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) 64 Edited by B Rozovski˘ ı G Grimmett Advisory Board M Hairer I Karatzas F.P Kelly A Kyprianou Y Le Jan B Øksendal G Papanicolaou E Pardoux E Perkins For other titles in this series, go to http://www.springer.com/series/602 Eckhard Platen r Nicola Bruti-Liberati Numerical Solution of Stochastic Differential Equations with Jumps in Finance Eckhard Platen Nicola Bruti-Liberati (1975–2007) School of Finance and Economics Department of Mathematical Sciences University of Technology, Sydney PO Box 123 Broadway NSW 2007 Australia eckhard.platen@uts.edu.au Managing Editors Boris Rozovski˘ ı Division of Applied Mathematics Brown University 182 George St Providence, RI 02912 USA rozovsky@dam.brown.edu Geoffrey Grimmett Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB UK g.r.grimmett@statslab.cam.ac.uk ISSN 0172-4568 ISBN 978-3-642-12057-2 e-ISBN 978-3-642-13694-8 DOI 10.1007/978-3-642-13694-8 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010931518 Mathematics Subject Classification (2010): 60H10, 65C05, 62P05 © Springer-Verlag Berlin Heidelberg 2010 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: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface This research monograph concerns the design and analysis of discrete-time approximations for stochastic differential equations (SDEs) driven by Wiener processes and Poisson processes or Poisson jump measures In financial and actuarial modeling and other areas of application, such jump diffusions are often used to describe the dynamics of various state variables In finance these may represent, for instance, asset prices, credit ratings, stock indices, interest rates, exchange rates or commodity prices The jump component can capture event-driven uncertainties, such as corporate defaults, operational failures or insured events The book focuses on efficient and numerically stable strong and weak discrete-time approximations of solutions of SDEs Strong approximations provide efficient tools for simulation problems such as those arising in filtering, scenario analysis and hedge simulation Weak approximations, on the other hand, are useful for handling problems via Monte Carlo simulation such as the evaluation of moments, derivative pricing, and the computation of risk measures and expected utilities The discrete-time approximations considered are divided into regular and jump-adapted schemes Regular schemes employ time discretizations that not include the jump times of the Poisson jump measure Jump-adapted time discretizations, on the other hand, include these jump times The first part of the book provides a theoretical basis for working with SDEs and stochastic processes with jumps motivated by applications in finance This part also introduces stochastic expansions for jump diffusions It further proves powerful results on moment estimates of multiple stochastic integrals The second part presents strong discrete-time approximations of SDEs with given strong order of convergence, including derivative-free and predictor-corrector schemes The strong convergence of higher order schemes for pure jump SDEs is established under conditions weaker than those required for jump diffusions Estimation and filtering methods are discussed The third part of the book introduces a range of weak approximations with jumps These weak approximations include derivative-free, predictor-corrector, and VI Preface simplified schemes The final part of the research monograph raises questions on numerical stability and discusses powerful martingale representations and variance reduction techniques in the context of derivative pricing The book does not claim to be a complete account of the state of the art of the subject Rather it attempts to provide a systematic framework for an understanding of the basic concepts and tools needed when implementing simulation methods for the numerical solution of SDEs In doing so the book aims to follow up on the presentation of the topic in Kloeden & Platen (1999) where no jumps were considered and no particular field of application motivated the numerical methods The book goes significantly beyond Kloeden & Platen (1999) It is covering many new results for the approximation of continuous solutions of SDEs The discrete time approximation of SDEs with jumps represents the focus of the monograph The reader learns about powerful numerical methods for the solution of SDEs with jumps These need to be implemented with care It is directed at readers from different fields and backgrounds The area of finance has been chosen to motivate the methods It has been also a focus of research by the first author for many years that culminated in the development of the benchmark approach, see Platen & Heath (2006), which provides a general framework for modeling risk in finance, insurance and other areas and may be new to most readers The book is