 75 Flesaker-Hughston, 566 Heath-Jarrow-Morton, 420 Jamshidian’s, 530 mixture, 287 of implied volatility, 303 of LIBOR, 486–497 bond option, 513 cap, 506 empirical study, 450 extension, 516 of local volatility, 303 of short-term interest rate affine, 404 Brennan and Schwartz, 392 Cox-Ingersoll-Ross, 392 Dothan, 392 extended CIR, 406 Hull and White, 395 lognormal, 398 Longstaff, 394 Merton, 384 Vasicek, 386 of stochastic volatility Heston, 299 Hull and White, 294 SABR, 301 Wiggins, 293 of stock price ARCH, 308 autoregressive, 308 Bachelier, 155 Black-Scholes, 114 Cox (CEV), 274 Cox-Ross-Rubinstein, 36 dividend-paying, 147, 148 GARCH, 308 jump-diffusion, 309 Lévy process, 310 stochastic volatility, 291 subordinated process, 312 variance gamma, 312 712 Index of swap rates, 530–550 co-initial, 541 co-sliding, 546 co-terminal, 530 Jamshidian’s, 537 Money market account, see savings account Mortgage-backed security, 359 Multi-asset option, see exotic option N-cap, see cap, dual strike Natural filtration, 47 Nominal annual rate, 470 Nominal value bond, 351 option, 188 Notional principal caplet, 477 swap, 518 Numeraire asset, 17, 51, 78, 327, 380 Open interest, Optimal stopping problem, 90 Option American, see American option at-the-money, 140 Bermudan, 222 bond, 435, 513 CMS spread, 528 coupon-bearing bond, 441 currency, 188, 582 exercise price, exotic, see exotic option expiry date, foreign asset, 198, 583 forward contract, 165 game, 102 in-the-money, 140 Israeli, 102 out-of-the-money, 140 replication, 446 sensitivity delta, 141, 158 gamma, 143 theta, 143 vega, 143, 266 standard call or put European, 8, 43 futures, 6, 161 parity, 9, 19, 134 stock, stochastic interest rate case, 438 strike price, Ornstein-Uhlenbeck process, 386 OTC, see over-the-counter Over-the-counter, Package option, see exotic option Partial differential equation Bachelier, 158 Black, 163 Black-Scholes, 127 Black-Scholes-Barenblatt, 178 bond price, 389, 392, 396, 400 forward bond price, 486 futures derivative, 461–464 spot derivative, 457–461 Partial equilibrium approach, 13, 46 Passport option, see exotic option Payer swap, see fixed-for-floating swap PDE, see partial differential equation Perpetual put, 220 Position delta, 143 long or short, 19 Predictable representation property, 635 Present value of basis point, see PVBP Price ask or bid, forward, 374, 376, 379, 503–505 seller, 178 Probability actual, 10 martingale, 12 real-world, 10 risk-neutral, 13 statistical, 10 subjective, 10 Probability measure absolutely continuous, 70 equivalent, 70 Process adapted, 35, 42, 65 arbitrage price, 82 Bessel, 245, 393, 407 bond price, 424 Brownian motion, 631 density, 639 diffusion, 369 expected value, 320 Index forward risk-adjusted, 375 gains, 67, 81, 319, 325 Itô, 629 level, 531 Lévy, 310 multidimensional diffusion, 403 Ornstein-Uhlenbeck, 386 predictable, 83 progressively measurable, 623 RCLL, 207 replicating, 69, 318 squared Bessel, 407 squared-Gauss-Markov, 403 subordinated, 312 variance gamma, 312 wealth, 66, 81 Put-call parity, 19, 51, 134, 164, 440, 456 PVBP, 531 Q-cap, see cap, cumulative Quadratic covariation, see cross-variation Quadratic variation Brownian motion, 618 continuous local martingale, 626 semimartingale, 331 Quanto caplet, 604 forward, 192 option, 200, 585 roll bond, 589 Radon-Nikodým derivative, 51, 78, 639 Random walk arithmetic, 38 exponential, 38 geometric, 38 Rational lognormal model, 566–571 RCLL, 207 Real-world probability, 10 Realized variation, 313 Receiver swap, see floating-for-fixed swap Region continuation, 216 stopping, 216 Reward function, 209 Riccati equation, 392 Risk credit, 607 