<s> QUANTITATIVE ANALYSIS,DERIVATIVES MODELING,AND TRADING STRATEGIESIN THE PRESENCE OF COUNTERPARTYCREDIT RISK FOR THE FIXED-INCOME MARKET
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QUANTITATIVE ANALYSIS,
DERlVATlVES MODELING,AND TRADING STRATEGIESIN THE PRESENCE OF COUNTERPARTYCREDIT RISK FOR THE FIXED-INCOME MARKET
Y I TANGGoldman, Sachs & Co., Inc., USA
BIN LIWestport Financial, LLC, USA
NEW JERSEY
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vp World ScientificLONDON * SINGAPORE
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BElJlNG * S H A N G H A I * H O N G KONG
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TAIPEI * CHENNAI
Published by
World Scientific Publishing Co. Re. Ltd.
5 Toh Tuck Link, Singapore 596224USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-PublicationDataTang, Yi.Quantitative analysis, derivatives modeling, and trading strategies : in the presenceof counterparty credit risk for fixed-income market / by Yi Tang & Bin Li.p. cm.Includes bibliographical references and index.ISBN-13 978-981-02-4079-0 -- ISBN-I0 981-02-4079-1 (alk. paper)1. Derivative securities--Mathematical models. 2. Finance--Mathematical models.3. Speculation--Mathematicalmodels. I. Li, Bin.HG6024.A3 T33 2007332.64'570151--d~222006042114
British Library Cataloguing-in-PublicationDataA catalogue record for this book is available from the British Library.
Copyright 0 2007 by World Scientific Publishing Co. Re. Ltd.All rights reserved. This book, or parts thereof;may not be reproduced in any form or by any means,electronic or mechanical, including photocopying, recording or any information storage and retrievalsystem now !mown or to be invented, without written permissionfrom the Publisher.
For photocopying of material in this volume, please pay a copying fee through the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission tophotocopy is not required from the publisher.
Printed in Singapore by World Scientific Printers (S) Pte Ltd
To Our FamiliesAmy, Jonathan, Rebecca, and the rest of the Tang FamilyTao, Conan, Jasmine, and the rest of the Li Family
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Preface
This book addresses selected practical applications and recent developments in the areas of quantitative financial modeling in derivatives instruments, While the primary scope of this book is the fixed-incomemarket (with further focus on the interest rate market), many of themethodologies presented can also be applied to other financial markets,such as the equity, credit, and foreign exchange markets.This book is written from the point of view of financial engineers orpractitioners, and, as such, it puts more emphasis on the practical applications of financial mathematics in the real market than the mathematicsitself with precise (and often tedious) technical conditions. It does notaim to be comprehensive; rather, it focuses on selected areas based onthe authors’ experience and expertise. It also attempts to combine economic and financial insights with mathematics and modeling so as tohelp the reader develop intuitions.This book also presents various exploratory (and possibly thoughtprovoking) ideas and practical issues that are beneficial to academics andpractitioners as the leads for further research. In terms of a popular saying that “the engineers have problems, but no solutions, and the mathematicians have solutions, but no problems,” we are trying to supply somesolutions, as well as some problems as part of the book.While the primary targeted readers are Quant modelers or those whoare studying to become Quant modelers, many parts of this book are alsobeneficial to Quant developers, Quant structurers, Quant traders, andpossibly senior managers. This book assumes that the reader is familiarwith the basics of stochastic calculus and derivatives modeling.The following are a few examples of the practical applications of
