Bielecki credit risk frontiers; subprime crisis, pricing and hedging, CVA, MBS, ratings, and liquidity (2011)

P1: OTA/XYZ P2: ABC FM JWBT388-Bielecki December 16, 2010 19:33 Printer Name: Yet to Come i P1: OTA/XYZ P2: ABC FM JWBT388-Bielecki December 16, 2010 19:33 Printer Name: Yet to Come CREDIT RISK FRONTIERS i P1: OTA/XYZ P2: ABC FM JWBT388-Bielecki December 16, 2010 19:33 Printer Name: Yet to Come Since 1996, Bloomberg Press has published books for financial professionals on investing, economics, and policy affecting investors Titles are written by leading practitioners and authorities, and have been translated into more than 20 languages The Bloomberg Financial Series provides both core reference knowledge and actionable information for financial professionals The books are written by experts familiar with the work flows, challenges, and demands of investment professionals who trade the markets, manage money, and analyze investments in their capacity of growing and protecting wealth, hedging risk, and generating revenue For a list of available titles, please visit our Web site at www.wiley.com/go/ bloombergpress ii P1: OTA/XYZ P2: ABC FM JWBT388-Bielecki December 16, 2010 19:33 Printer Name: Yet to Come CREDIT RISK FRONTIERS Subprime Crisis, Pricing and Hedging, CVA, MBS, Ratings, and Liquidity Tomasz R Bielecki, Damiano Brigo, and Fred Patras ´ eric ´ iii P1: OTA/XYZ P2: ABC FM JWBT388-Bielecki Copyright December 16, 2010 C 19:33 Printer Name: Yet to Come 2011 by Tomasz R Bielecki, Damiano Brigo, and Fr´ed´eric Patras All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance 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loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books For more information about Wiley products, visit our web site at www.wiley.com Library of Congress Cataloging-in-Publication Data: Bielecki, Tomasz R., 1955– Credit risk frontiers : subprime crisis, pricing and hedging, CVA, MBS, ratings, and liquidity / Tomasz R Bielecki, Damiano Brigo, and Fr´ed´eric Patras p cm Includes index ISBN 978-1-57660-358-1 (hardback); ISBN 978-0-47087-923 (ebk); ISBN 978-1-11800-382-4 (ebk); ISBN 978-1-11800-383-1 (ebk) Credit derivatives—United States Global financial crisis, 2008–2009 I Brigo, Damiano, 1966– II Patras, Fr´ed´eric III Title HG6024.U6B54 2011 332.64'57—dc22 2010045236 Printed in the United States of America 10 iv P1: OTA/XYZ P2: ABC FM JWBT388-Bielecki December 16, 2010 7:49 Printer Name: Yet to Come Contents Foreword ix Greg M Gupton Introduction Tomasz R Bielecki, Damiano Brigo, and Fr´ed´eric Patras PART I: EXPERT VIEWS CHAPTER Origins of the Crisis and Suggestions for Further Research Jean-Pierre Lardy CHAPTER Quantitative Finance: Friend or Foe? 19 Benjamin Herzog and Julien Turc PART II: CREDIT DERIVATIVES: METHODS CHAPTER An Introduction to Multiname Modeling in Credit Risk 35 Aur´elien Alfonsi CHAPTER A Simple Dynamic Model for Pricing and Hedging Heterogeneous CDOs 71 Andrei V Lopatin CHAPTER Modeling Heterogeneity of Credit Portfolios: A Top-Down Approach 105 Igor Halperin CHAPTER Dynamic Hedging of Synthetic CDO Tranches: Bridging the Gap between Theory and Practice 149 Areski Cousin and Jean-Paul Laurent CHAPTER Filtering and Incomplete Information in Credit Risk 185 Răudiger Frey and Thorsten Schmidt v P1: OTA/XYZ P2: ABC FM JWBT388-Bielecki December 16, 2010 7:49 Printer Name: Yet to Come vi Contents CHAPTER Options on Credit Default Swaps and Credit Default Indexes 219 Marek Rutkowski PART III: CREDIT DERIVATIVES: PRODUCTS CHAPTER Valuation of Structured Finance Products with Implied Factor Models 283 Jovan Nedeljkovic, Dan Rosen, and David Saunders CHAPTER 10 Toward Market-Implied Valuations of Cash-Flow CLO Structures 319 Philippos Papadopoulos CHAPTER 11 Analysis of Mortgage-Backed Securities: Before and After the Credit Crisis 345 Harvey J Stein, Alexander L Belikoff, Kirill Levin, and Xusheng Tian PART IV: COUNTERPARTY RISK PRICING AND CREDIT VALUATION ADJUSTMENT CHAPTER 12 CVA Computation for Counterparty Risk Assessment in Credit Portfolios 397 Samson Assefa, Tomasz R Bielecki, St´ephane Cr´epey, and Monique Jeanblanc CHAPTER 13 Structural Counterparty Risk Valuation for Credit Default Swaps 437 Christophette Blanchet-Scalliet and Fr´ed´eric Patras CHAPTER 14 Credit Calibration with Structural Models and Equity Return Swap Valuation under Counterparty Risk 457 Damiano Brigo, Massimo Morini, and Marco Tarenghi CHAPTER 15 Counterparty Valuation Adjustments 485 Harvey J Stein and Kin Pong Lee CHAPTER 16 Counterparty Risk Management and Valuation Michael Pykhtin 507 P1: OTA/XYZ P2: ABC FM JWBT388-Bielecki December 16, 2010 7:49 Printer Name: Yet to Come Contents vii PART V: EQUITY TO CREDIT CHAPTER 17 Pricing and Hedging with Equity-Credit Models 539 Benjamin Herzog and Julien Turc CHAPTER 18 Unified