HANDBOOKS IN OPERATIONS RESEARCH AND MANAGEMENT SCIENCE VOLUME 15 Handbooks in Operations Research and Management Science Advisory Editors M Florian Université de Montréal A.M Geoffrion University of California at Los Angeles R.M Karp University of California at Berkeley T.L Magnanti Editors J.K Lenstra Centrum voor Wiskunde en Informatica, Amsterdam G.L Nemhauser Georgia Institute of Technology J.G Dai Georgia Institute of Technology Massachusetts Institute of Technology D.G Morrison University of California at Los Angeles S.M Pollock Volume 15 University of Michigan at Ann Arbor A.F Veinott, Jr Stanford University P Whittle University of Cambridge Amsterdam – Boston – Heidelberg – London – New York – Oxford Paris – San Diego – San Francisco – Singapore – Sydney – Tokyo North-Holland is an imprint of Elsevier Financial Engineering Edited by John R Birge University of Chicago, IL, USA Vadim Linetsky Northwestern University, IL, USA Amsterdam – Boston – Heidelberg – London – New York – Oxford Paris – San Diego – San Francisco – Singapore – Sydney – Tokyo North-Holland is an imprint of Elsevier North-Holland is an imprint of Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK First edition 2008 Copyright © 2008 Elsevier B.V 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 or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: permissions@elsevier.com Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-444-51781-4 ISSN: 0927-0507 For information on all North-Holland publications visit our web site at books.elsevier.com Printed and bound in The Netherlands 08 09 10 11 12 10 Contents I Introduction Introduction to the Handbook of Financial Engineering John R Birge and Vadim Linetsky References 11 CHAPTER An Introduction to Financial Asset Pricing Robert A Jarrow and Philip Protter 13 Introduction Introduction to derivatives and arbitrage The core of the theory American type derivatives Acknowledgements References 13 14 21 60 67 67 II Derivative Securities: Models and Methods CHAPTER Jump-Diffusion Models for Asset Pricing in Financial Engineering S.G Kou 73 Introduction Empirical stylized facts Motivation for jump-diffusion models Equilibrium for general jump-diffusion models Basic setting for option pricing Pricing call and put option via Laplace transforms First passage times Barrier and lookback options Analytical approximations for American options 10 Extension of the jump-diffusion models to multivariate cases References 73 75 84 89 92 94 96 100 103 108 113 CHAPTER Modeling Financial Security Returns Using Lévy Processes Liuren Wu 117 Introduction 117 v vi Contents Modeling return innovation distribution using Lévy processes Generating stochastic volatility by applying stochastic time changes Modeling financial security returns with time-changed Lévy processes Option pricing under time-changed Lévy processes Estimating Lévy processes with and without time changes Concluding remarks Acknowledgements References 120 127 133 144 155 159 159 160 CHAPTER Pricing with Wishart Risk Factors Christian Gourieroux and Razvan Sufana 163 Introduction Wishart process Pricing Examples Concluding remarks References 163 167 172 175 181 181 CHAPTER Volatility Federico M Bandi and Jeffrey R Russell 183 Introduction A model of price formation with microstructure effects The variance of the equilibrium price Solutions to the inconsistency problem Equilibrium price variance estimation: directions for future work The variance of microstructure noise: a consistency result The benefit of consistency: measuring market quality Volatility and asset pricing Acknowledgements References 183 184 186 191 202 210 210 216 217 217 CHAPTER Spectral Methods in Derivatives Pricing Vadim Linetsky 223 Introduction Self-adjoint semigroups in Hilbert spaces One-dimensional diffusions: general results One-dimensional diffusions: a catalog of analytically tractable models Symmetric multi-dimensional diffusions Introducing jumps and stochastic volatility via time changes Conclusion References 224 230 237 253 285 288 294 294 Contents vii CHAPTER Variational Methods in Derivatives Pricing Liming Feng, Pavlo Kovalov, Vadim Linetsky and Michael Marcozzi 301 Introduction European and barrier options in the Black–Scholes–Merton model American options in the Black–Scholes–Merton model General multi-dimensional jump-diffusion models Examples and applications Summary References 302 305 315 320 329 339 340 CHAPTER Discrete Barrier and Lookback Options S.G Kou 343 Introduction A representation of barrier options via the change of numeraire argument Convolution, Broadie–Yamamoto method via the fast Gaussian transform, and Feng–Linetsky method via Hilbert transform Continuity corrections Perturbation method A Laplace transform method via Spitzer’s identity Which method to use Appendix A Proof of (1) Appendix B Calculation of the constant β References 343 348 350 355 361 363 365 366 368 370 III Interest Rate and Credit Risk Models and Derivatives CHAPTER Topics in Interest Rate Theory Tomas Björk 377 Introduction Basics Forward rate models Change of numeraire LIBOR market models Notes Geometric interest rate theory Consistency and invariant manifolds Existence of nonlinear realizations 10 Potentials and positive interest References 377 378 381 387 390 400 400 401 411 419 434 viii Contents CHAPTER 10 Calculating Portfolio Credit Risk Paul Glasserman 437 Introduction Problem setting Models of dependence Conditional loss distributions Unconditional loss distributions Importance sampling Summary References 437 439 444 451 457 462 467 468 