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financial mathematics in the real market covered by this book.For instance, this book addresses various practical applications ofmodeling, such as martingale arbitrage modeling under real market situations (such as using the correct risk-free interest rate, revised put-callparity, defaultable derivatives, hedging in the presence of the volatilityskew and smile, secondary model calibration for handling the unhedgeable variables, models for pricing and models for hedging). Thisbook also presents practical numerical algorithms for the model implementation, such as martingale interpolation and resampling for enforcingdiscrete martingale relationships in situ in numerical procedures, modeling of the volatility skew and smile, and a nonexploding bushy tree(NBT) technique for efficiently solving non-Markovian models under thebackward induction framework. In addition, this book also covers thebasics of the interest rate market, including various yield curve modeling,such as the well-known Orthogonal Exponential Spline (OES) model.This book also discusses various trading and structuring strategies,including those for market making and those for proprietary trading including hedging, yieldcoupon enhancement, funding cheapening, andleveraging.' For instance, this book touches upon various popular derivatives instruments, such as TArget Redemption Notes (TARNS), (callable, knock-out, or TAR) inverse floaters, (callable, knock-out, or TAR)capped floaters, (callable, knock-out, or TAR) CMS (Constant MaturitySwaps) spread floaters, (callable, knock-out, or TAR) (CMS spread)range accrual notes, (callable, knock-out, or TAR) Snowballs and Snowbears, (callable, knock-out, or TAR) Power Reverse Dual Currency(PRDC) notes, some credit/IR/FX hybrids (such as CCDS (contingentCDS) and credit extinguishers), as well as asset swaps or repackaging.As we shall see, these strategies essentially are trading activities betweenthe arbitrage-free probability measure and the real world probabilitymeasure or between the arbitrage models and the forecasting models.Furthermore, this book addresses the counterparty credit risk modeling, pricing, trading/arbitraging strategies, and risk management that arerecent developments and are of increasing importance. The counterpartycredit risk comes from the fact that as one enters into an OTC (over-thecounter) derivatives transaction, one (implicitly or explicitly) grantsone's counterparty an option to default and, at the same time, one alsoreceives an option to default oneself.
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It is important to point out that not all the trading strategies are necessarily legal in alljurisdictions and, obviously, one should use them within the legal boundaries.
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The term of counterparty credit risk may remind the reader of a backoffice risk management functionality or a treasury function. We emphasize that it should be an integral part of the front office derivatives pricing and hedging activities. This is the approach that we shall adopt inthis book. With this approach, a large part of the counterparty credit riskbecomes the market risk that can be traded and hedged with market oriented trading strategies (in addition to the classical strategies for counterparty credit risk management).In essence, pricing the counterparty credit risk is the same as pricingdefaultable derivatives or the default premium in OTC derivatives instruments, which is a relatively new area of derivatives modeling andtrading. In some cases, the counterparty credit risk is more significantthan any other risks and a derivatives instrument becomes a hybrid derivatives instrument with credit or a compound credit derivatives instrument. For instance, a plain vanilla interest rate (IR) swap becomes a defaultable IR swap. Also, in the prevailing credit derivatives, to price acredit default swap (CDS) is essentially to price the default premium inthe underlying bond, whereas to price the counterparty credit risk is toprice the default premium of the default protection seller of the CDS or,more precisely, the premium of the joint default of the underlying assetand the default protection seller. Additionally, while cash settled IRswaptions and physically settled IR swaptions can have identical pricesfrom a non-defaultable IR model, their associated counterparty defaultpremiums can be significantly different. Furthermore, the exerciseboundaries of Bermudan and American options can be significantlychanged by the counterparty default risk.For IR derivatives, in particular, the counterparty credit risk modeling is essentially the same as the WCredit hybrid modeling.The PV (present value) of the counterparty credit risk can be a significant percentage of the overall P&L (profit and loss) of the derivativesdesks and can reach many hundreds of millions of dollars for a top investment bank. The PV of the counterparty credit risk needs to be actively hedged and risk managed, just like other risks in derivatives trading.There are also arbitrage strategies that one can employ to monetizethe counterparty credit risk by bringing in cashflows in some cases andby reducing the outgoing cashflows in some other cases. These strategies can also be applied to achieve trade price cheapening to one’s counterparties and, thus, can bring in more business. In other words, while