written at a level that is appropriate for a reader with an engineer’s or similar undergraduate training in mathematical methods It is readily accessible to many who only require numerical recipes Together with Nicola Bruti-Liberati we had for several years planned a book to follow on the book with Peter Kloeden on the “Numerical Solution of Stochastic Differential Equations”, which first appeared in 1992 at Springer Verlag and helped to develop the theory and practice of this field Nicola’s PhD thesis was written to provide proofs for parts of such a book It is very sad that Nicola died tragically in a traffic accident on 28 August 2007 This was an enormous loss for his family and friends, his colleagues and the area of quantitative methods in finance The writing of such a book was not yet started at the time of Nicola’s tragic death I wish to express my deep gratitude to Katrin Platen, who then agreed to typeset an even more comprehensive book than was originally envisaged She carefully and patiently wrote and revised several versions of the manuscript under difficult circumstances The book now contains not only results that we obtained with Nicola on the numerical solution of SDEs with jumps, but also presents methods for exact simulation, parameter estimation, filtering and efficient variance reduction, as well as the simulation of hedge ratios and the construction of martingale representations I would like to thank several colleagues for their collaboration in related research and valuable suggestions on the manuscript, including Kevin Burrage, Leunglung Chan, Kristoffer Glover, David Heath, Des Higham, Hardy Hulley, Constantinos Kardaras, Peter Kloeden, Uwe Kăchler, Herman Lukito, u Preface VII Remigius Mikulevicius, Renata Rendek, Wolfgang Runggaldier, Lei Shi and Anthony Tooman Particular thanks go to Rob Lynch, the former Dean of the Faculty of Business at the University of Technology Sydney, who made the writing of the book possible through his direct support Finally, I like to thank the Editor, Catriona Byrne, at Springer for her excellent work and her encouragement to write this book as a sequel of the previous book with Peter Kloeden It is greatly appreciated if readers could forward any errors, misprints or suggested improvements to: eckhard.platen@uts.edu.au The interested reader is likely to find updated information about the numerical solution of stochastic differential equations on the webpage of the first author under “Numerical Methods”: http://www.business.uts.edu.au/ finance/staff/Eckhard/Numerical Methods.html Sydney, January 2010 Eckhard Platen Contents Preface V Suggestions for the Reader XV Basic Notation XIX Motivation and Brief Survey XXIII Stochastic Differential Equations with Jumps 1.1 Stochastic Processes 1.2 Supermartingales and Martingales 1.3 Quadratic Variation and Covariation 1.4 Itˆ Integral o 1.5 Itˆ Formula o 1.6 Stochastic Differential Equations 1.7 Linear SDEs 1.8 SDEs with Jumps 1.9 Existence and Uniqueness of Solutions of SDEs 1.10 Exercises 1 16 23 26 34 38 45 53 57 59 Exact Simulation of Solutions of SDEs 2.1 Motivation of Exact Simulation 2.2 Sampling from Transition Distributions 2.3 Exact Solutions of Multi-dimensional SDEs 2.4 Functions of Exact Solutions 2.5 Almost Exact Solutions by Conditioning 2.6 Almost Exact Simulation by Time Change 2.7 Functionals of Solutions of SDEs 2.8 Exercises 61 61 63 78 99 105 113 123 136 X Contents Benchmark Approach to Finance and Insurance 139 3.1 Market Model 139 3.2 Best Performing Portfolio 142 3.3 Supermartingale Property and Pricing 145 3.4 Diversification 149 3.5 Real World Pricing Under Some Models 158 3.6 Real World Pricing Under the MMM 168 3.7 Binomial Option Pricing 176 3.8 Exercises 185 Stochastic Expansions 4.1 Introduction to Wagner-Platen Expansions 4.2 Multiple Stochastic Integrals 4.3 Coefficient Functions 4.4 Wagner-Platen Expansions 4.5 Moments of Multiple Stochastic Integrals 4.6 Exercises 187 187 195 202 206 211 230 Introduction to Scenario Simulation 5.1 Approximating Solutions of ODEs 5.2 Scenario Simulation 5.3 Strong Taylor Schemes 5.4 Derivative-Free Strong Schemes 5.5 Exercises 233 233 245 252 266 271 Regular Strong Taylor Approximations with Jumps 273 6.1 Discrete-Time Approximation 273 6.2 Strong Order 1.0 Taylor Scheme 278 6.3 Commutativity Conditions 286 6.4 Convergence Results 289 6.5 Lemma on Multiple Itˆ Integrals 292 o 6.6 Proof of the Convergence Theorem 302 6.7 Exercises 307 Regular Strong Itˆ Approximations o 7.1 Explicit Regular Strong Schemes 7.2 Drift-Implicit Schemes 7.3 Balanced Implicit Methods 7.4 Predictor-Corrector Schemes 7.5 Convergence Results 7.6 Exercises 309 309 316 321 326 331 346 Author Index Math´, P., 785, 812 e Matsumoto, H., 71, 804 Matsumoto, M., 716, 785, 822, 825 Mattingly, J C., 790, 818 Mauthner, S., 789, 822 McLeish, D L., XXVII, 788, 822 McNeil, A., XXVII, 65, 66, 97, 783, 822 McNeil, K J., 787, 