CHAPTER 11 Valuation of Basket Credit Derivatives in the Credit Migrations Environment Tomasz R Bielecki, Stéphane Crépey, Monique Jeanblanc and Marek Rutkowski 471 Introduction Notation and preliminary results Markovian market model Changes of measures and Markovian numeraires Valuation of single name credit derivatives Valuation of basket credit derivatives Model implementation References 472 476 481 485 492 497 500 507 IV Incomplete Markets CHAPTER 12 Incomplete Markets Jeremy Staum 511 Introduction The over-the-counter market Causes of incompleteness Pricing and optimization Issues in pricing and expected utility examples Quadratics Entropy and exponential utility Loss, quantiles, and prediction Pricing kernel restrictions 10 Ambiguity and robustness 11 Calibration 12 Conclusion Acknowledgements Appendix A Definition of incompleteness and fundamental theorems Appendix B Financial perspectives on incompleteness References 511 513 516 518 528 533 536 537 540 544 550 551 554 554 556 558 Contents ix CHAPTER 13 Option Pricing: Real and Risk-Neutral Distributions George M Constantinides, Jens Carsten Jackwerth and Stylianos Perrakis 565 Introduction Implications of the absence of arbitrage Additional restrictions implied by utility maximization Special case: one period without transaction costs Special case: one period with transaction costs and general payoffs Special case: two periods without transaction costs and general payoffs Special case: two periods with transaction costs and general payoffs Multiple periods without transaction costs and with convex payoffs Multiple periods with transaction costs and with convex payoffs 10 Empirical results 11 Concluding remarks Acknowledgements References 566 567 570 574 578 579 580 581 583 585 588 589 589 CHAPTER 14 Total Risk Minimization Using Monte Carlo Simulations Thomas F Coleman, Yuying Li and Maria-Cristina Patron 593 Introduction Discrete hedging criteria Total risk minimization in the Black–Scholes framework Total risk minimization in a stochastic volatility framework Shortfall risk minimization Conclusions References 593 599 603 618 625 632 634 CHAPTER 15 Queuing Theoretic Approaches to Financial Price Fluctuations Erhan Bayraktar, Ulrich Horst and Ronnie Sircar 637 Introduction Agent-based models of financial markets Microstructure models with inert investors Outlook and conclusion Acknowledgements References 638 639 649 671 674 674 V Risk Management CHAPTER 16 Economic Credit Capital Allocation and Risk Contributions Helmut Mausser and Dan Rosen Introduction Credit portfolio models and general framework 681 682 684 1000 P Boyle, S Feng and W Tian Pukhalskii, A.A (1991) On functional principle of large deviations In: Sazonov, V., Shervashidze, T (Eds.), New Trends in Probability and Statistics VSP Moks’las, Moskva, pp 198–218 Roll, R (1992) A mean/variance analysis of tracking errors Journal of Portfolio Management 18 (4), 13–22 Sadowsky, J (1996) On Monte Carlo estimation of large deviations probabilities Annals of Applied Probability 6, 399–422 Schilder, M (1966) Some asymptotic formulas for Wiener integrals Transactions of the American Mathematical Society 125 (1), 63–85 Sharpe, W.F (1966) Mutual fund performance Journal of Business 1966, 119–138 Stein, E., Stein, J (1991) Stock price distributions with stochastic volatility: An analytic approach Review of Financial Studies 4, 727–752 Sornette, D (1998) Large deviations and portfolio optimization Physica A 256, 251–283 Stutzer, M (1995) A Bayesian approach to diagnosis of asset pricing models Journal of Econometrics 68, 367–397 Stutzer, M (2000) A portfolio performance index Financial Analyst Journal May/June, 52–61 Stutzer, M (2003) Portfolio choice with endogenous utility: A large deviation approach Journal of Econometrics 116, 365–386 Varadhan, S.R (1984) Large Deviations and Applications, second ed SIAM, Philadelphia Williams, N.M (2004) Small noise asymptotics for a stochastic growth model Journal of Economic Theory 119 (2), 271–298 Subject Index 3/2 activity rate dynamics 3/2 dynamics 133 145 approximation error 892, 898, 901, 902 – approximate dynamic programming 926 approximation result for stochastic integrals 654 arbitrage 345, 566, 768 – free 27, 728 – free market 733 – opportunity 18, 27, 649, 733 – strategies 655 Asian options 343 ask price 513, 515, 519, 521, 524, 553 asset 868, 870, 882, 955 – allocation – – models 908 – – problems 902 – – rules 878 – price 183, 869 – return 870, 874 asset-liability 770 asymmetric information 214 asymmetric Laplace distribution 109, 110 asymptotic – analysis 78, 360 – approximations 201 – behavior 892, 896 – confidence intervals 895, 902 – convergence results 905 – covariance matrix 895 – coverage probability 895 – distribution 901 – efficiency 988 – error behavior 898 – error distribution 884, 887, 892, 893, 895, 898, 900, 901 – expansions 346, 361 – MCMD error distribution 901 – properties 867, 869 – second-order bias 895 – variance 894 asynchronous order arrivals 646 autocorrelation 79 autocorrelation function 80 autocovariance 79 α-admissible 29, 734 α-stable Lévy process 124 absolute aggregator 836 absolute risk tolerance 876 acceptance set 521, 522, 524, 525, 537, 549 accumulated gain 599 accuracy 908 ACF 80 ACF plot 80 action functional 995 actuarial reserving 770 actuaries 763 adapted 846 additive dual 935 admissible 29, 734 adverse selection 764 affine 132, 145 – diffusion process 169 – jump-diffusion 85 – models 909 – processes 903 – stochastic-volatility 85 agent-based models 639, 645 aggregate excess demand 647 aggregator 