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one usually needs to charge a premium from a low credit rated counterparty due to the counterparty’s default risk, one can actually structure atrade whereby one pays a premium to one’s counterparty due to thecounterparty’s likelihood of default. The reader may want to think abouthow to structure such a trade and we shall come back to this point lateron.All these have created and will likely create more trading opportunities in the market place, especially after the implementation of the Base1I1 Accord, which will be an important driving factor for the developmentof the credit hybrid market and modeling. Without proper modeling andmanagement of the counterparty credit risk, one is missing a significantpart of the big picture of the OTC derivatives modeling and business.As we shall see that one challenging and interesting aspect of pricingthe counterparty default risk premium for a trade is that it depends on theportfolio that the trade is in and has non-linear portfolio effects due to thenetting agreement and the default treatment conventions.This is the major motivation for us to divide derivatives models intoseparate categories, such as trade-level or micro models and portfoliolevel or macro models (where the portfolio information acts as an input).Most of the prevailing derivatives models are trade-level or micro models, focusing on the modeling of a specific trade regardless of the portfolio it is in. However, as we shall see, there are several categories of risksthat (directly or indirectly) contribute to the price (or the funding cost) ofa specific trade have non-linear portfolio effects. Aside from the counterparty credit risk, other examples of these types of risks include, theunexpected risks, such as VAR (Value at Risk) and PE (Potential exposure), and the risks associated with the un-hedgeable variables.More specifically, this book is divided into two parts: Part I - Theoryand Applications of Derivatives Modeling and Part I1 - Interest RateMarket Fundamentals and Proprietary Trading Strategies. Various practical numerical implementation techniques and selected topics in financial stochastic calculus are also presented.Part I provides a framework for general derivatives modeling, fixedincome derivatives modeling (with emphasis on interest rate derivativesmodeling), and counterparty credit risk modeling.For general derivatives modeling, it covers Harrison-Pliska, martingale arbitrage modeling framework, martingale arbitrage modeling underreal market conditions (such as using the correct risk-free interest rate,revised put-call parity, defaultable derivatives, hedging in the presence of
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the volatility skew and smile, secondary model calibration for handlingthe un-hedgeable variables, models for pricing and models for hedging),and martingale interpolation and resampling for enforcing discrete martingale relationships in situ in numerical procedures. Black-Scholesframework and its extensions derived from martingale modeling are discussed. The impact of volatility skews and smiles on hedging is alsodiscussed.For interest rate derivatives modeling, it covers various interest rateterm structure models, such as the Heath-Jarrow-Morton (HJM) framework, the Brace-Gatarek-Musiela (BGM) market model, and MarkovianHJM models, such as the multi-factor Ritchken-Sankarasubramanian(RS) model. The details of the nonexploding bushy tree (NBT) technique for efficiently solving multi-factor full yield curve interest ratemodels, as well as the tree and Monte Car10 based numerical techniquesin general are discussed with various volatility structures such as lognormal, shifted lognormal, Constant Elasticity of Variance (CEV), aswell as stochastic volatilities.For counterparty credit risk modeling, it provides a review of counterparty credit risk and covers how to price the counterparty default options, how to incorporate such risk into the price (or spread) of the OTCderivatives, how to derive and apply various trading and structuringstrategies, and how to manage the risk associated with such options. Italso touches upon general credit risk modeling and pricing, as well as arisky market model for credit spread modeling.Part I1 covers selected areas of the