790, 822 Mei, C., 786, 789, 833 Melchior, M., 791, 822 M´l´ard, S., 75, 783, 785, 786, 789, 813 ee Meng, X., 787, 790, 834 Mercurio, F., XXVII, 789, 798 Merener, N., 57, 534, 786, 789, 808 Merton, R C., XXIII, 14, 37, 38, 41, 47, 50, 52, 158, 161, 162, 166, 414, 565, 822 Messaoud, M., 791, 796 Metwally, S A K., 786, 789, 822 Meyer, P A., 784, 812 Michna, Z., 786, 813 Mikhailov, G A., 638, 791, 806 Mikosch, T., XXVII, 805, 822 Mikulevicius, R., 482, 517, 518, 525, 534–537, 620, 728, 748, 785, 786, 789, 822 Miller, S., 41, 158, 164, 166, 168, 170–172, 784, 812, 822 Milne, F., 53, 820 Milstein, G N., XXVI, XXVII, 61, 256, 261, 278, 316, 318, 322, 324, 352, 358, 477, 483, 499, 528, 572, 576, 585, 586, 638, 642, 643, 645, 784–791, 808, 822, 823, 833 Miltersen, K R., 434, 823 Misawa, T., 784, 823 Mitsui, T., 572, 574, 786, 787, 789, 790, 799, 817, 823, 829 Miyahara, Y., 53, 823 Mohammed, S.-E A., 785, 812 Monfort, A., 398, 788, 809 Moore, J B., XXVII, 309, 434–437, 445, 788, 805 Mora, C M., 786, 789, 823 Mordecki, E., 789, 823 Mortimer, I K., 787, 790, 804 Morton, K., 734, 740, 791, 828 Mostafa, O L., 786, 805 841 Muirhead, R J., 71, 72, 824 Măller-Gronbach, T., 530, 786, 787, u 789, 812, 824 Musiela, M., XXVII, 163, 824 Nahum, E., 56, 829 Nakao, S., 785, 815 Nakayama, T., 785, 824 Nelsen, R., 783, 824 Nelson, D B., 50, 824 Neuenkirch, A., 786, 816 Newton, N J., 425, 430, 638, 642, 785–788, 791, 824 Nguyen, T T., 787, 819 Niederreiter, H., XXVII, 638, 716, 785, 824 Nikitin, N N., 786, 824 Nikitopoulos-Sklibosios, C., 56, 309, 347, 784, 786, 787, 798, 799 Ninomiya, S., 789, 824 Nishimura, T., 716, 785, 822 Novikov, A A., 53, 135, 647, 787, 823, 824 Ocone, D L., 601, 606, 824 Ogawa, S., 786, 789, 817, 824 Øksendal, B., XXVII, 783, 824 Olkin, I., 66, 821 Ombach, J., 785, 788, 803 Osborne, M F M., 47, 825 ă Ottinger, H., 791, 822 Ozaki, T., 786, 788, 789, 814, 825, 830 Pag´s, G., 716, 785, 825 e Palleschi, V., 785, 821 Pan, J., XXIII, 804 Pang, S., 787, 790, 825 Panneton, F., 716, 785, 825 Papaspiliopoulos, O., 784, 796 Pardoux, E., 785, 787, 789, 790, 825 Paskov, S., 785, 825 Pasquali, S., 457, 788, 801 Pearson, N., 42, 825 Pearson, R A., 785–787, 816 Pedersen, A R., 390, 788, 825 Pelsser, A., 789, 791, 825 Peng, H., 417, 793 Penski, C., 787, 803 Petersen, W P., 785–787, 789, 790, 816, 825 842 Author Index Petryakov, M G., 786, 817 Petterson, R., 638, 786, 791, 817, 825 Petzold, L., 787, 790, 800 Picard, J., 429, 825 Piccardi, M., 528, 785, 789, 798, 799 Pietersz, R., 789, 825 Pironneau, O., XXVII, 788, 793 Piterbarg, V., 783, 794 Pitman, J., 786, 794 Pivitt, B., 246, 785, 825 Pliska, S R., 791, 827 Polson, N., 414, 805 Pope, S B., 789, 810 Portait, R., 784, 795 Potters, M., 790, 827 Poulsen, R., 390, 398, 788, 801, 827 Pozo, R., 742, 796 Prakasa Rao, B L S., 389, 390, 788, 827 Press, S J., 415, 827 Press, W H., 64, 716, 827 Privault, N., 598, 790, 805 Protter, P., XXVII, 13, 14, 23–25, 27, 29–31, 33–35, 37, 43, 44, 58, 59, 75, 379, 573, 783–786, 789, 802, 804, 813, 814, 817, 818, 820, 827 Qian, M P., 786, 806 Qian, X., 791, 834 Qin, C., 790, 828 Rachev, S T., 786, 807 Ramkrishna, D., 786, 827 Randall, C., 734, 744, 791, 832 Rao, C R., 64, 827 Rao, N J., 786, 827 Rascanu, A., 786, 828 Rathinam, M., 787, 790, 800 Rathinasamy, A., 787, 790, 828 Raviart, P A., 740, 828 Razevig, V D., 786, 824, 828 Rebolledo, R., 40, 747, 786, 827 Rebonato, R., 789, 828 Reimer, M., 791, 819 Renault, E., 398, 788, 809 Rendek, R., 61, 154, 158, 427, 781, 783, 788, 827 Repin, Y M., 785, 823 Revuz, D., 62, 89, 90, 168, 169, 783, 828 Rib´mont, B., 787, 819 e Richtmeyer, R., 734, 740, 791, 828 Riera, J J., 787, 797 Rindisbacher, M., 790, 803 Ripley, B D., 638, 791, 828 Ritchken, P., 703, 815 Ritter, K., 530, 786, 787, 789, 812 Roberts, G., 61, 783, 784, 796, 800 Rochet, J C., 783, 808 Rodkina, A., 787, 790, 828 Rogers, L C G., 784, 785, 789, 828 Romine, C., 742, 796 Rămisch, W., 788–790, 828 o Roncoroni, A., XXIII, 56, 808 Ronghua, L., 790, 828 Rootz´n, H., 786, 829 e Rosa, A M., 786, 808 Rosenfeld, E R., 42, 821 Rosi´ski, J., 787, 794 n Ross, S A., 41, 146, 181, 533, 791, 802, 829 Ross, S M., 638, 639, 641, 669, 791, 829 Răssler, A., 786, 789, 819 o Rozovski, B., 457, 802 Rubenthaler, S., 75, 783, 829 Rubinstein, M., 181, 791, 802, 829 Rubinstein, R Y., 638, 791, 829 Rudd, A., 702, 814 Rămelin, W., 785, 787, 829 u Runggaldier, W J., XXIII, 45, 56, 456, 457, 472, 473, 784, 788, 791, 796, 797, 801, 804, 806, 807, 809, 827, 829 Rutkowski, M., XXVII, 163, 797, 824 Ryashko, L B., 787, 790, 829 Ryd´n, T., 784, 829 e Sabelfeld, K K., 791, 829 Sagsyan, K V., 786, 804 Saisho, Y., 787, 815 Saito, Y., 572, 574, 786, 787, 790, 817, 829 Salminen, P., XXVII, 132, 783, 797 Samuelides, Y., 56, 829 Samuelson, P A., 14, 40, 47, 162, 829 Sandmann, K., 42, 829 Author Index Santa-Clara, P., 390, 788, 789, 798 Sanz-Sol´, M., 787, 806 e San Martin, J., 785, 820 Schachermayer, W., XXVII, 147, 783, 