Credit-Equity Modeling 553 Vadim Linetsky and Rafael Mendoza-Arriaga PART VI: MISCELLANEA: LIQUIDITY, RATINGS, RISK CONTRIBUTIONS, AND SIMULATION CHAPTER 19 Liquidity Modeling for Credit Default Swaps: An Overview 587 Damiano Brigo, Mirela Predescu, and Agostino Capponi CHAPTER 20 Stressing Rating Criteria Allowing for Default Clustering: The CPDO Case 619 Roberto Torresetti and Andrea Pallavicini CHAPTER 21 Interacting Path Systems for Credit Risk 649 Pierre Del Moral and Fr´ed´eric Patras CHAPTER 22 Credit Risk Contributions 675 Dan Rosen and David Saunders Conclusion 721 Tomasz R Bielecki, Damiano Brigo, and Fr´ed´eric Patras Further Reading 725 About the Contributors 727 Index 729 P1: OTA/XYZ P2: ABC FM JWBT388-Bielecki December 16, 2010 7:49 Printer Name: Yet to Come viii P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 Printer Name: Yet to Come 740 Generalized Poisson loss (GPL) model, 47–49 General Poisson cluster loss model See GPCL (general Poisson cluster loss) model Genetic-type algorithms, 651–652 Geometric Brownian motion (GBM), 460 GEV (generalized extreme value) law, 20–23, 26 Giesecke, K., 49, 106 Giesecke-Goldberg formula, 114 Ginnie Mae, 350, 351 Girsanov’s theorem, 161, 166, 230, 261, 263, 268, 269, 447 Glasserman, P P., 650, 667 Global risk management, 14–16 Globoxx, 631, 632, 635–636, 647 Goldberg, L., 49, 114 Gordy, M., 691–692 Gourieroux, C., 687, 689 Government National Mortgage Association (Ginnie Mae), 351 Government-sponsored enterprise (GSE), 297, 351, 359 GPCL (general Poisson cluster loss) model, 627, 632, 637–641, 642–646 at work, 639–641 GPL (generalized Poisson loss) model, 47–49 GPUs (graphical processing units), 345, 384–387 Granularity adjustment, 686, 687, 688, 713 Graphical processing units (GPUs), 345, 384–387 Greeks, 387–390 Green’s function, 574, 576 GSE (government-sponsored enterprise), 297, 351, 359 G2++ model, 372 Guess, initial, 116–118 Index H Haircuts, 342, 343 Halperin, I., 56, 58, 105–147, 174 Hard default, 542, 544–546 Hawkes process, 49–51, 52 Hazard process approach, 164, 222, 558, 595 Hazard rate, 541, 545, 604 HCFs (heterogeneity correction factors), 133–134 Heath-Jarrow-Morton (HJM) model, 58, 179 Hedge funds, 9–10 Hedge ratios, 151, 173, 291 Hedging: assessing performance, 170 a CDO tranche, 83–91 with CDO tranches, 168–170 computation of strategies in a homogeneous Markovian setting, 158–160 and counterparty valuation adjustments, 500–501 of credit default swaptions, 229–232, 239–243 and credit derivatives, 211–212 credit portfolio derivatives in multivariate structural models, 161–163 effectiveness, 166–177 efficiency, 173 with equity-credit models, 539–551 heterogeneous CDOs, 71–103 instruments, 154, 156, 165–166 models, 19–20 perfect, 156–158 ratios in the dynamic model, 85–86 ratios in the Gaussian copula model, 83–84 static, 168 and swaptions, 500 of synthetic CDO tranches, 149–184 theoretical issues with CDO tranches, 152–166 P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 Index Herfindahl index, 694 Herzog, Benjamin, 19–31, 539–551 Heston model, 543–544, 546 Heterogeneity, 177 bias, 131 of credit portfolios, 105–147 notional, 108–109, 128–138 numerical examples, 134–138 recovery, 108–109 spread, 107–108 Heterogeneity correction factors (HCFs), 133–134 Heterogeneous model, 90 Heterogeneous recovery coefficients, 91–92 H-hypothesis, 164 Higher-dimensional default risk, 440–441 High mean-reversion factor, 382 Historical probability measure, 161 HJM (Heath-Jarrow-Morton) model, 58, 179 Hoeffding decomposition, 677, 714–715 Homeowner default, 351–352, 354 Homogeneity, 177, 316, 704 Homogeneous Markovian setting, 158–160 Homogeneous model, 213, 688–703 Homogeneous recovery coefficients, 91 Homogenization, 128, 129, 133–134 Housing: bubble, 358–359 prices, 304, 305, 346, 349, 359–360 turnover, 356–357 Hull-White model, 371–372, 375 Hybrid derivatives, 554 Hybrid securities, 202 I IASB (International Accounting Standards Board), 486 Idiosyncratic factors, 292, 686–688, 714 Printer Name: Yet to Come 741 Idiosyncratic gamma, 180 Idiosyncratic procedure, 88, 89, 90, 171 Idiosyncratic risk, 86–88, 181, 293, 442, 686, 692 iGamma, 180 Illiquidity, 587 See also Liquidity discount rate, 608 premium, 606 spread, 598, 605 Imbalance, macroeconomic, Immersion property, 164, 166 Implied Archimedean copula, 287 Implied copula model, 213, 287 Implied factor models, 287 calibration, 298–315 valuation of structured finance products with, 283–344 and weighted Monte Carlo methods, 292–295 Implied forward rates, 375 Implied risk-neutral default probabilities, 227 Implied systematic factor distribution, 293–294 Implied volatility, 540, 579 Importance sampling (IS), 663 Incomplete markets, 30–31 Incremental risk contributions, 683–684 Indenture, 321, 322 Independent factor decomposition, 713 Index basis points, 647 Index