807 allocate capital 441 ambiguity 516, 517, 528, 544–549, 554 ambiguity aversion 789 American option 60, 74, 96, 103, 315, 343, 926 – American-type call option 16, 62 – American-type put option 63, 66 – pricing 869, 902, 906 American-style exercise 301 ancestor scenario 848 annuity rates 782 approximate MCMD estimators 884 approximately complete 728, 739 approximation 868, 869, 881–884, 887, 888, 890, 898 1001 1002 autoregressive gamma process 167 auxiliary – parameter 904, 905 – process 882, 884 average computation time 905–908 axe 530 axiom 79 Azéma martingales 38 backward – differences 884 – finite differences (MCBFD) 904 – induction 353 – stochastic differential equation 63, 792, 902 barrier 86, 100, 989 – correction 356 – crossing 74 – option 74, 96, 108, 113, 343 Basel 79 Basel Committee 459 Basel II 693 basis functions 853, 902 basket default swap 442 behavioral finance 79, 87, 639 Bellman equation 885 benchmark complete model 994 benchmark portfolio 767 Benders’ decomposition 857 bequest 872 Bermuda options 343 Bernoulli trials 975 bias corrected estimators 895 bid price 513, 515, 519, 521, 524, 553 bid–ask spreads 83, 366, 517 bid–price control 851 binomial approximation 21 binomial model 569 binomial tree 345, 347, 366 bisection method 880 Black–Scholes 17, 50, 123, 566 – formula 742 – hedge 738 – measure 778 – model 50, 146, 910 Black’s Formula 392 Bochner’s subordination 288 bonus 766 Borel sigma field 912 boundary conditions 875 bounded total variation 912 boundedness 880 bracketing method 880 Bromwich inversion integral 454 Subject Index Brownian 979 – bridge 347, 991 – functional 911 – increment 898, 912 – innovation 911 – motion 85, 741, 868, 870, 902–904, 909– 914 budget constraint 917 budget constraint multiplier 882 budget equation 796 bundle method 859 business time sampling 195 calculus of variations 909 calendar time sampling 195 calibration 514, 527, 550, 552, 553 call 15, 74, 741 capital allocation 689 capital diversification factor 698 capital diversification index 698 CAPM 543, 544 cardinal utility 796 cash delivery 736 cash flow 961 Cauchy principle value 355 CDS swaptions 496 central difference approximation 883 central differences 884 central finite differences (MCCFD) 904 central limit theorem 156, 345, 766, 902, 908, 972 certainty equivalent 815, 961 CEV 84 CGMY model 124 chain rule 873, 913, 916 change of numeraire 349, 387 change of variables 868, 880 chaos theory 84 characteristic exponent 121, 145 characteristic function 110 chartists 640 Clark–Ocone formula 872, 873, 913, 916 coefficient of absolute risk aversion 876 coefficient of relative risk aversion 817, 823 coherent risk measure 521 collateralized debt obligation 438, 443, 474 collective risk theory 768 common jumps 109, 111–113 commonotonic subadditivity 79 commutativity condition 881 comparative risk aversion 810 comparison lemma 808 comparison principle 808 Subject Index compensated Poisson process 38 complete 36, 737, 768 complete markets 869, 908 complete preferences 522, 548 compound autoregressive (Car) process 169 compound Poisson jump 120, 123 computation times 904 concavity adjustment 134, 146, 148 concavity-adjusted return innovation 146 conditional – independence framework 685 – Laplace transform 168, 170, 171 – moments 886 – tail expectation 775 – variance potentials 427 conditionally independent 96, 98, 438 conditionally independent migrations 485 confidence interval 893–895 constant – absolute risk aversion 534, 536 – elasticity of variance 84 – percentage strategies 982 – relative risk averse utility functions 882 – relative risk aversion 529, 874, 881, 903, 917 constrained problems 909 consumption 89, 870, 872 consumption-portfolio choice model 869, 875 consumption-portfolio policy 872 consumption-terminal wealth plan 871 contingent claim 736 continuation region 106 continuity correction 347, 355, 356 continuous compounding 77 continuously compounded forward rate 379 continuously compounded spot rate 379 contraction principle 977 convergence 902, 904 – analysis 902 – behavior 867, 869, 902 – issues 887 – parameter 884, 897, 901 – properties 882, 884, 898 – rate 902 – results 895, 902 – studies 902 convergent approximation 884 convergent estimator 894 convex normal integrands 847 convex risk measure 521 convexity 770 convolution 346 copula function 446 1003 copula models 446 corporate default 123, 131 correlation 97 cost of consumption 879 counterparties 439 covariance matrix 895 covariation 868 coverage probability 893 Cox–Ingersoll–Ross process 167, 979 CPU time 366 Cramér’s theorem 974 credit – default swaps 440, 442, 493 – derivatives 75 – migrations 477 – portfolio 986 – portfolio models 684 – ratings transition intensities 484 – risk 75, 344, 647 – risk models 180 CreditRisk+ 450, 462 cross-variation 912 CRRA preferences 906 cubic splines 597 cumulant exponent 122 cumulant generating function 453, 974 cumulative cost 599 cumulative Gaussian distribution function 894 curse of dimensionality 908 cutting plane 854 dampened power law 137, 141, 148 Dantzig–Wolfe decomposition (inner linearization) 857 death benefit 764 default 224 – intensity 180, 439 – payment leg 493 – swaps 438 defaultable bond 473 delta 96, 779 derivative 14, 15, 34 – asset 224 – pricing 566 – security 301 descendant scenario 848 deterministic finite difference methods 884 diagonal quadratic approximation 860 diffeomorphism 915 differencing scheme 900 differentiability 880 diffusion process 869, 873, 888–890, 898, 900, 914, 915 1004 diffusion state variables 908 Dirac measure 975 discount rate 90 discounted