784, 803, 804 Schăer, S., 786, 787, 803 a Schein, O., 786, 789, 819, 829 Schmitz-Abe, K., 789, 829 Schnell, S., XXIII, 375, 787, 833 Schoenmakers, J., 789–791, 814, 823, 829 Scholes, M., 14, 37, 40, 47, 136, 797 Schănbucher, P J., XXIII, XXVII, 56, o 167, 829 Schoutens, W., 784, 789, 830 Schrage, L., 638, 716, 785, 791, 798 Schroder, M., 40, 830 Schurz, H., 61, 245, 322, 326, 358, 390, 393, 429, 477, 576, 585, 586, 785–788, 790, 814, 816, 817, 823, 828–830 Schwartz, E., 434, 823 Schwartz, E S., 638, 713, 789, 820 Schweizer, M., 40, 128, 473, 475, 593, 638, 645, 668, 671, 682, 687, 784, 791, 806, 811–813, 830 Seneta, E., 53, 75, 783, 820 Sestovic, D., 790, 827 Setiawaty, B., 440, 788, 830 Sevilla-Peris, P., 786, 802, 814 Seydel, R., XXVII, 788, 830 Shahabuddin, P., 791, 808 Shardlow, T., 785, 799 Shaw, W., 734, 741, 744, 789, 791, 829, 830 Shedler, G S., 61, 783, 819 Shen, J., 785, 820 Shen, Y., 790, 821 Shephard, N., 14, 53, 75, 105, 390, 414, 783, 788, 795, 805 Sheu, S.-J., 40, 806 Shi, L., 571, 587, 588, 781, 790, 827 Shimbo, K., 784, 813, 814 Shimizu, A., 786, 830 Shinozuka, M., 786, 830 Shirakawa, H., 40, 94, 803 Shiryaev, A N., XXVII, 1, 4, 5, 23, 25, 389, 421, 460–463, 609, 612, 615, 843 631, 632, 747, 786, 788, 813, 819, 830 Shkurko, I O., 787, 790, 794 Shoji, I., 785, 788, 789, 830 Shreve, S E., XXVII, 20, 27, 596, 610, 815, 830 Sickenberger, T., 787, 830 Sin, C A., 784, 830 Singleton, K., XXIII, 788, 804 Siopacha, M., 789, 831 Situ, R., 58, 831 Skeel, R., 791, 834 Sklar, A., 783, 831 Skorokhod, A V., 785, 808 Sloan, I H., 785, 831 Slominski, L., 786, 831 Smereka, P., 786, 804 Smith, A M., 787, 790, 831 Smith, G D., 734, 791, 831 Smith, R D., 61, 63, 107, 783, 831 Sole, J L., 791, 819 Sondermann, D., 42, 829 Sørensen, H., 390, 831 Sørensen, M., 390, 393, 394, 397, 398, 402, 411–413, 417, 788, 795–797, 801, 816–818, 831 Sorini, L., 789, 809 Sottinen, T., 784, 796 Spigler, R., 572, 578, 787, 790, 811 Stacey, A M., 789, 814 Stahl, G., 145, 151, 784, 827 Stegun, I A., 78, 793 Stentoft, L., 789, 831 Stramer, O., 785, 801 Stricker, C., 146, 469, 794 Strittmatter, W., 786, 809 Stroud, A H., 784, 831 Stroud, J R., 789, 798 Stuart, A M., 785, 787, 790, 812, 818 Studer, M., 194, 549, 784, 789, 830, 831 Sufana, R., 70, 89, 111, 112, 784, 809 Sugiura, H., 786, 787, 790, 817 Sulem, A., XXVII, 783, 824 Sun, M., 788, 831 Sun, T S., 42, 825 Sussmann, H J., 785, 831 Symens, S., 789, 830 Szepessy, A., 789, 823, 831 844 Author Index Takahashi, A., 784, 791, 831 Takeuchi, A., 790, 831 Talay, D., 75, 316, 318, 495, 556, 565, 783, 785–790, 795, 797, 809, 825, 827, 828, 832 Talkner, P., 787, 816 Tan, K S., 638, 785, 814 Tanaka, H., 791, 832 Tankov, P., XXIII, XXVII, 73, 75, 98, 565, 783, 788, 802 Taraskin, A F., 787, 832 Tauchen, G., 398, 414, 788, 801, 807 Tausworthe, R C., 717, 832 Tavella, D., 734, 744, 791, 832 Tebaldi, C., 112, 784, 791, 803, 809, 832 Teichmann, J., 784, 789, 791, 796, 831, 832 Tempone, R., 789, 823, 831 Tetzlaff, U., 786, 833 Teukolsky, S A., 64, 716, 827 Thalmaier, A., XXVII, 785, 788, 789, 791, 802, 821 Thitsiklis, J., 789, 833 Thomas, J M., 740, 828 Tian, T., 785, 787, 790, 800 Tocino, A., 784, 787, 789, 790, 833 Toft, K J., 791, 798 Tărok, C., 786, 833 o Torres, S., 785, 820 Touzi, N., 598, 638, 645, 785, 790, 791, 797, 806, 807 Toy, W., 42, 791, 797 Traub, J., 785, 825 Tretjakov, M V., XXVII, 528, 784–789, 823 Tschitashvili, R J., 786, 804 Tubaro, L., 495, 789, 832 Tudor, C., 785, 786, 828, 833 Tudor, M., 785, 786, 833 Tullie, T A., 791, 806 Turnbull, S., XXIII, 813 Turner, T., XXIII, 375, 787, 833 Tysk, J., 784, 805 Unny, T E., 786, 833 Uppal, R., 154, 803 Utzet, F., 791, 819 Valdes, P A., 787, 797 Valkeila, E., 784, 796 van Dijk, D., 63, 783, 820 van Regenmortel, M., 789, 825 Vanden-Eijnden, E., 787, 790, 833 van der Hoek, J., XXVII, 457, 788, 791, 805, 833 van der Vorst, H., 742, 796 Van Roy, B., 789, 833 Vasicek, O A., 3, 41, 46, 833 Ventzel, A D., 789, 791, 833 Vetterling, W T., 64, 716, 827 Vetzal, K R., XXVI, 803 Victoir, N., 784, 789, 790, 820, 824, 833 Vigo-Aguiar, J., 789, 833 Villafuerte, L., 786, 814 Vives, J., 791, 819 Voltchkova, E., XXVI, 802 Vora, D., 638, 809 Vorst, T., 791, 825 Wadewitz, A., 786, 809 Wagner, W., 187, 261, 784, 785, 789, 791, 819, 827, 833 Wakolbinger, A., 789, 828 Wang, H., 791, 805 Wang, L., 786, 789, 833 Wang, P., 787, 834 Wang, Z., 790, 834 Warin, X., 785, 808 Watanabe, S., XXVII, 21, 44, 58, 59, 278, 783, 786, 813 Webber, N., XXIII, 795 Weeks, D., 638, 809 Weron, A., 785, 786, 813 White, A., 42, 638, 791, 812 Whitlock, P A., 638, 791, 814 Widom, H., 612, 834 Wiktorsson, M., 784, 829 Willard, G A., 784, 811, 820 Wilmott, P., 734, 743, 791, 834 Winkler, R., 786, 790, 799, 828 Wong, H Y., XXVII, 788, 801 Wonham, W M., 425, 788, 834 Wărner, J H., 414, 834 o Wozniakowski, H., 785, 831 Wright, D J., 785–787, 834 Wright, I., 784, 817 Author Index Wu, K., 790, 804 Wu, S., 787, 790, 834 Xia, X., 787, 790, 820 Xiao, Y J., 716, 785, 825 Xu, C., 791, 834 Xue, H., 786, 789, 833 Yamada, T., 785, 834 Yamato, Y., 785, 834 Yan, F., 785, 812 Yannios, N., 786, 834 Yen, V V., 784, 834 Yin, G., 786, 789, 790, 821 Yingmin, H., 790, 828 Yong, J M., 785, 820 Yor, M., XXVII, 14, 53, 62, 71, 75, 89, 90, 105, 168, 169, 783, 800, 804, 808, 814, 828 845 Yoshida, M., 791, 831 Yuan, C., 785, 786, 789, 790, 821, 834 Zakai, M., 425, 788, 834 Zanette, A., 598, 790, 795 Zeitouni, O., 438, 788, 803 Zhang, C., 790, 834 Zhang, H., 790, 834 Zhang, Q., 788, 815 Zhao, X., 786, 789, 808 Zhao, Y., 785, 820 Zhaoguang, C., 790, 828 Zheng, Z., 785, 832 Zhou, G., 707, 709–711, 719, 791, 811 Zou, G., 791, 834 Zouraris, G E., 789, 823, 831 Zschiesche, H.