default, 607 713 market, 607 premium, 417 Risk-free portfolio, 113, 137 Risk-neutral economy, 13 individual, 70 probability, 13, see martingale measure valuation Black-Scholes model, 124 Cox-Ross-Rubinstein model, 50 discrete-time case, 84 stochastic interest rate, 377 world, 13 Russian option, see exotic option Saddle point, 104, 227 Sandmann-Sondermann model, 398 Savings account, 36, 83, 114, 354 SDE, 114 Security derivative, primary, Semimartingale canonical decomposition, 331 continuous, 316, 628 canonical decomposition of, 630 special, 332 Separating hyperplane theorem, 77 Settlement daily, in advance, 477, 520 in arrears, 477, 518 Short-term interest rate Brennan and Schwartz, 392 CIR model, 392 Dothan’s model, 392 Hull and White model, 395 lognormal model, 398 Longstaff’s model, 394 Merton’s model, 384 Vasicek’s model, 386 Smooth fit principle, 220 Snell envelope, 90, 211 Spread bear, bull, butterfly, calendar, Statistical probability, 10 714 Index Stock common, dividend-paying, 63, 147, 148, 224 index option, 340 preferred, Stock price Black-Scholes model multidimensional, 333 one-dimensional, 114 Cox (CEV) model, 274 CRR model, 36 stochastic volatility model, 291 Stopped process, 621 Stopping time, 30, 60, 87, 208 Strike price, 4, Subjective probability, 10 Super-replication, 174, 177 Swap accrual period, 518 annuity (level process), 531 CMS spread, 528 co-initial, 541 co-sliding, 546 co-terminal, 530 constant maturity, 525 cross-currency, 588–602, 606 forward-start, 363, 518 length, 518 notional principal, 364 payer, 364 rate, 363, 520, 546 receiver, 364 reset date, 518 settlement date, 518 swap option, see swaption valuation, 471–474, 518–520 yield curve, 529 Swaption Bermudan, 526, 541 co-initial, 545 co-sliding, 551 co-terminal, 539 cross-currency, 588, 599 forward, 524 payer, 522 receiver, 522 valuation formula Flesaker-Hughston model, 571 Gaussian HJM model, 527 lognormal LIBOR model, 555 market, 530 Tenor, 470 Term structure affine, 390 foreign, 574 of interest rates, 352 Trading and consumption strategy admissible, 208 self-financing, 207 Trading strategy, 118 admissible, 123, 318 bonds, 422 buy-and-hold, 209 currency, 188 futures, 81, 160, 454 instantaneously risk-free, 138 mean self-financing, 339 replicating, 43, 68 self-financing, 40, 42, 67, 119, 138, 454 simple predictable, 333 Transaction costs Black-Scholes model, 145, 347 Treasury bill, 360 bond, 359 Utility-based valuation, 179, 345 Variance-covariance matrix, 441 Variational inequality, 221 Vasicek’s model, 386, 432 Volatility historical, 255 implied, 256–264 asymptotic behavior, 261 Black-Scholes, 256 Brenner-Subrahmanyam formula, 259 CEV model, 276 Corrado-Miller formula, 259 smile effect, 257 surface, 259 local, 278–290 Dupire’s formula, 284 matrix, 334 of stock price, 114 stochastic, 291–314 uncertain, 169 Black-Scholes-Barenblatt equation, 178 Index seller price, 171 super-replication, 174 Warrant, Wiener process, see Brownian motion Yield curve fitting, 397 initial, 352 swap, 529 Yield-to-maturity, 356 715 ... few sections that, in our opinion, were of marginal importance for the understanding of the fundamental principles of financial modelling of arbitrage valuation of derivatives In view of the abundance... new results in the area, it would be in any case unimaginable to cover all existing approaches to pricing and hedging financial derivatives (not to mention all important results) in a single book,... book, considerable progress was achieved in the area of financial modelling and pricing of derivatives Needless to say, it was our intention to incorporate into the second edition at least the most