wealth 882 discrete – American options 344 – Asian option 343 – barrier options 344 – difference 898 – Girsanov theorem 349 – lookback options 344 – time portfolio choice problems 885 – trading strategies 743 discrete-time Wishart process 173 discretely compounded simple returns 78 discretization 881 – points 879, 884, 888, 895, 901, 902 – scheme 879, 894 – step 907 – values 904 discretized SDE 898 distortion 769 distribution of the price process 657 diversifiable 765 dividend process 473 dividend yield 870 dividend–price ratio 870 Doss transformation 868, 880, 895 double – auctions 646 – barrier options 344 – exponential distribution 77, 92, 109, 110 – exponential jump-diffusion model 93 doubling strategy 28 down-and-in call 356 down-and-out call 356 downcrossing 356 drift 870, 972 dual problem 909 duality 525, 926 Duffie–Epstein utility 817 Duffie–Pan–Singleton model 334 duration 770 dynamic – consistency 157, 547, 797 – consumption-portfolio problem 874 – hedging 765 – portfolio choice 789, 867 – portfolio problem 868, 885 – programming 868, 869 – programming algorithm 885 – programming principles 875 – trading 565 – utility 807 Subject Index early exercise region 106 economic credit capital 682 effective coverage probability 895 efficiency 904, 908 efficiency comparisons 895 efficiency ratio 904 efficient – estimators 905 – Monte Carlo estimators 890, 901 – price 213 – scheme 895 – twist 989 eigenfunction expansion 228 embedded options 764 empirical mean 888, 901 empirical probability measure 729 end effects 850 endogenous default 345 endowment 89 endowment policy 765 entropy 74, 527, 534, 536, 541, 542, 546, 548 entropy-Hellinger process 537 Epstein–Zin utility 818 equality in distribution 136 equidistant discretization 888 equilibrium 513, 543, 545, 548, 550, 551, 557 equilibrium price 513 Equitable Life 764 equity indexed annuity 767 equity premium 545, 547, 556 equity tranche 444 equivalent local martingale measure 734 equivalent martingale measure 30, 32, 728, 734, 992 ergodic risk-sensitive control 983 error components 900 error distribution 902 error term 891, 902 Esscher transforms 537 essentially bounded functions 846 estimation error 892 estimator error 896 Euler 884 – algorithm 93, 95 – approximation 881 – equation 89 – inversion 103, 365 – scheme 879, 888, 895, 904 European – call 15, 45, 94, 736 – derivative 172, 173 – put 15, 46, 94 Subject Index European-style exercise 301 excess returns 885 exchange options 108, 113 execution times 904 expected – approximation error 888, 889, 893, 895 – errors 890 – shortfall 438, 683 – utility 523, 526, 527, 871, 903 – value principle 769 exponential 123 – approximation 108 – decay 973 – martingale 138, 145 – principle 769 – tightness 974 – utility 534, 536, 542, 835, 951, 952, 956 – type distribution 73, 74, 96 – type tails 73, 77 exponentially dampened power law 124 exponentially twisted 989 extended Black–Scholes 741 external law enforcement 79 external risk management 74 external risk regulations 79 extrapolation 74 extrapolation scheme 313 extreme tail dependence 449 factor model 163, 164, 457 factor structure 448, 449 fair hedging price 595, 616 false position method 880 fast Fourier transform (FFT) 149, 153, 351, 951 fast Gaussian transform 346, 347, 351 FBSDE 813 feedback 639 feedback effects 655 Fenchel–Legendre transformation 974 Feynman–Kac 876 Feymann–Kac formula 74, 98, 99 Feynman–Kac semigroup 226 Filipovi´c state space approach 408 filtration 21 financial engineering 764 financial engineers 764 financial market 869, 870 finite – activity 122 – difference 869, 884, 908 – difference approximation 882–884, 900, 901 – difference methods 303, 882, 884 1005 – difference schemes 303 – dimensional realization 411 – element method 303 – element schemes 908 – quadratic variation 123 – sample properties 202 – variation 122 first fundamental theorem of asset pricing 525, 555, 728, 733 first passage time 96, 109, 113, 346, 350 first variation process 876, 915, 918 first-order conditions (FOC) 885 first-order risk aversion 820 first-to-default swap 442 fixed income derivatives 74 fixed mix 855 fixed point algorithm 906 fixed transaction costs 745 flat boundary 96 Fleming–Viot process 979 Flesaker–Hughston fractional model 421 floating lookback 344 fluid limit 658 force of mortality 777 forward – backward stochastic differential equations 792 – CDS 495 – curve manifold 402 – differences 884 – finite differences (MCFFD) 904 – kth-to-default CDS 500 – measures 389 – rate equation 400 – rate models 381 – volatilities 392 Fourier inversion 148, 150, 152, 353 Fourier transform 94, 122, 145, 149, 152, 352 fractal Brownian motion 84 fractional – Brownian motion 639, 648 – Fourier transform (FRFT) 154 – Ornstein–Uhlenbeck process 667, 669 – volatility 671 Fréchet differentiable 58 free boundary problem 103 free lunch with vanishing risk 555, 734 Freidlin–Wentzell 977 Frobenius 414 full-information price 213 functional central limit theorem 653 functional law of large numbers 660 fundamental pricing PDE 305 Subject Index 1006 fundamental theorem of asset pricing fundamentalists 640 future consumption 872 28 gain process 26 gain–loss ratio 541, 961 Galerkin finite element method 301 Gamma 96 GARCH models 84 Gärtner–Ellis 976 Gaussian – copula 446, 465 – distribution 353 – investment 960 – martingale 893, 894, 897, 900, 901 – process 893 Gaver–Stehfest 102, 103 generalized hyperbolic distribution 110 geometric Brownian motion 20, 50, 77, 902 Girsanov theorem 354 Girsanov transformation 994 good deal bounds 521, 524–526, 541–543, 549, 552 