-U., 786, 833 Index absolute error, 251 actuarial pricing formula, 149 adapted, additive diffusion coefficient, 287 additive noise, 45, 258, 493 affine diffusion, 404 affine model, 42 almost exact approximation, 63 simulation, 361 American option, 700 antithetic variates, 638 appreciation rate, 141 approximate estimator, 396 GOPs, 153 Wonham filter, 428 approximating probabilities, 251 approximation almost exact, 63 discrete-time, 247 finite difference, 737 jump-adapted, 348 Markov chain, 722 pathwise, 251 regular weak, 507 strong, 251 weak, 251, 478 arbitrage strong, 147 ARCH diffusion model, 50, 113 multi-dimensional, 103, 115, 156 Archimedian copula, 65 asset-or-nothing binary, 163, 170 asymptotically p-stable, 573 stable, 575 augmented sigma-algebra, Bachelier model, 40 balanced implicit Euler method, 585 implicit method, 322, 451 basket option, 621 Bayes rule, 134 benchmark, 145 benchmark approach, 139, 456 benchmarked price, 145, 734 Bessel function modified, 71 binary asset-or-nothing, 163, 170 bond-or-nothing, 165, 171 binomial model, 176 binomial tree, 697 CRR, 698 Black model, 41 Black-Derman-Toy model, 42 Black-Karasinski model, 42 Black-Scholes formula, 136 model, 40, 47 multi-asset, 49 multi-dimensional, 102 modification, 706 bond-or-nothing binary, 165, 171 E Platen, N Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Stochastic Modelling and Applied Probability 64, DOI 10.1007/978-3-642-13694-8, © Springer-Verlag Berlin Heidelberg 2010 847 848 Index Brownian motion, geometric, 47 Cauchy-Schwarz inequality, 21 CEV model, 40 chain rule classical, 34 change of measure, 435 Chebyshev inequality, 23 CIR model, 41 classical chain rule, 34, 192 Clayton copula, 65 coefficient function Itˆ, 204 o commutativity condition diffusion, 258, 516 jump, 286 compensated drift coefficient, 196 Poisson measure, 195, 211 Wagner-Platen expansion, 196, 206 weak Taylor scheme, 519 compound Poisson process, 414 concatenation operator, 197 conditional density, expectation, iterated, variance, 472, 639 conditionally Gaussian model, 460 confidence interval, 480 consistent, 240 contingent claim, 470, 592 continuation region, 128, 592 continuous uncertainty, 139 control variate, 670 general, 671 convergence order, 238 order of strong, 292 strong order, 251 theorem jump-adapted, 535 Markov chain, 744 strong, 291 weak, 544 weak order, 519, 724 copula, 65 Archimedian, 65 Clayton, 65 Gaussian, 65, 66 correlation property, 29 counting process, 139, 193 Courtadon model, 42 covariance, covariation, 24 approximate, 24 property, 29 Cram´r-Lundberg model, 12, 52 e Crank-Nicolson method, 741 credit worthiness, 10 CRR binomial tree, 698 d-dimensional linear SDE, 48 defaultable zero coupon bond, 166 degree of implicitness, 239 density Gaussian, 66 derivative-free scheme, 267, 309, 334 strong order 1.0 scheme, 310 deviation, 479 difference method, 234 differential equation ordinary, 233 diffusion coefficient, 28, 248 additive, 287 multiplicative, 288 commutativity condition, 258, 516 stationary, 409 Dirac delta function, 131 discount rate process, 125 discounted payoff with payoff rate, 126 discrete-time approximation, 247 discretization spatial, 735 distribution generalized hyperbolic, 73 generalized inverse Gaussian, 80 non-central chi-square, 78 normal inverse Gaussian, 74 Student-t, 74 variance-gamma, 74 Wishart non-central, 80 Diversification Theorem, 153 diversified portfolio, 152 Index drift coefficient, 28, 248 compensated, 196 drift-implicit Euler scheme, 316 simplified Euler scheme, 497 strong order 1.0 scheme, 318 strong scheme, 337 equidistant time discretization, 246 ergodicity, 132 error absolute, 251 covariance matrix, 420 function, 78 global discretization, 234 local discretization, 234 roundoff, 234 statistical, 478 strong, 368 systematic, 478 weak, 478, 699 estimate least-squares, 16 Monte Carlo, 478 estimating function, 397 martingale, 398 estimation maximum likelihood, 389 parameter, 390, 413 estimator approximate, 396 HP, 678 maximum likelihood, 392, 394, 417 unbiased, 600 Euler method, 234 balanced implicit, 585 Euler scheme, 207, 246, 253, 276 fully implicit, 239 jump-adapted, 351 predictor-corrector, 576 simplified symmetric, 