gamma, 180 Index options, 209 Index projection, 345, 374–375 Industry shock, 639 Information, incomplete, 185–218 Information-theoretic interpretation, 118–120 Inhomogeneity, 671 Initial guess, 116–118 Innovation, 188 Instantaneous digital credit default swap, 156, 158 P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 Printer Name: Yet to Come 742 Integer-value default process, 106 Intensity, 477, 595 Intensity-based formulation, 113–114, 164, 220 Intensity models, 38–39, 53–54, 152, 154, 166, 458, 459 for CDSs, 594–596 for liquidity, 596–600 Intensity of default, 38, 45, 222 Intensity reduced-form models, 477 Interacting particle systems (IPS), 650–662 for multiname credit derivatives, 663–665 for rare event analysis, 663–667 selection and mutations, 652–654 Interacting path systems (IPaS), 649–673 for credit portfolios, 669–670 introduction, 649–651 for multiname credit risk, 667–672 Interest coverage (IC) tests, 625 Interest coverage ratios, 325–326 Interest-only loans, 350 Interest payments, senior bonds, 328–329 Interest rate models/modeling, 364, 368–374 validation, 390 Interest rate risk, 354 Interest rates, 343, 345 and CPDOs, 631 and real estate bubble, Interest rate swap, 399, 487, 499–500 International Accounting Standards Board (IASB), 486 International Swaps and Derivatives Association (ISDA), 296, 463, 509 Intex, 316 IO (interest-only) tranches, 354 IPaS (interacting path systems) See Interacting path systems (IPaS) IPS (interacting particle systems) See Interacting particle systems (IPS) Index ISDA See International Swaps and Derivatives Association (ISDA) Iterative scaling (IS), 108, 115–120 Itˆo formula, 159, 189, 230, 233, 263 ITraxx, 125, 145, 169, 175, 176, 243–244, 633, 635, 639, 647 calibration, 290 Europe, 150, 168 index, 41 Main, 21–22 Series tranches, 82 standard calibration to, 213–214 tranches, 174, 180 J Jackson, A., 257–258 Jamshidian’s model, 272 JDCEV (jump-to-default extended CEV) model, 553, 555–556, 562–566 numerical illustration, 578–580 time-changing, 575–578 JDED (jump-to-default extended diffusion), 553, 556–557, 557–562 Jeanblanc, Monique, 397–436 Jensen’s inequality, 131, 201 Joint credit migration, 163 Joint probability distribution, 92 Jump-diffusion process, 163, 164, 212 Jump intensity, 52, 549 Jump process, 191, 221, 458, 480 Jump risk, 554 Jumps, 63, 106 and time changes, 567–578 Jump-to-default extended Black-Scholes, 555 Jump-to-default extended CEV model See JDCEV (jump-to-default extended CEV) model Jump-to-default extended constant elasticity of variance See JDCEV (jump-to-default extended CEV) model P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 Printer Name: Yet to Come Index Jump-to-default extended diffusion See JDED (jump-to-default extended diffusion) Jump-to-default extended Heston model, 556 Jump-to-default risk, 539, 544, 546 Junior mezzanine tranche, 169 K Kalkbrener, M., 677, 689 Kalman-Bucy filter, 188–189, 192, 217 Kalman filter, 605 Kernel estimators, 718, 719 Killing boundary, 567 KMV method, 200–201 Knockout feature, 224, 227, 250 Kolmogorov equation, 55, 159, 160 Koyluoglu, U., 677, 690 Kullback-Leibler (KL) divergence, 119 Kummer confluent hypergeometric function, 565 Kushner-Stratonovich equation, 191, 192, 202, 205, 213 L Lagrangian function, 119 Lando, D., 164, 193, 194–196, 634 Laplace transforms, 208, 569, 573, 576, 577 Lardy, Jean-Pierre, 7–18 Lardy-Finkelstein model, 441–443 Large homogeneous portfolio (LHP), 43 Laurent, Jean-Paul, 149–184, 687 Law of the default time, 193 Lee, Kin Pong, 485–506 LEG (loan equivalent exposure), 534–535 Legal risks, 13–14 Lehman Brothers, 10, 457, 467–470, 722 SBTV case study, 471–474 Lending risk, 508 743 Leverage, 10–11, 626 Leveraged super-senior ratings, 625–626 Leverage ratio, 11 Levin, Kirill, 345–394 L´evy subordinator, 556–557, 567, 569 LGD (loss given default), 30, 43, 95–96, 106, 128, 139–140, 141, 293 LGM (linear Gaussian Markovian) models, 371–374, 378–383 Liabilities, 322, 324, 325, 504 Liability benefit, 407 Liability cash flows, 321 LIBOR (London Interbank Offered Rate), 12, 316, 350, 589, 628 vs 10-year swap rate, 370 and Treasury rates, 365 Linden, A., 632, 635–636, 642 Linear factor approximations, 711–713 Linear Gaussian Markovian (LGM) models, 371–374 Linear homogeneous model, 688–703 Linear nonhomogeneous functions, 703–709 Linetsky, Vadim, 553–583 Linux clusters, 383 Lipton, A., 440–441 Lipton-Sepp model, 666, 670 Liquidity, 316 See also Illiquidity and CDS pricing, 591 intensity models for, 596–600 introduction, 588–591 modeling for CDSs, 587–617 as pricing component, 589–590 regression-based approaches for measuring CDS liquidity, 612–614 in risk measurement, 590–591 as a spread in reduced-form models, 592–609 through the CAPM framework, 609–612 P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 744 Liquidity intensity, 