good semimartingales 654 granularity adjustment 460, 695 Gronwall’s lemma 662 growth-maximum 980 guaranteed annuity options 765 Guaranteed Minimum Accumulation Benefit 776 Guaranteed Minimum Death Benefit 776 Guaranteed Minimum Income Benefit 776 Guaranteed Minimum Maturity Benefit 776 Guaranteed Minimum Withdrawal Benefit 776 guarantees 764 habit formation 838 HARA 871 hazard rate 439 heat equation 362 heavy tails 76 heavy-tailed sojourn time 651 hedge 770 hedging – component 898 – demand 519, 520, 876, 878, 879, 892, 894, 907 – error 744, 773 – motive 874, 876 – parameters 365, 366 – term 868, 877, 882–884, 893, 898 Hermite expansion 352 Hermite functions 352 Heston model 139, 146 high peak 76 high-frequency data 183 higher order polynomial-regression methods 904 Hilbert transform 346, 347, 352, 353, 355, 366 historical volatilities 83 history process 846 hitting time 990 HJB equation 876 HJM drift condition 383, 385 homeomorphism 915 homothetic 813 Hurst parameter H 648 hybrid approach 774 hyperbolic models 84 IAT 904, 905 idiosyncratic risk 217 (il-)liquidity risk 217 illiquidity effects 644 immunization 770 imperfect markets 588 implementable 846 implied – binomial trees 85 – Black volatilities 392 – correlation 449 – volatility 82, 83 – volatility smile 84 importance sampling 462, 988 imprecise probability 544 Inada conditions 887, 888 incomplete 764, 992 incomplete market 566, 826, 908, 972 incomplete preferences 548, 549 incremental perturbation 876 independent components analysis 950, 957, 959 index options 566 indexes 166 indifference prices 521, 522, 524, 525, 530, 531, 536, 553 indifference pricing 74, 522, 523, 526, 537, 545, 548, 551, 552 indirect utility function 876 individual jump 109, 111, 113 inertia 638, 647 infinite activity 122 infinite expectation 78 infinite variation 122 infinitely divisible 43 Subject Index infinitesimal generator 93, 100, 323 infinitesimal perturbation 898 information ratio 985 initial – auxiliary parameter 905 – Brownian increment 907 – condition 870, 914, 915 – MPR 904 – perturbation 877 – shadow price of wealth 888 – wealth 879 inner and outer solutions 361 instantaneous activity rate 129 instantaneous forward rate 379 instantaneous mean–variance efficiency 824, 827 institutional investors 674 insurer insolvency 766 integral equation 99, 107 integrated process 170, 171 integrated Wishart process 174 integration by parts 911, 912 integration by parts formula 912, 921 integro-differential 99 – equation 74, 98 integro-differential free boundary problems 104 integro-differential variational inequality 301 interacting agents 639 interactive Markov processes 641 interest rate 870, 883, 885, 903, 916 intermediate consumption 881, 891, 892, 903 intermediate utility 885 internal risk management 79 intertemporal hedging demand 874, 903 intertemporal hedging terms 868 intertemporal marginal rate of substitution 568 invariant manifold 401 inverse average time (IAT) 904 inverse marginal utilities 883, 884 inverse marginal utility functions 873 invertibility 880 investment horizon 904 investment opportunity set 868 investor inertia 647 investor sentiment 87, 639, 656 Itô – formula 74, 99, 919 – integral 912 – lemma 879, 915 – price processes 868, 873 1007 Jacobian matrix 873 joint error 891 jump diffusion 85, 96, 301, 345, 360, 363, 998 jumps 516 Kelly investment strategy 980 kinked proportional aggregator Knightian uncertainty 544 knock-out options 306, 989 Kou’s model 332 Kreps and Porteus utility 816 kth-to-default CDS 474 kurtosis 76, 949, 953 831 L-estimators 703 L-shaped method 857 L1 -strategy 597 L2 -hedging 597 L2 errors 902 ladder height 357 Laplace inversion 93, 95, 365 Laplace transform 74, 94, 101, 145, 147, 148, 346, 354, 363, 366, 454 lapsing 767 large deviation principle 972 large deviation techniques 972 large-scale problems 885 lattice methods 346, 347, 908 law of large numbers 972 Leibniz’s formula 364 leptokurtic distribution 76 leptokurtic feature 84, 86 Lévy – characteristics 121 – density 121 – process 43, 81, 84, 120, 352, 355, 357, 953 Lévy subordinator 128 Lévy–Khintchine Theorem 121 LIBOR – forward rate 379 – market models 390 – rate 379 – spot rate 379 Lie algebra 414 life annuity 782 life insurance 763 likelihood ratio 463 limit order markets 646 limited liability law 79 limiting loss distribution 459 linear – approximations 904 – BSDE 799 1008 Subject Index – complementarity problem 315 – SDE 914 – supply curve 748 Liouville transformation 247 Lipschitz and growth conditions 888 liquidity 214 liquidity cost 733 liquidity risk 727 local – martingale 29 – mean–variance hedging 74 – risk minimization 534, 536, 600 – utility 527 – utility maximization 520, 542 – volatility 517 log-RSPD 882 logarithmic – moment generating function 974 – state price density (SPD) 884, 904 – utility 868 lognormal distribution 774 long range dependence 639 long term investors 868 lookback options 58, 74, 86, 96, 100, 102, 343 loss-given-default 180 lower bounds 937 Lugannani–Rice approximation 456 Malliavin – calculus 868, 869, 873, 909, 913, 914, 916 – derivative operator 913 – derivative representation 876 – derivatives 60, 868, 869, 874, 877, 880–882, 884, 892, 908–916, 918 marginal capital contributions 689 marginal utility 884, 887, 888 marginal utility functions 871 marked-to-market value 732 market – completion 527, 826 – depth 644 – imbalance 650, 663 – incompleteness 565 – microstructure 638 – microstructure noise 183 – price of risk 138, 142, 550, 870, 884, 903, 916, 995 – quality 210 market maker 513, 519, 552 market price of risk process 