587 Euler-Maruyama scheme, 246 European call option, 166, 172 event driven uncertainty, 140 exact simulation, 64 almost, 361 jump-adapted, 361 exact solution 849 almost, 105 exercise boundary, 592 expansion compensated Wagner-Platen, 196, 206 general Wagner-Platen, 206 stochastic Taylor, 187 Stratonovich-Taylor, 191 Wagner-Platen, 187 Expectation Maximization algorithm, 438 expectation, conditional, explicit filter, 430 solution, 45 strong order 1.5 scheme, 269 weak order 2.0 scheme, 491 exponential L´vy model, 14, 53 e extrapolation, 240, 495 Richardson, 241, 710 weak order 2.0, 495 weak order 4.0, 496 Făllmer-Schweizer decomposition, 475 o fair price, 471 fair zero coupon bond, 149 Feynman-Kac formula, 124, 128 filter, 436 distribution, 460 explicit, 430 finite-dimensional, 460 implicit, 432 Kalman-Bucy, 420 SDE, 421 filtered probability space, filtering, 419, 456 filtration, natural, finite difference approximation, 737 difference method, 734, 738 variation property, 29 finite-dimensional filter, 460 finite-state jump model, 460 first exit time, 128 first hitting time, 19 first jump commutativity condition, 515 850 Index first moment estimate, 211 first strong order balanced implicit method, 326 Fokker-Planck equation, 130 forward contract, 162 forward price, 163 fraction, 142, 474 free lunches with vanishing risk, 147 fully implicit Euler scheme, 239 method, 741 fundamental matrix, 48 fundamental solution, 45 gamma function incomplete, 107 multi-dimensional, 71 gamma process, 97 Gaussian copula, 65, 66 density, 66 elimination method, 742 transition density, 66 general affine process, 96 general market risk, 151 general method of moments, 415 general Wagner-Platen expansion, 206 generalized hyperbolic distribution, 73 inverse Gaussian distribution, 80 method of moment, 398 MMM, 123 volatility process, 141 generator random bit, 713, 716 random number linear congruential, 716 geometric Brownian motion, 47 multi-dimensional, 67 geometric OU-process, 69 matrix, 96 Girsanov theorem, 436 transformation, 134, 643 global discretization error, 234 GOP, 144 approximate, 153 Gronwall inequality, 22 growth optimal portfolio, 115, 144, 457 growth rate long term, 145 Heath-Platen method, 677 hedgable part, 474 hedge ratio, 178, 591, 595, 634 hedging strategy, 473 Hermite polynomial, 202, 262 Heston model, 106, 111, 686 multi-dimensional, 108, 155 Heston-Zhou trinomial tree, 708 Heun method, 239 Heyde-optimal, 403 hidden Markov chain, 425 hidden state, 420 hierarchical set, 206, 290, 363, 519 high intensity, 372, 378 Ho-Lee model, 41 Hălder inequality, 21, 22 o HP estimator, 678 Hull-White model, 42 implicit filter, 432 methods, 583 weak Euler scheme, 498 weak order 2.0 scheme, 500 incomplete gamma function, 107 market, 458, 472 indirect inference, 398 indistinguishability, 57 inequality Cauchy-Schwarz, 21 Chebyshev, 23 Doob, 20 Gronwall, 22 Hălder, 21, 22 o Jensen’s, 22 Lyapunov, 22 Markov, 23 maximal martingale, 20 information matrix, 393 information set, insurance premium, 12 integrable random variable, integral Itˆ, 31 o multiple stochastic, 279 Index Riemann-Stieltjes, 28 Stratonovich, 33 symmetric, 33 intensity, 139 high, 372, 378 measure, 13, 32 interest rate model, 41 interpolation, 247 inverse Gaussian process, 97 inverse transform method, 64, 77 iterated conditional expectations, Itˆ coefficient function, 204, 363 o compensated, 205 Itˆ differential, 34 o Itˆ formula, 34, 36, 100 o jump process, 36 multi-dimensional, 35 Itˆ integral, 27, 31 o equation, 39 multiple, 253 Itˆ process with jumps, 44 o Jensen’s inequality, 22 jump coefficient, 376, 468 jump commutativity condition, 286 first, 515 second, 515 jump diffusion, 44, 413 jump martingale, 140 jump measure Poisson, 44 jump model finite-state, 460 jump ratio, 51 jump size, 24 mark-independent, 511 jump time, 9, 11 jump-adapted approximation, 348 derivative-free scheme, 355 drift-implicit scheme, 357 Euler scheme, 351, 524 simplified, 524 exact simulation, 361 predictor-corrector Euler scheme, 531 scheme, 359 strong order γ Itˆ scheme, 367 o strong order γ Taylor scheme, 363 851 Taylor scheme weak order 2.0, 525 time discretization, 348, 377, 523 weak order 3.0 scheme, 527 weak order β Taylor scheme, 535 weak scheme exact, 533 Kalman-Bucy filter, 420 knock-out-barrier option, 128 Kolmogorov backward equation, 125, 130, 598 forward equation, 130 stationary, 131 Law of the Minimal Price, 147 least-squares estimate, 16 Monte Carlo, 638 leverage effect, 41 L´vy area, 261 e L´vy measure, 14 e L´vy process, 14, 31 e matrix, 97 L´vy’s Theorem, 26 e Lie group symmetry, 75 likelihood ratio, 390 linear congruential random number generator, 716 linear growth condition, 58 linear SDE, 101 linear test SDE, 572 linearity, 29 linearized SDE, 603, 629 stochastic differential equation, 598 Lipschitz condition, 58 local discretization error, 234 local martingale, 25 property, 29 local risk minimization, 473 