606–607 Liquidity premium, 603–604 Liquidity risk, 605, 722 Liquidity spread, 598 Liquid market, 167 Loan equivalent exposure (LEQ), 534–535 Loan-level characteristics, 360–361 Loans, 350 attributes, 360 failure to make payment on, 359 origination, Local intensity models, 53–54, 55, 73, 79, 151, 160, 179 Local volatility models, 53, 93, 151, 160, 544, 551, 556 LOGIT model, 717 Lognormal dynamics, 666 Lognormality assumption, 215 Lognormal processes, 375, 389 Lognormal short-rate models, 365, 369–370 London Interbank Offered Rate See LIBOR (London Interbank Offered Rate) Longstaff, F A., 638–639 Long-Term Capital Management (LTCM), Lopatin, Andrei V., 57–58, 71–103, 106 Loss: distribution, 53, 627 models, 47–52, 67, 133, 354 portfolio, 106 process, 54–56, 63–66 rate, 642–644 Loss-adjusted CDIS, 251–253 Loss-adjusted spread, 215 Loss given default See LGD (loss given default) Low mean-reversion factor, 382 LTCM (Long-Term Capital Management), LTV (loan to value), 360–361 Printer Name: Yet to Come Index M Macroconvexity, 180 Macroeconomic imbalance, Madan, D., 678, 710–711 Management fees, 325, 337–339 Mapping models, 287, 341 Margin agreements, 444, 509, 515–516 and CVA contributions, 699–703, 706–709 Marginal capital allocation, 691–699 Marginal contributions, 688–703 and axiomatic capital allocation, 689 and game theory, 690 for linear, nonhomogeneous functions, 703–709 for nonlinear risk functions, 709–716 and performance measurement, 690 Marginal diversification factor, 692–693 Marginalization, 110 Marginal risk contributions, 677, 684, 685 Margin calls, 516 Margining, 425 Margin period of risk (MPR), 518–519 Market completeness, 165 Market information, 202 Market liquidity, 588 Market-making activities, 27–29 Market model, 540, 541–542, 546–550 of CDS spreads, 259–278 Market payoff, 249–250 Market prices, 204–206, 258–259 Market risk, 12, 93, 291, 443 Market size, 346–349 Market state process, 206 Market structure, 346–354 Market value, 541–542 Market-value CDOs, 624–625 Markit indexes, 295, 343 Markov chain, 55, 60, 73, 160, 162, 163, 190, 191–192, 196, 319, 656, 658, 660, 668, 670 continuous time, 159 discrete-time finite-state, 197 P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 Printer Name: Yet to Come Index finite-state, 202, 204 Markovian projection onto, 99–100 Markov copula model, 160 Markovian contagion model, 179 Markovian loss models, 167, 170, 171, 173–174, 179 Markovian model, 417–418 Markovian processes, 572–575 Markovian projection, 53–56, 78, 158–160, 162 onto default contagion model, 92–95 onto one-dimensional Markov chain, 99–100 Markovian property, 332, 341 Mark-to-market accounting, 505 Mark-to-market risk, 539, 623 Mark-to-market triggers, 625 Mark-to-market value, 121, 316, 398, 443 Marshall-Olkin model, 666 Martingale measure, 37, 155, 157, 158, 161, 164, 179, 222, 229, 256, 265, 266, 268, 269, 271, 275–276, 378 Martingale representation theorem, 157, 165 Matrix pricing, 286 Maturities, short-term, 439 Mausser, H., 677 Maximum likelihood estimation (MLE), 600, 603, 605 MBSs (mortgage-backed securities), 621, 676 before and after credit crisis, 345–394 pools, 351–353 two ways to value, 361–364 MC simulations See Monte Carlo simulation Mean, approximate, 140–141 Mean parameter set, 632 Mean reversion, 372 Mean-reversion parameters, 642 745 Media effect, 358 Mendoza-Arriaga, Rafael, 553–583 Mercurio, F., 372 Merton, R., 458 Merton model, 541, 542, 554–555 Merton-type structural model, 438–439, 460–461 Mezzanine bond interest payments, 329–330 Mezzanine fees, 330, 339 Mezzanine tranches, 41, 181, 285 Microconvexity, 180 Milbourne, T., 622 Minimum entropy, 287 Minimum transfer amount (MTA), 444, 516 Misirpashaev, T., 57–58, 106 MLE (maximum likelihood estimation), 600, 603, 605 Model risk, 479 Models, 11–12 See also specific types, e.g., Stochastic correlation model calibration, 210 collateralization, 409–411 collateralized exposure, 521–522 counterparty valuation adjustment (CVA), 487–489 credit, 66–67 credit risk, 678–681 of default dates, 150 default events, 36, 544–546 design problems, 151 effect of credit crisis on, 345 equity-credit, 539–551 extreme risk, 20 heterogeneity of credit portfolios, 105–147 interest rate, 368–374 loss, 51–52 multiname modeling in credit risk, 35–69 notional heterogeneity, 128–138 parametric, 150–151 prepayment, 345 P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 Printer Name: Yet to Come 746 Models (Continued ) pricing, 27–31 proper use of risk models, 25–27 quantitative, 19 rating, 619 unified credit-equity, 553–583 using to complete markets, 29–30 Molchanov, S A., 678 Moneyness matching approach, 169, 172 Monte Carlo analysis, 345, 375–383, 390 controlling the error term, 388–389 weighted, 292–295 Monte Carlo integration, 134 Monte Carlo method, 42, 73, 363 Monte Carlo scheme, 140, 141, 