fixed income market, such as theintroduction to some basic interest rate products, the yield curve construction and modeling, including the well-known Orthogonal Exponential Spline (OES) model. It also covers proprietary trading strategiesusing statistical arbitrage - another dimension of Quant modeling ascompared to derivatives modeling for market making purposes. Parts ofthe materials presented in this part can be used as a review on the fundamentals of the interest rate market, which would be beneficial to novicereaders.Significant amount of the materials included in this book are theoriginal research of the authors’. Some have been published by the authors as working papers. Some have been presented by the authors inconferences, particularly, various RISK conferences, such as RISK USA,Math Week, and Credit Risk Summit USA.To provide convenience to the reader, we have provided various
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Web links for downloading some of the papers and software, etc., referenced in this book.If there are future editions of this book, we will provide more numerical examples and possibly Excel spread sheets illustrating the methodologies presented in this book. The reader is encouraged to submit tothe authors relevant numerical examples and questions and answers forpossible inclusion in the future editions of the book. All included submissions will be acknowledged in the future editions of the book.Yi Tang is grateful to Goldman Sachs for granting the permission forthe publication of this book. The views expressed in this book are solelythe authors' own and do not necessarily reflect the views of the entitiesthat the authors are affiliated to. The authors are solely responsible forthe errors, if any, in the book.The authors would like to thank Peter Can- of Bloomberg and PeterRitchken of Case Western Reserve University for discussions and writing the book cover reviews.The authors are grateful to Jeffrey Lange and Andrew Lawrencefrom Longitude for contributions and support of the development of thenonexploding bushy tree (NBT) technique. The authors would like tothank Hong Amy Yang of Bloomberg and Gang Liu of Citi Group forvarious contributions to the book, and Eugene Guillemette of Bloombergfor proof-reading the entire book and providing helpful comments.The authors would also like to thank the following people for beneficial communications and interactions (in alphabetical order in the affiliation and then in the last name): Leif Andersen and Michael Sotiropolisfrom Banc of America; Anlong Li from Barclays Capital; David Bai,Rupert Cox, Thomas Fox, Man-Wing Li, Gregory Pelts, DmitryPugachevsky, Jin Qian, Asli Rustemli, Wei-Tong Shu, Zhenyu Yuan,Harry Zhang, Lizeng Zhang, and Robert Zvan from Bear Steams; PatrickHagan from Bloomberg; Vladimir Finkelstein from Citadel; L. Sankarasubramanian and Frank Xia from Citi Group; Mark Broadie, SidneyBrowne, and Emanuel Derman from Columbia University; Dino Buturovic, William Kearstead, Alfred0 Pastro, Klaus Toft, and Paul Youngfrom Goldman Sachs; Lane Hughston from King's College; Eduardo Canabarro, Kanglin (Kenneth) Huang, Fei Zhou, and Qun Zuo from Lehman Brothers; Fong Liu, and Jane Yu from Merrill Lynch; Rui Kan,Jacob Xia, and Guang Yang from Morgan Stanley; David Cai from NewYork University; Jingzhi (Jay) Huang from Pennsylvania State University; Riccardo Rebonato form Royal Bank of Scotland; Darrel Duffie and
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Ming Huang from Stanford University; Francis Longstaff, Pedro SantaClara, and Didier Sornnette from UCLA; Stuart Turnbull from University of Houston; and Susanta Basu and Judith Yang formerly fromGoldman Sachs. The authors have presented at various RISK conferences and were also benefited from discussions with the conference attendees.The authors are also grateful to Cheong Chean Chian, Juliet Lee,Kim Tan, Yubing Zhai, and others from World Scientific Publishing Co.for their help, encouragement, and patience in the publication process ofthe book.The authors would also like to thank Vasuki Balasubramaniam andLaurie Donaldson from Risk Books and Journals for granting the reprintpermission of the paper entitled “A Nonexploding Bushy Technique andIts Application to the Multifactor Interest Market Model” (Tang andLange, 2001) published in the Journal of Computational Finance (Vol. 4,No. 4, pp 5-31). This paper is included in Section 7.9 of this book withmore details added.Yi Tang would also like to thank the following people: Rubin Braunstein, Maha Ashour-Abdala, and Roberto Peccei from UCLA; DietrichBelitz from University of Oregon; as well as Chuni Gosh, Minwen Hong,Xiao-Dong Su, Pai (Patrick) Tang, Changjiamg Wu, Minlong Xu,Hanxiang Yu, and Tao Zhan.Yi TangTang-Yi@yahoo.comBin LiBinLi@EntropyFund.com