799 marking to market 513, 515 Markov – chain 347 – chain approximations 908 – potential approach 428 – process 45, 223 – switching 671 Markovian market model 481 Markovian numeraires 488 Markowitz 950 Markowitz theory 957 martingale 74, 91, 99, 872, 898, 912, 913 martingale representation theorem 913 mathematical finance 971 maturity benefit 767 maturity date 378 max–min utility 860 maximum likelihood 155, 157, 159 MCBFD 900, 901, 907 MCBFD estimators 884, 907 MCC estimators 882, 884, 895, 900, 901 MCC (Monte Carlo Covariation) method 869, 882, 887, 901, 904–908 MCCFD 884, 900, 901, 907 MCCFD estimators 884, 907 MCCN 905, 907 MCCO 905, 907 MCFD 869, 901, 904, 905, 908 MCFD estimators 884, 898, 900, 901, 907 MCFD methods 904, 907, 908 MCFD portfolio estimators 887 MCFFD 884, 900, 901, 907 McKean–Vlasov limit 977 MCMD 867, 869, 887, 898, 900, 901, 904, 905, 908, 909 MCMD estimators 884, 895, 901 MCMD portfolio estimator 891 MCMD-Doss estimator 881 MCR 887, 904, 905, 908 MCR error 903 MCR methods 905 MCR portfolio estimator 902 MCR-lin methods 904, 906 mean–variance 876, 892 – analysis 868 – component 868, 893, 898, 906 – demand 874, 903 – hedging 74 – optimal 534 – optimizers 868 – portfolio 884 – portfolio rules 868 – ratio 519 – trade-off 868 mean-squared error 190 memoryless property 96, 97 merger 79 Subject Index Merton point 855 Merton’s (1974) model 445 Merton’s model 332 Merton’s solution 875 method of van Wijngaarden–Dekker–Brent 880 Milshtein scheme 879, 881, 884, 895 minimal entropy martingale measure 536 minimal martingale measure 534, 537, 542 minimax martingale measure 527 minimum-distance 552 minimum-distance measure 526, 527, 534, 536, 541, 553 misspecifications 79 mixed Poisson model 450, 460 model – error 773 – estimation 155 – misspecification 993 – of Black and Cox 445 – risk 553 – specification 992 modified MCC method 906, 907 moment generating function 93 moments 540, 543, 972 money account 380 money market account 22, 35 monitoring points 355, 363 Monte Carlo – averaging procedure 894 – error 902 – estimators 901, 905 – finite difference (MCFD) method 867, 869, 882 – Malliavin derivative (MCMD) 867, 868 – Malliavin derivative method with Doss transformation (MCMD-Doss) 868 – MCC 867, 869 – method 366, 867, 878, 901, 902, 904 – regression (MCR) method 867 – regression (MCR) scheme 869 – replications 895, 901, 902 – simulation 345–347, 457, 701, 771, 869, 875, 882, 885, 907, 908, 972 moral hazard 764 MPR 906 MPR diffusion 907 multi-currency models 425 multi-factor model 696 multiple exercise opportunities 936 multiple priors 545, 547, 548 multiplicative dual 934 multiplier 874, 878, 879, 881 multivariate 74 – asymmetric Laplace distribution – diffusion 880, 881 – normal distribution 110, 350 Musiela equation 386 Musiela parameterization 386 1009 109 Nash equilibrium 673, 674 negative dependence 180 Nelson–Siegel family 406 nested decomposition 858 nested decomposition method 854 neutralization 532 neutralizing 533 Newton–Raphson procedure 880 no arbitrage condition 30 no free lunch with vanishing risk 30, 735 no-arbitrage assumption 172 no-arbitrage bounds 515, 521, 522, 525, 531, 537 no-arbitrage price bounds 521, 540, 549 noise traders 640 nominal size 894, 895 non-Gaussian 950, 959 non-Gaussian investment 960 non-Gaussianity 951 non-stationary Gaussian process 662 nonanticipative 846 noncentered error distributions 901 nondiversifiable risk 765 nonsmoothness 99 nontradeable income 838 nontradeable income stream 835 normal – approximation 356 – copula model 685 – distribution 92, 345 – inverse Gaussian distribution 449 – jump-diffusion model 93 nth-to-default swap 442 numéraire 32, 536 numéraire invariance 32 numerical – approximation 891, 900 – discretization scheme 884 – experiments 908 – fixed point scheme 901 objective function 917 obligors 439 observable factors 165 observational equivalence 547 opportunity set 870, 874 optimal 1010 Subject Index – consumption 875, 876, 888 – consumption policy 872, 874, 875, 917 – convergence rate 908 – dynamic portfolio policies 874 – exercise boundary 106 – future consumption 872 – policy 882 – portfolio 868, 869, 872, 875, 881, 908 – – calculations 869 – – policy 867, 874, 875, 903 – – rule 868, 908 – stock demand 903 – stopping 315 – stopping problem 60, 359, 902 – wealth 874, 878, 916, 921 – wealth process 881, 915, 919 optimality conditions 812 option bounds 577 option pricing 209, 565, 728, 882, 971 optional σ-algebra 23 optional sampling theorem 96 order arrivals 646 order books 646 order rates 650 ordinal utility 797 ordinary differential equation (ODE) 361, 880, 884 Ornstein–Uhlenbeck (OU) process 667, 671, 903, 996 orthonormal basis 906 outer linearization 853 outperformance event 985 over-the-counter (OTC) derivatives 513 overreaction 87 overshoot 96, 356 Panjer recursion 462 parameters 972 partial barrier option 356 partial differential equation (PDE) 302, 347, 361, 868, 875 partial integro-differential equation 301 partial lookback options 344 path-dependent 100 path-dependent options 74, 343 penalization 318 pension plans 764 percentage lookback options 344 performance indices 984 perpetual American options 86 perturbation 877, 878, 907, 910 – analysis 346 – method 347, 359 – parameter 901 physical delivery 736 piecewise exponential approximation 105 planning horizon 869 Poisson process 85 polynomial 887, 902 polynomial basis 887, 902 polynomial estimators 902 portfolio 22, 885 – allocation 914 – choice 206 – choice models 908 – choice problem 868, 885, 902 – components 868, 891, 892 – constraints 908, 937 – credit