144, 180, 208, 680 brute-force, 649 Monte Carlo simulation, 112, 215, 316, 398, 477–478, 512–513, 521, 652, 663, 670, 691, 694, 698 numerical results, 418–435 Monte Carlo valuation model, 290–292 Moody’s, 621, 622 Moody’s KMV, 163 Morini, Massimo, 251, 257, 457–484 Mortgage-backed securities See MBSs (mortgage-backed securities) Mortgage debt, 346–347 Mortgages, 349, 350 MTA (minimum transfer amount), 444, 516 Multilevel splitting model, 668 Multiname credit derivatives, 36, 649, 663–665 Multiname credit risk, 650, 667–672 Multiname modeling, 35–69 Multiple-jump approach, 332 Multivariate discriminant analysis, 623–624 Multivariate Markovian default model, 415–418 Multivariate structural models, 161–163, 167, 168 Index N Naive homogenization rule, 131 Nationally Recognized Statistical Rating Organizations (NRSROs), 647 NAV See Net asset value (NAV) Nedeljkovic, Jovan, 283–344 Net asset value (NAV), 316, 627 of a CPDO, 630–637 Net debt/EBITDAR financial ratio, 548, 549 Net present value (NPV), 289, 291, 630 Netting agreements, 425, 486, 504, 509, 511–512, 516, 517, 701, 702–703 New Century, 306 Next-neighbor dynamics, 203, 216 No-Armageddon event, 244 Noisy accounting reports, 194, 211 Noisy Greeks, 387–389 Nonagency deals, 353, 354 Nonconforming loans, 350 Noneligible probability mass, 627 Nonhomogeneous functions, 703–709 Nonlinear filtering, 202–212 Nonlinear risk functions, 709–716 Nonobservable inputs, 27 Normal-inverse Gaussian (NIG) processes, 180 Notations, 37 Notional heterogeneity, 108–109, 128–138 Notional reduction, scheduled, 326–327 NPV See Net present value (NPV) Number of outstanding contracts (NOC), 614 Num´eraire, 371, 378, 393 asset, 265, 267 change of, 261–264 O OAS (option adjusted spread), 361–364, 386, 492 convexities, 392 methodology, 363 P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 Printer Name: Yet to Come Index and prepayment, 363–364 tracked over time, 390 Observable inputs, 27 Observation errors, 211 Observation process, 187 Observations, as jump process, 190–191 One-factor Gaussian copula model, 162–163, 167–168, 171 One-factor Hull-White model, 375 One-factor lognormal model, 375, 376–378 One-factor model, 203 One-period forward CDS spreads, 272–278 On-the-fly probability transitions, 650 On-the-run ABX indexes, 300–307 On-the-run Globoxx indexes, 632 Option adjusted spread See OAS (option adjusted spread) Option-based valuation, 288 Options: on credit default indexes, 243–259 on credit default swaps, 227–233 on traded assets, 208–209 Option-theoretic models, 290 Ordinary differential equations (ODEs), 50 Orthogonal decompositions, 713–716 OTC derivatives, 398, 399, 485, 508, 676 Out-of-the-money puts, 544, 554 Overcollateralization (OC): tests, 625 triggers, 337–339 P Packer, F., 621, 648 PACs (planned amortization classes), 354 Pallavicini, Andrea, 619–648 Papadopoulos, Philippos, 319–344 Parallel gamma credit spread risks, 171, 180 Parallelization, 383–387 747 Parametric model, 150–151 Parametrized family, 45 Parmalat, 470 Par rate, 365, 375 Partial differential equations (PDEs), 162, 555 Particle filtering, 211 Pass-through securities, 351 Path-dependent contracts, 341 Path space measures, 658–661 Patras, Fr´ed´eric, 437–455, 649–673 Payment waterfall, 326, 328–331 Payoffs: default-contingent, 155 discounted, 223, 245 PCA (principal component analysis), 21, 379 Pedersen, L H., 609–611 Pedersen’s formula, 251, 257 Perfect hedging, 156–158 Performance measurement, 690 PFE See Potential future exposure (PFE) Picoult, E., 700 Pipeline effect, 358 Piterbarg, V., 63–66 Plain-vanilla credit default swaption, 220 Planned amortization classes (PACs), 354 PO (principal-only) tranches, 354 Point process observation, 205 Poisson process, 47, 164, 190, 200, 442, 501–502, 555, 558, 595, 620 default time, 458 Polyhedral domain, 453–454 Pool losses, 631–632, 639 simulation, 633–635 summary of, 636–637 Portfolio: asset, 322 credit derivatives, 320 credit loss, 533 defaults, 109 P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 748 Portfolio (Continued ) general credit-risky, 411–412 with heterogeneous recovery coefficients, 91–92 mapping, 341 replication, 28 risk, 676 trading, 342 Portfolio credit derivatives, 320 Portfolio credit risk, 676 Portfolio loss, 106, 533–535 distribution, 91 and risk contributions, 682 time resolution of, 108–109 Position-level contributions, 684–685 Position risk contributions, 677 Postponed running CDS (PRCDS), 465 Potential future exposure (PFE), 401, 435, 508, 513–514, 523–525 PRCDS (postponed running CDS), 465 Precollapse event, 244 Precollapse fair CDIS spread, 247–248 Precollapse loss-adjusted fair CDIS spread, 253 Precollapse present value, 247 Predefault fair forward CDS spread, 225–227 Predescu, Mirela, 587–617 Predictable representation theorem, 155–156, 157, 158, 162 Premium payments leg, 154, 