lognormal model, 42 long term growth rate, 145 Longstaff model, 42 lookback option, 627 Lyapunov inequality, 22 852 Index mark, 10, 11 mark set, 13 mark-independent jump size, 511 market activity, 119 incomplete, 458, 472 price of event risk, 142 of risk, 141, 468 time, 120 market risk general, 151 specific, 151 Markov chain approximation, 722 second order, 730 third order, 731 convergence theorem, 744 hidden, 425 Markov inequality, 23 Marsaglia-Bray method, 716 martingale, 16, 133 estimating function, 398 jump, 140 local, 25 measure minimal equivalent, 593 property, 29 representation, 473, 591, 595, 596, 600, 605, 646 representation theorem, 607 strict local, 25 matrix commute, 48, 101 error covariance, 420 fundamental, 48 Riccati equation, 421 maximum likelihood estimation, 389 estimator, 392, 394, 417 mean, reverting, 451 measure change, 435 intensity, 13, 32 L´vy, 14 e Poisson, 55 transformation, 645 Merton model, 37, 50, 52, 55, 158, 161, 414, 562 method balanced implicit, 322, 451 Crank-Nicolson, 741 Euler, 234 finite difference, 734, 738 first strong order balanced implicit, 326 fully implicit, 741 Gaussian elimination, 742 Heath-Platen, 677 Heun, 239 inverse transform, 64 Marsaglia-Bray, 716 theta, 740 trapezoidal, 238 weak order 1.0 predictor-corrector, 501 2.0 predictor-corrector, 502 Milstein scheme, 256, 258 minimal equivalent martingale measure, 593 market model, 40, 115, 168 mixture of normals, 80 MMM, 40 generalized, 123 multi-dimensional, 117 stylized, 116 model affine, 42 ARCH diffusion, 50, 113 Bachelier, 40 binomial, 176 Black, 41 Black-Derman-Toy, 42 Black-Karasinski, 42 Black-Scholes, 40, 47 CEV, 40 CIR, 41 conditionally Gaussian, 460 constant elasticity of variance, 40 Courtadon, 42 Cram´r-Lundberg, 12, 52 e exponential L´vy, 14, 53 e Heston, 106, 111, 686 Ho-Lee, 41 Hull-White, 42 lognormal, 42 Index Longstaff, 42 Merton, 37, 50, 52, 55, 158, 414, 562 minimal market, 40, 115, 168 multi-factor, 458 Ornstein-Uhlenbeck, geometric, 40 Pearson-Sun, 42 Platen, 42 Sandmann-Sondermann, 42 Vasicek, 41, 209 modification Black-Scholes, 706 modified Bessel function, 71 moment estimate for SDEs, 59 Monte Carlo estimate, 478 least-squares, 638 method, 477 simulation, 477 multi-dimensional ARCH diffusion, 156 Heston model, 155 MMM, 117 multi-factor model, 458 multi-index, 196, 206, 363 multiple Itˆ integral, 253 o approximate, 263 multiple stochastic integral, 194, 197, 218 mixed, 283 multiple Stratonovich integral, 253 approximate, 265 multiplicative diffusion coefficient, 288 natural filtration, non-central chi-square distribution, 78, 107, 171 Wishart distribution, 80 normal inverse Gaussian distribution, 74 Novikov condition, 135, 646 numerical instability, 244 numerical stability, 571, 740 numerically stable, 244 optimal estimating equation, 403 option American, 700 basket, 621 European call, 166, 172 knock-out-barrier, 128 lookback, 627 path dependent, 638 Optional Sampling Theorem, 20 order of convergence, 238 order of strong convergence, 251 order of weak convergence, 491 Ornstein-Uhlenbeck process, 46, 69 orthogonal, 421 parameter estimation, 390, 413 partial differential equation, 123 integro differential equation, 124 path dependent option, 638 pathwise approximation, 251 payoff continuous, 617 non-smooth, 556, 633 rate, 126 smooth, 549, 707 Pearson-Sun model, 42 pentanomial tree, 709 Platen model, 42 Platen scheme, 267 Poisson jump measure, 44 measure, 12, 14, 32, 55 compensated, 195, 211 process, compensated, 17 compound, 10, 414 time transformed, portfolio, 142, 468 diversified, 152, 158 predictable, 19 predictor-corrector Euler scheme, 327, 576 price benchmarked, 734 pricing formula binomial option, 181 Black-Scholes, 184 real world, 148, 177 risk neutral, 148 primary security account, 140 probability measure change, 133 space, filtered, 853 854 Index probability (cont.) unnormalized conditional, 426 process counting, 139, 193 gamma, 97 general affine, 96 inverse Gaussian, 97 L´vy, 14 e Ornstein-Uhlenbeck, 46, 69, 85 multi-dimensional, 86 Poisson, predictable, 19 pure jump, 32, 193 Radon-Nikodym derivative, 133 square root, 62, 72, 259, 409 squared Bessel, 77, 116, 210 multi-dimensional, 89 stationary, VG-Wishart, 98 Wiener, Wishart, 70, 91, 94 property correlation, 29 covariation, 29 finite variation, 29 local martingale, 29 martingale, 29 regularity, 152 supermartingale, 146 pure jump process, 32, 193 SDE, 375 quadratic variation, 23 approximate, 23 Radon-Nikodym derivative, 133 random bit generator, 713, 716 measure, 12 number,quasi, 638 variable,integrable, real world pricing formula, 148, 177 recovery rate, 10 regular time discretization, 275 weak approximation, 507 regularity property, 152 remainder set, 206, 363 Richardson extrapolation, 241, 710 Riemann-Stieltjes integral, 28 risk neutral pricing formula, 148 roundoff error, 234 sample mean, 480 space, variance, 480 Sandmann-Sondermann model, 42 savings bond, 