risk 437, 505 – decomposition 874 – demand components 891 – estimator 867, 869, 895, 902 – formula 868, 872, 874, 880 – hedging components 884 – measurement 972 – optimization 937, 972 – policy 881 – problem 885 – rules 868 – selection 971 – theory 874, 950 – weight 906 positive interest 419 positively homogeneous risk measures 441 potential 419, 421 power (constant relative risk averse) class 887 power-type distributions 73, 92 power-type right tail 77 power-type tails 73, 77 predictable σ-algebra 23 predictable process 872 predictable representation property 37 prediction intervals 539 preference function 522 preferences 876, 882, 901, 917 premium 765 premium payment leg 494 premium principles 765 price system 870 price takers 727 pricing error 212 pricing kernel 515, 540–542, 552, 555, 565, 566 pricing operator 223 pricing semigroup 223, 225 probabilistic representation 875, 876, 890 Subject Index probability 972 probability of ruin 768 probability space 21, 729 programming effort 366 progressive hedging algorithm 859 progressively measurable 24 projection basis 902 projections 902 proportional aggregator 813, 829 proportional hazard risk 770 proportionate transaction costs 745 prospect theory 79 protection buyer 442 protection seller 442 proximity 997 put 74 put option 15, 781 put–call parity 15, 83 quadratic 132, 145 – approximation 104 – BSDE 828 – hedging 533 – model 257 – problem 886 – variation 893, 897, 900 quadrature method 347 quadrature schemes 908 quantile 78, 79 – estimation 463 – hedging 539 – principle 769 quantization 908 quartic equation 93 quasi-exponential 415 quasi-Monte Carlo methods 850 quasi-quadratic absolute aggregator 836 quasi-quadratic proportional aggregator 825 queuing models of investor behavior 638 Radon–Nikodym derivative 51, 988 random Riemann integral 911 random walk 356 random walk hypothesis 81 rare events 986 rare-event simulation 462 rate function 976, 977 rate of convergence 895 ratings transitions 440 rational expectations 74, 89 real options 516, 551 real probability measure 566 real-world models 771 realized covariance 206 1011 realized volatility 183 reasonable price process 51 recovery 224 recovery rate 180 recursive convolution 452 recursive utility 790, 804 redundant derivative 34–36 reflection principle 96, 97, 357 regime switching lognormal 774 regression 869, 887 – based computation 885 – based Monte Carlo methods 902 – method 869, 887, 906 – parameters 887 – simulation method 869 relative efficiencies 901 relative errors 904 relative risk aversion 828, 868, 876 relative risk aversion coefficients 873 relative risk-aversion process 823 relative state price density (RSPD) 870 renewal 74 renewal arguments 99 renewal equation 100 renewal-type integral equation 100 replication strategy 45 reservation prices 519 reset option 776 resolvent 428 retirement 838 return volatilities 870 returns 886 reversionary bonuses 766 Riccati equations 171 Riccatti ordinary differential equations 903 Ridder’s method 880 Riemann and Itô integrals 912 Riemann zeta function 355, 359 Riesz decomposition 426 right stochastic exponential 889 risk – aversion 789, 814, 904 – aversion coefficients 882 – contributions 438, 441, 462 – factors 172 – management 79, 206, 764 – management problems 869 – measures 79, 765 – neutral probability 30 risk-free asset 885 risk-neutral distribution 566 risk-neutral pricing measure 74 risk-neutral probability 565 1012 Subject Index riskfree rate 870 riskless asset 870, 903 riskless interest rates 378 risky asset returns 870 risky assets 869, 870, 885 risky stock 903 RMSRE 905, 906 robust 79, 546, 549, 551, 552 – estimation 860 – risk measures 74, 79 – utility 545, 548, 549, 552, 553 robustness 79, 528, 544, 545, 553, 789 root mean square relative error (RMSRE) 904–908 roots 93 running consumption 893, 894 running utility 887 σ martingale 29 saddlepoint 455, 464 saddlepoint approximation 455 sample sizes 79 Sanov’s theorem 975 scale invariance 821 scale-invariant 813 scenario 848 scenario generation 853 scenario reduction 853 Schilder 978 scores 166 secant method 880 second fundamental theorem of asset pricing 39, 555, 728, 735 second-order – approximation 662, 663 – bias 890, 893–895, 901, 902, 904, 907 – bias corrected estimators 890, 901 – bias function 893 – bias term 898, 900, 901 segregated funds 767 self-financing 26, 599 self-financing trading strategy 731 semi-Markov process 639, 651 semi-martingale 21, 650 sequential analysis 356 shadow price 888 shadow price of optimal wealth 888 shadow price of wealth 901, 902, 917 Shareownership2000 647 sharp large deviations 989 Sharpe ratio 541, 544, 795, 824, 961, 980, 985 short rate 379 short term investors 868 shortfall 537 shortfall risk 593, 598, 625 shortfall strategy 982 sigma martingale 32 signal processing 950 simple forward rate 379 simple spot rate 379 simulation 868, 869, 879, 881, 882, 884, 901, 903 – approach 878, 881 – method 868, 869, 908 – schemes 908 single backward difference 883 single forward difference 883 size distortion 894, 895 skewness 76, 614, 953 small investors 647, 648 smooth Brownian functional 909–911 Snell Envelopes 61 solvency 764 source-dependent first-order risk aversion 831 source-dependent risk aversion 818, 836 sovereign default 123 S&P 500 index 648 special semimartingale 55 spectral representation 228 Spectral Theorem 242 speed of computation 908 speed of convergence 907 speed-accuracy trade-off 904–908 speeds of convergence 901 Spitzer function 348, 359, 362 Spitzer’s identity 346, 347, 364, 366 spline approximation 853 spot rate of interest 