161–162, 208 Prepayment risk, 285, 291, 354 Prepayments, 342, 355–361 modeling, 345, 356, 363 and negative convexity, 391 and OAS, 363–364 predictors of behavior, 360 speeds, 355–356 and yields, 355–361, 362 Prepayment speed assumption (PSA), 355 Printer Name: Yet to Come Index Present value, precollapse, 247 Present value of a basis point (PVBP), 225 Price: of a CDS, 449–450 determining for CDOs, 42–43 fair, 42, 245–246 manipulation, 13 market, 36, 204–206 single-name, 36 Price-yield calculator, 288 Pricing: bespoke products, 286–287, 290 of bespoke tranches, 169 cash structured finance instrument, 312–315 CDOs, 665 corporate debt, 570–572 counterparty credit risk, 529–531 counterparty risk, 395–536 counterparty risk in equity return swaps, 474–480 of credit derivatives, 206–209, 663 of credit index options, 215–216 equity, 199–200 with equity-credit models, 539–551 heterogeneous CDOs, 71–103 of hybrid securities, 202 in incomplete markets, 30–31 and liquidity, 589–590 models, 19–20, 29, 540 risk-neutral, 203 structured finance products, 286 Pricing measure, 156–158, 166 Pricing models, 542 associated with implied parameters, 167 dynamic arbitrage-free, 175 future of, 27–31 Prime loans, 350 Principal component analysis (PCA), 21, 379 Priority of payments, 328–331 Prior model, 299 P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 Index Probability: distribution of the default rate, 335 historical, 161 implied risk-neutral default, 227 Probability-based formalism, 114 Profit and loss (P&L), 125 Propagation of chaos, 654 Protection buyer, 219–220, 296–297, 443, 463–464 Protection leg, 246 Protection payment, 220 Protection premium, 631 Protection seller, 36, 219–220, 296–297, 443, 463–464, 605 PSA (prepayment speed assumption), 355 PTGLKR (Predescu et al.) study, 591, 609, 612–614 Pure death process, 159 PVBP (present value of a basis point), 225 Pykhtin, Michael, 507–536, 677, 692, 701, 702 Q QMC (quasi Monte Carlo) methods, 316 Quasi Monte Carlo (QMC) methods, 316 R Radon-Nikod´ym density, 228, 255, 261, 263, 267, 270, 274, 277 Radon-Nikod´ym derivative, 155, 157, 166, 378 Rajan, A., 638–639 Random-barrier structural model, 164–165 Random factor loadings (RFL)/random recovery model, 138, 177 Random recovery, 49, 138–144 Random thinning procedure, 51–52, 72, 107, 108, 127, 174, 176 Random time, 220, 221 Printer Name: Yet to Come 749 Ranieri, Lewis S., 353 Rare event analysis, 650, 651, 663–667 Rate-dependent volatility models, 365–366 Rate of return, expected, 161 Rating agencies, 9, 646, 647 Rating arbitrage, 624 Rating criteria, 619–648 Rating models, 619 Ratings, 620–626 structured financial issues, 644–645 RDB (roll-down benefit), 635–637, 642 Real economy, 8–11 Real estate bubble, Receiver credit default index swaption, 250 Recovery, 293, 343 heterogeneity, 108–109 on principal, 486 process, 411 random, 138–144 risk, 501–502 stochastic, 109, 139 Recovery convexity correction, 108 Recovery rate markdown, 173 Reduced-form credit risk models, 192, 193–194 constructing via nonlinear filtering, 202–212 Reduced-form default models, 38–39 Reduced form loss models, 47–52 Reduced-form models, 220, 458, 477, 592–609 Reduced models, 438–440, 650 Reference credit, 463–464 Reference filtration, 163–164, 221, 244–245 Refinancing, 357–359, 361 Reflexivity, 28 Regimes of volatility, 167 Regression, 612–614 analysis, 297 linear, 168 P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 750 Regularization procedures, 210 Regulation, 8, 9, 10, 14 Regulatory capital, 20 Regulatory management, 663 Regulatory risks, 354 Relative value model, 541, 542, 548–550 Remunerations, 10 Replicating strategy, 230, 231 Replication portfolios, 28, 30–31 Replication risk, 31 Reremics, 353 Residential mortgage-backed securities (RMBSs) See RMBSs (residential mortgage-backed securities) Residual factors, 713 Residual risk, 713 Resolvent operators, 572–574, 575 Reversion to the mean, 372 Right-way risk, 531–532 Risk, 291 See also specific types, e.g., Counterparty credit risk (CCR) analysis, 345 associated with contracts, 398 global risk management, 14–16 idiosyncratic, 293 legal, 13–14 mark-to-market value of, 443 measurement, 590–591 minimization of, 212 models, 25–27 policy, 15–16 premium, 636 Risk contributions: and capital allocation, 681–688 for CDO tranches, 704–706 of positions, 676 types of, 683–688 Risk factor contributions, 685–688 Risk management, 15, 174, 663 of bespoke tranches, 169 of CDOs, 107, 150 counterparty risk, 507–536 models, 20 Printer Name: Yet to Come Index quantitative, 24 of synthetic CDOs, 149 Risk-neutral default intensity, 193 Risk-neutral measure, 166 Risk-neutral pricing, 203 RMBSs (residential mortgage-backed securities), 295, 312–315, 620, 722 Roll-down benefit (RDB), 635–637, 642 Rosen, Dan, 283–344, 398, 675–720 Rutkowski, Marek, 219–279 S Saunders, David, 