172 scale measure, 132 scenario simulation, 233, 248, 251 scheme derivative-free, 267, 309, 334 strong order 1.0, 310 drift-implicit Euler, 316 simplified Euler, 497 strong, 337 strong order 1.0, 318 Euler, 207, 246, 253, 276 Euler-Maruyama, 246 explicit strong order 1.5, 269 explicit weak order 2.0, 491 explicit weak order 3.0, 493 implicit weak Euler, 498 implicit weak order 2.0, 500 jump-adapted derivative-free, 355 drift-implicit, 357 Euler, 351 predictor-corrector, 359 strong order γ Itˆ, 367 o strong order γ Taylor, 363 Milstein, 256, 258 Platen, 267 predictor-corrector Euler, 327 simplified Euler, 482 strong order γ Itˆ, 331 o weak Taylor, 481 score vector, 394 SDE filter, 421 linear, 45, 48, 101 linearized, 603, 629 pure jump, 375 Stratonovich, 253 with jumps, 52, 54 Index second jump commutativity condition, 515 second moment estimate, 212 self-financing, 142, 468 semimartingale, 30 special, 30 sequence of approximate GOPs, 153 diversified portfolios, 152 JDMs regular, 152 sigma-algebra, 1, augmented, simplified Euler scheme, 482 symmetric predictor-corrector Euler scheme, 587 weak order 2.0 Taylor scheme, 484 3.0 Taylor scheme, 486 weak schemes, 712 Sklar’s theorem, 65 smooth payoff, 707 sparse matrix, 742 spatial discretization, 735 specific generalized volatility, 151 specific market risk, 151 square root process, 62, 72, 259, 409 squared Bessel process, 77, 116, 210 SR-process correlated, 89 multi-dimensional, 72, 87 stability region, 574 state space, stationary density, 131 diffusion, 409 independent increment, process, statistical error, 478 stochastic differential equation, 38 linearized, 598 vector, 43 integral compensated multiple, 198 multiple, 194, 197, 218 process, 855 stationary independent increments, 14 Taylor expansion, 187 time change, 113 stopping region, 592 stopping time, 19 inaccessible, 19 stratified sampling, 640 Stratonovich equation, 191 Stratonovich integral, 33 multiple, 253 Stratonovich SDE, 253 Stratonovich-Taylor expansion, 191 strong approximation, 251 arbitrage, 147 error, 368 solution, 57 strong order γ Itˆ scheme, 331 o γ Taylor scheme, 290 1.0 Taylor scheme, 278 1.0 compensated Taylor scheme, 284 1.5 Taylor scheme, 261 2.0 Taylor scheme, 264, 266 Student-t distribution, 74 stylized MMM, 116 sub-sigma-algebra, submartingale, 17 subordination, 80 supermartingale, 17, 25, 147 property, 146 strict, 17 symmetric integral, 33 systematic error, 478 Taylor scheme simplified weak order 2.0, 484 weak order 3.0, 486 strong order γ, 290 strong order 1.0, 278 strong order 1.0 compensated, 284 strong order 1.5, 261 strong order 2.0, 264, 266 weak compensated, 519 weak order β, 519 weak order 2.0, 483, 508, 512 weak order 3.0, 485 856 Index Taylor scheme (cont.) weak order 4.0, 488 theorem martingale representation, 607 weak convergence, 746 theta method, 740 time change stochastic, 113 time discretization equidistant, 23, 246 jump-adapted, 348, 377 regular, 275 time set, tracking rate, 153 transfer function, 575 transform function, 398, 399 transformed Wiener process, 39 transition density, 75 Gaussian, 66 transition distribution, 63 trapezoidal method, 238 tree binomial, 697 pentanomial, 709 trinomial, 704 Heston-Zhou , 708 tridiagonal solver, 742 trinomial tree, 704 unbiased estimator, 600 unhedgable part, 474 unique strong solution, 58 uniqueness, 57 of strong solution, 57 unnormalized conditional probability, 426 variance, conditional, 472, 639 variance reduction, 637 by conditioning, 640 technique, 480, 638 variance-gamma distribution, 74 Vasicek model, 3, 41, 209 extended, 41 vector stochastic differential equation, 43 VG-Wishart process, 98 volatility, 468 Wagner-Platen expansion, 187 weak approximation, 251, 478 convergence criterion, 478 convergence order, 491 convergence theorem, 746 error, 478, 699 relative, 552 weak order β Taylor scheme, 519 1.0 predictor-corrector method, 501 2.0 Taylor scheme, 483, 508, 512 2.0 extrapolation, 495 2.0 predictor-corrector method, 502 3.0 Taylor scheme, 485 4.0 Taylor scheme, 488 4.0 extrapolation, 496 convergence, 519, 724 weak scheme derivative-free, 529 simplified, 712 weak Taylor scheme, 481 Wiener process, correlated, 82 matrix, 83 multi-dimensional, time changed, 85 transformed, 39 Wishart process, 70, 91, 94 Wonham filter approximate, 428 problem, 425 Zakai equation, 426 zero coupon bond defaultable, 167 ... Perkins For other titles in this series, go to http://www.springer.com/series/602 Eckhard Platen r Nicola Bruti-Liberati Numerical Solution of Stochastic Differential Equations with Jumps in Finance. .. (1.1.1) and tij ∈ T determines its probability E Platen, N Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Stochastic Modelling and Applied Probability... uniqueness of solutions of SDEs These tools and results provide the basis for the application and numerical solution of stochastic differential equations with jumps 1.1 Stochastic Processes Stochastic