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Contents
Preface
vii
PART I
THEORY AND APPLICATIONS OFDERIVATIVES MODELING
1
Introduction to Counterparty Credit Risk
3
Credit Charge, Credit Benefit, and Credit PremiumCredit Cost, Accrued Funding Cost, and AccruedFunding BenefitTrading Strategies and OpportunitiesComparison with Bond Credit RiskPrevailing Strategies for Counterparty Credit RiskManagementWrong-way and Right-way Exposures or TradesIntroduction to Modeling and Pricing of CounterpartyCredit Risk
8
Chapter 1
1.11.21.31.41.5
1.6I .7
Chapter 22.12. I .I2.I .22.22.2.12.2.22.2.3
Martingale Arbitrage Pricing in Real Market
Basics of ArbitrageArbitrage Opportunity and Arbitrage PricingSelf Financing Trading Strategies and ArbitrageSubtleties in Arbitrage Pricing in Real MarketCounterparty Credit RiskThe Risk-free Interest RateBid/Ask Spread
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3738384245454549
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2.2.42.2.5
Contents
Un-hedgeable VariablesPrimaiy Model Calibration and Secondary ModelCalibration2.2.6Models for Pricing, Models for Hedging, and HedgingCalibration2.2.7Incomplete Market and Completing the Market2.3Arbitrage Models and Non-arbitrage ModelsArbitrage Models and Non-arbitrage Models2.3.1Financial Market Participants and Financial Activities2.3.22.4Trading Opportunities and Strategies2.4.1Simple Bonds and IR SwapsCallable Bonds and Cancelable IR Swaps2.4.22.4.3Examples of Practical Complications2.4.4Structured Notes and Exotic DerivativesIWFX Hybrid Notes and Derivatives2.4.5Asset Swaps and Repackaging2.4.6Credit Hybrid Derivatives2.4.7Capital Structure Arbitrage2.4.8Quasi-arbitrage Opportunities2.4.92.4.10 Why Should Derivatives Instruments ExistMartingale Arbitrage Modeling2.5Harrison-Pliska Martingale No-arbitrage Theorem2.5.1MartingaleDerivatives Pricing in a Binomial2.5.2EconomyHarrison-Pliska Martingale No-arbitrage Theorem2.5.3for Assets with Intermediate Cashflows or IncomeFoundationfor Arbitrage Pricing2.5.4Examples of Martingales and Equivalent Martingale2.5.5MeasuresMartingale Representation and SDE for Derivatives2.5.6PricingChange of Probability Measure and Importance2.5.7SamplingPDE for Derivatives Pricing and PQL Decomposition2.5.8SABR Stochastic Volatility Model2.5.92.5.10 An Example of Martingale Modeling in Real MarketProblems2.6
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Contents
The Black-Scholes Framework and Extensions
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More on Martingale ModelsSingle State Variable and Single NumeraireSingle State Variable and Multiple NumerairesBlack's ModelPut-Call Parity RevisedReplication ModelImpact of Volatility Skews and Smiles on Hedge Ratiosand Hedging StrategiesOther Extensions of Black-Scholes Framework
123124133142143147
Chapter 3
3.13.1.13.1.2
3.23.33.43.53.6
Martingale Resampling and Interpolation
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Martingale InterpolationBrownian Bridge InterpolationMoment Matching in One-factor CaseQuadratic ResamplingMoment Matching for All Odd Moments and KurtosisMoment Matching for Higher Order MomentsConditional Quadratic ResamplingMoment Matching in Multi-factor CaseMartingale ResamplingUnconditional Martingale Resampling at the StateVariable LevelConditional Martingale Resampling at the StateVariable LevelBrownian Bridge Resampling at the State VariableLevelMartingale Control Variate at the UnderlyingInstrument LevelMartingale Resampling at the Derivatives Price LevelApplication to Secondary Model CalibrationOther Applications of Martingale ResamplingModeling of Multiple IndicesJLT Risk Neutralization of Credit Rating TransitionProcessCalibration of Credit Spread ProcessesRisk Neutralization of Mortgage Prepayment ModelAccuracy and Precision TestsExamples of Numerical Results
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Chapter 4
4.14.24.34.3.14.3.24.3.34.3.4
4.44.54.5.14.5.24.5.34.5.44.5.54.5.6
4.64.6.14.6.24.6.3
4.6.44.7
4.8
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Chapter 55.15.25.2.15.2.25.2.35.35.4
Interest Rate Models ClassificationShort Rate ModelsGaussian Short Rate ModelsLognormal Short Rate ModelsConstant Elasticity of VarianceModelsAffine Models and Quadratic ModelsWhat Interest Rate Models Should One Use?