283–344, 675–720 SBTV (scenario barrier time-varying volatility) model, 457, 459 Brigo and Morini, 470–471 Lehman Brothers case study, 471–474 Scaillet, O., 687 Scandolo, G., 591 Scenario: definition, 289 discount rates, 375 generation, 290, 374 Scenario barrier time-varying volatility model See SBTV (scenario barrier time-varying volatility) model Scheduled debt service reduction, 327 Scheduled notional reduction, 326327 Scheicher, M., 297 Schmidt, Thorsten, 185218 Schăonbuchers forward model, 58–62 S-curve, 357, 358, 361 SDE (stochastic differential equation), 376 Sector capital contributions, 698 Securities, hybrid, 202 Securities financing transactions (SFTs), 508 Securitization, 8, 284, 622 Self-amortizing loans, 350 Self-consistency, 78, 100–101 P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 Printer Name: Yet to Come Index Semianalytic dynamic model, 74–78, 100–101 Senior bonds, 328–329 Senior fees, 328 Senior investors, 624–625 Seniority, 319 Senior tranches, 41, 181, 285 Sensitivities, single-name, 120–127 Sepp, A., 666 Sequential-pay bonds, 354 SFTs (securities financing transactions), 508 Share price, 199, 542 Shifted square-root diffusion (SSRD) model, 39 Short sales, 13 Short-term CDS contracts, 539 Short-term credit spreads, 474 Short-term default probability model, 541 Short-term maturities, 439 Sidenius, J., 63–66, 138, 177 Signal process, 187 SIMD (Single Instruction, Multiple Data) architecture, 384–385 Simulation: credit exposure, 512–513 model, 417–418 reducing errors in, 479 valuation, 288 Single defaults, 51–52 Single-factor credit model, asymptotic, 691–692 Single Instruction, Multiple Data (SIMD) architecture, 384–385 Single-jump approach, 332 Single monthly mortality (SMM), 355–356 Single-name credit derivatives, 35 Single-name credit portfolios, 105 Single-name default probability, 114 Single-name sensitivities, 120–127 Single-scenario valuation, 286, 288–290 Skew, 215, 365, 579 751 Skodeberg, T M., 634 Skreta, V., 623 Smile curve, 365–368, 579 Smile to credit (S2C) pricing model, 542–550 SMM (single monthly mortality), 355–356 Smoothing, 198–199 Sobol points, 316 Sobol sequences, 382 Soft default risk, 544, 545–546 Sovereign bonds, 404 S&P: CDO Evaluator, 633 and CDOs, 638 ratings, 621, 622, 644 SPA model, 63–66 Special purpose vehicles (SPVs), 627 Spectral expansion model, 574–575 Splitting technique, 650, 667, 668 Spread: excess, 330 heterogeneity, 107–108 illiquidity, 598, 605 liquidity, 598 loss, 106 loss-adjusted, 215 option adjusted, 361–364, 492 predefault fair forward CDS, 226 in reduced-form models, 592–609 risk, 162, 204 short-term credit, 474 SPVs (special purpose vehicles), 627 Stand-alone contract-level exposure, 509–511 Stand-alone risk contributions, 683, 684 Standard calibration, 210, 213–214 Stanton, R., 297 Static hedge ratio, 168 Static hedging, 168 Static models, 287 Statistical analysis, 345 Stein, Harvey J., 345–394, 485–506 P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 Printer Name: Yet to Come 752 Sticky delta, 171, 172, 173 Sticky strike, 171–172 Stochastic correlation model, 176 Stochastic default intensity process, 532 Stochastic differential equation (SDE), 376 Stochastic filtering, 186–192 Stochastic i.i.d recovery, 106 Stochastic intensity, 233, 429–434, 458, 477 Stochastic local intensity (SLI) models, 53–54, 56–58 Stochastic pricing methods, 286, 287, 288, 290–292 Stochastic recovery, 95–97, 109, 139, 173 Stochastic volatility, 29, 543, 556 and local volatility, 551 models, 366 and time changes, 567–578 Stoker, J., 677, 690 Stressed case, 641–642 Stress tests, 26, 290 Strong random thinning, 51 Structural counterparty risk valuation, 437–455 Structural dynamic credit risk models, 192–193 Duffie and Lando, 194–196 Frey and Schmidt, 196–202 Structural models, 438–440, 458–459, 541 basic, 460–462 calibration to CDS data, 463–467 for credit calibration, 457–484 for credit risk, 665–667 main drawbacks, 474 Structured credit products, 676, 677 Structured finance products: defined, 284 and implied factor models, 283–344 introduction, 284–288 pricing, 286, 312–315 Index ratings, 622–624 valuation of, 288–292 Sturm-Liouville equation, 573 S2C pricing model, 542–550 Subordinated/equity tranches, 285 Subordinated notes, 326 Subordination, 319–320 Subprime crisis, 305–306 calibration to the ABX indexes after, 307–311 financial techniques, 11–14 origins, 7–18, 721 Subprime loans, 350–351, 352 Super-senior investors, 625–626 Survival claim, 220 Survival probability, 79–80, 123, 469, 559, 570, 578 Survival process, 220, 221 Sverdlove, R., 591, 600–604 Swap annuity, 225 Swap counterparty exposure, 486 Swap exposure, 486 Swap market calibration, 364–365 Swap portfolio, 229 Swap rate correlation, 28 Swaps, counterparty valuation adjustment calculations, 493–497 Swaptions, 365, 493, 495 accounting