21 spot volatilities 392 standard deviation principle 769 state dependent Markovian service networks 655 state dependent queuing networks 639 state price density process 420 state price density (SPD) 566, 798, 870, 876, 882, 904 state variables 868–870, 874, 876–878, 882, 884, 885, 887, 898, 901, 903, 908, 918 state-dependent queuing networks 646 static budget constraint 872, 878 static consistency 157 statically hedge 596 stationary on/off process 669 step-up corporate bonds 474 Stirling’s formula 368 Subject Index stochastic – central tendency 132, 147 – correlation 180 – covolatility 166, 180 – differential equation 44, 870, 873, 874, 877, 888, 890, 894, 900, 914–916, 972 – discount factor 172–174, 419, 420, 567, 870, 873 – discounting 801 – dominance 543, 544 – dominance bounds 566 – dynamic programs 845 – elasticity 650 – exponential 42 – flow 914, 915 – flow of diffeomorphisms 915 – flow of homeomorphisms 915 – game 673 – integral 881, 911 – intensity 439 – investment opportunity set 824 – process limit theorems 638 – processes 971 – risk aversion 824 – simulation 775 – skewness 128, 130 – time changes 128 – volatility 82, 84, 128, 164, 166, 334, 516, 593, 993 – volatility model 164, 177 – volatility model of Heston (1993) 135 – Wiener integral 911 stock price 910 stock return 903 stocks 869 stopping rule 17 Stratonovich integral 404 strong approximation 657 strong Markov process 45 structural model 179 structural model of default 445 structured product 784 Sturm–Liouville problem 242 sub-discretizations 881 subadditivity 79 subgradient method 859 subjective discount factor 874 subsistence consumption 871 sum assured 765 sunspot equilibrium 557 super-replicating trading strategy 61 super-replication 594, 744 supergradient density 798 superreplication 522, 539 1013 supply curve 728, 729, 747 surplus 766 survival claims 492 survival function 769 swaptions 784 symmetric Markov process 223 synthetic CDO 444 t-copula 449 t-distribution 92 tail conditional expectations 79 tail conditional median 79 tail distributions 79 – distinguishing 79 tail risk 972 tailweight 73 tangent process 869, 876, 877, 882–884, 889, 898, 900, 901, 915, 918 Taylor approximation 902 Taylor series 885 Taylor series approximation 885 t distribution 77 temporal utility 847 term insurance 765 term structure density 432 term structure models 75 terminal measure 395 terminal wealth 875, 876, 881, 886, 891–894, 903 Theta 96 thin-tailed 774 time changed Brownian motions 85 time discretization 850 time-additive utility 802 time-changed Lévy process 82, 85 time-separable von Neumann–Morgenstern representation 871 total cost 605 total risk 606 total risk minimization 593, 601 total variation 912 totally inaccessible stopping time 23 tracking error variance 985 trading – activity 650 – constraints 833 – costs 565 – strategy 22, 26, 599, 728, 731 tranches 444 transaction costs 183, 516, 517, 565, 566, 728, 743, 745, 838 transaction time sampling 195 transactions 773 1014 Subject Index transform analysis 178 transform inversion 453 translation invariance 835 transportation metric 853 trinomial tree 347 true coverage probability 894 truncation function 121 two-dimensional barrier options 108, 344, 356 two-dimensional Laplace inversion 102 two-dimensional Laplace transform 102 two-stage simulation procedure 879 uncertainty 544 underreaction 87 unhedged liability 779 uniform integrability conditions 890 unit linked 783 unit linked contracts 767 univariate diffusions 881 unobservable factors 165 unscented Kalman filter 157, 159 up-and-in put 356 up-and-out call 344, 348 up-and-out put 356 upcrossing 356 upper bounds 937 usual hypotheses 21 utility 813 – function 90, 871, 876, 885, 887 – maximization 89 – of intermediate consumption 882 – of terminal wealth 882, 887 – supergradient density 810 vague convergence 664 valuation measures 488 valuation of American contingent claims 902 value function 301, 875, 876, 885, 886, 902 value-at-risk (VaR) 74, 79, 438, 441, 682 Varadhan 977 variable 892 variable annuities 765 variance 894, 895, 912 – gamma (VG) model 124, 951, 953 – principle 769 – reduction techniques 706 – optimal martingale measure 534 variational inequality 315 variational methods 74, 301 Vasicek model 257 Vega 96 volatility 870, 881, 882, 903, 911, 913 – allocation 691 – clustering effect 81, 84 – coefficient 881 – risk 216 – smile 257, 566 – term 972 von Neumann–Morgenstern preferences 869 Walrasian auctioneer 646, 650 weak convergence 653, 888, 974 weakly stationary 79 wealth 867, 868, 870, 872, 885, 918 – derivative 876 – process 872, 881 – proportions 870 whole life 765 Wiener 912 – functional 909 – measure 912, 913 – (or Brownian) functionals 909 – space 909, 912 Wiener–Hopf equation 348, 359, 362 Wiener–Hopf factorization 97 Wilkie model 771 Wishart – factor models 165 – process 168, 169, 174 – quadratic term structure 176 – risk factor models 175 zero coupon bond 90, 378 zero utility principle 769 zonal polynomial 169 ... allocate investment funds to meet financial goals As financial models are stochastic, probability theory and stochastic processes play a central role in financial engineering Furthermore, in order... risk management, and survey financial engineering applications in insurance The authors compare the actuarial and financial engineering approaches to risk assessment and focus on the life insurance... Introduction to the Handbook of Financial Engineering John R Birge Graduate School of Business, University of Chicago, USA Vadim Linetsky Department of Industrial Engineering and Management Sciences,