considerations, 503 credit default, 227 and hedging, 500 plain-vanilla credit default, 220 volatilities, 389 Synthetic cash flow structure, 336–337 Synthetic CDOs, 290 Synthetic CDO tranches, 149–184 Synthetic tranches, 333–334 Systematic credit factors, 292–294, 679 Systematic diversification, 692 Systematic factors, 686–688 Systematic risk, 714 Systematic shock, 639 Systemic liquidity, 588–589 P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 Index T TACs (targeted amortization classes), 354 Tang, D Y., 591, 614–615 Tarenghi, Marco, 457–484 Targeted amortization classes (TACs), 354 Tasche, D., 677, 689, 704 Taylor expansion, 388, 687 TD matrices, 110–114 Tejwani, G., 624, 634 Tenor structure, 222 Termination clauses, 509 Term structure of volatility, 365–368 Thinning procedure, 51–52, 72, 107, 108, 115–120, 127, 174, 176 Tian, Xusheng, 345–394 Time-changed JDCEV model, 553 Time-changed Markov processes, 572–575 Time changes, influence on jumps and stochastic volatility, 567–578 Time-changing jump-to-default extended diffusions, 567–568 Tomecek, P., 106 Top-down CDO models, 175–176, 179, 287 Top-down CLO models, 321 Top-down credit risk model, 164 Top-down default time matrices (TD matrices), 110–114 Top-down models, 72 difference from bottom-up models, 145 modeling heterogeneity of credit portfolios, 105–147 random recovery in, 138–144 and spread heterogeneity, 107–108 Torresetti, Roberto, 619–648 Trade-to-quote (T2Q) ratio, 614 Trading liquidity, 588–589 Tranches, 41, 353 calibration to, 80–82 categories of, 285 Printer Name: Yet to Come 753 CDOs and risk management, 107 and default risk, 623 deltas, 88–91 derivatives-based market, 319 evaluating hedge ratios, 72 options, 106 price, 45 spreads, 134 synthetic, 333–334 Transition matrices, 332 Transmission errors, 211 Transparency, 284, 285–286 Trapp, M., 591, 605–609 Treasury market calibration, 364, 365 Trigger coverage ratios, 326 Turc, Julien, 19–31, 539–551 Turnover rate, 356–357 Two-dimensional default risk, 438–443 Two-factor LGM model, 371–374, 378–383 Two-point distribution, 139 TY (Tang-Yan) model, 591, 614–615 U Unconditional measure, 656–658 Unemployment rate, 359 Unified credit-equity modeling, 553–583 introduction, 553–557 Uniformization bound, 117 Uniform with antithetic noise (UWAN) method, 382 Unilateral credit valuation adjustment (CVA), 505, 529–530, 700 Unilateral margin agreements, 515, 516, 519–520 Unilateral portfolio CCR, 414–415, 422–424 Unilateral pricing, 528–530 Unwinding procedure, 627 UWAN (uniform with antithetic noise) method, 382 P1: OTA/XYZ P2: ABC ind JWBT388-Bielecki December 16, 2010 10:27 Printer Name: Yet to Come 754 V Valuation, 200 cash-flow CLOs, 334 CDIS, 248–249 of contingent claims on time-changed Markov processes, 572–575 of corporate debt, credit, and equity derivatives, 559–560 counterparty risk, 507–536 equity return swap, 457–484 market-implied, of cash-flow CLO structures, 319–344 of MBSs, 361–364 numerical examples, 335–339 single-scenario, 286 of structural counterparty risk, 437–455 of structured finance products with implied factor models, 283–344 synthetic tranche, 333–334 Value at risk (VaR), 20–27, 680–684, 686, 688, 722 Variance: approximate, 140–141 reduction in the one-factor lognormal model, 376–378 reduction in the two-factor LGM model, 378–383 reduction techniques, 649–650 VaR (value at risk), 20–27, 686, 722 Vasicek portfolio credit model, 298–300 Vega hedging, 83 Veldkamp, L., 623 Vestal, D., 666 Volatility, 263 Black-Scholes, 45, 365 of CDS spread, 236–239 and counterparty adjustment valuation, 494, 495 Index data, 365–368 deterministic, 232 equity, 540, 547, 554 exogenous, 466 implied, 540 instantaneous, 562 local, 551, 556 regimes of, 167 skews, 215 smile, 365 stochastic, 29, 551, 556 swaptions, 389 Volatility to volume (V2V), 614 W Wallace, N., 297 Wallenius’s noncentral multivariate hypergeometric distribution, 132, 134 Waterfall structures, 289, 290, 316, 328–329, 354, 623, 625 Wrong-way risk, 398, 501–502, 531–532 Wu, L., 591, 604–605 X XIBOR rates, 12 Y Yan, H., 591, 614–615 YASN function, 491, 492, 493 Yields, 361–364 Z Zero coupon bonds, 196 and CDSs, 207–208 defaultable, 203 and standard calibration, 210 Zero coupon CDSs, 162–163 ... JWBT388 -Bielecki December 16, 2010 19:33 Printer Name: Yet to Come CREDIT RISK FRONTIERS Subprime Crisis, Pricing and Hedging, CVA, MBS, Ratings, and Liquidity Tomasz R Bielecki, Damiano Brigo, and. .. Cataloging-in-Publication Data: Bielecki, Tomasz R., 1955– Credit risk frontiers : subprime crisis, pricing and hedging, CVA, MBS, ratings, and liquidity / Tomasz R Bielecki, Damiano Brigo, and Fr´ed´eric Patras... several urgent topics such as the subprime crisis, pricing and hedging of credit risk, CVA, CDO, CLO, MBS, ratings, and liquidity ix P1: OTA/XYZ P2: ABC FM JWBT388 -Bielecki x December 16, 2010 7:49