Chapter 66.16.26.36.46.4.16.4.26.4.3
6.56.5.16.5.2
7.5.2
7.67.6.17.6.2
The Heath-Jarrow-Morton Framework
The Heath-Jarrow-MortonModelThe Ritchken-SankarasubramanianModelThe Inui-Kijima ModelOverview of Numerical Implementations of the RS andthe IK ModelRecombining Trinomial Tree TechniqueAdaptive Recombining Trinomial Tree TechniqueOverview of Applications of the Adaptive TrinomialTree Technique to the RS Model and the IK ModelAppendixClosed-form Solutionsfor the RS ModelClosed-formSolutionsfor the IK Model
Chapter 77.17.27.37.47.57.5.1
Introduction to Interest Rate Term StructureModeling
The Interest Rate Market Model
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250BGM Model versus HJM Model252The Brace-Gatarek-MusielaOriginal Approach256Comparison Between HJM and BGM Models258Jamshidian’s Approach259Martingale ApproachThe LIBOR Market Model and the Black Formulafor259Caps/FloorsThe Swap Market Model and the Black Formula for266European SwaptionsOverview of Simultaneous and Globally Consistent273Pricing and HedgingSimultaneous Consistent Pricing Through Approximation 275More on Simultaneous Consistent Pricing279
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More on the Martingale or Full-dimensional LIBORMarket ModelModeling Interest Rate Volatility Skew and Smile7.8CEV and LCEV Models for Modeling the Volatility Skew7.8.Iof Volatility Skew for Caplets and SwaptionsExamples7.8.2The Nonexploding Bushy Tree Technique7.9Construction of a Nonexploding Bushy Tree7.9.17.9.2 Modeling Stochastic Processes on a NonexplodingBushy Tree7.9.3 Application of Martingale Control Variate Technique7.9.4 Numerical ResultsGeneral Framework for Multi-factor Modeling for7.10Hybrid MarketStochastic Volatility BGM Models7.11Examples of Stochastic Volatility BGM Model Results7.12Appendix7.137.13.1 More Numerical Results Obtained With the NBTTechnique7.13.2 SufJicient Conditionsfor Convergence7.13.3 Application of Girsanov 's Change of Measure Theoremto Derivation of the Martingale or Full-dimensionalLIBOR Market Model
7.7
Chapter 88.18.28.38.48.5
8.68.78.88.98.9.18.9.28.9.3
Credit Risk Modeling and Pricing
Pricing Simple Defaultable InstrumentsDefault Contingent InstrumentsA Simple Markov Chain ModelModeling Correlated Default Event Processes with aFactor ModelModeling Correlated Default Time Processes with theCopula ApproachRecovery Rate ModelingRisky Market Model for Credit Spread ModelingJoint Credit Spread and Default ModelingCounterparty Credit Risk Pricing in OTC DerivativesCredit Charge CalculationExpected and Potential Exposures and ExpectedShortfallCredit Benefit Calculation
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8.9.48.9.58.9.68.9.78.108.10.18.10.28.10.38.10.48.10.58.10.68.10.78.10.8
Collateral or Margin AgreementNet Credit Charge and Funding Spread CalculationMartingale Relationships in Credit Charge CalculationsClosed-form Solutions and ApproximationsFramework for Counterparty Credit Risk Modeling andPricingCentralized Market Process Modeling and ScenarioGeneration EngineExposure or MTM Modeling EngineNew Trade and Real-time Exposure or MTM ModelingEngineCounterparty Credit Process Modeling and ScenarioGeneration EnginePortfolio ESfect Handling and Aggregation EngineCounterparty Credit Risk Pricing EngineSensitivity and Scenario Analysis EngineUnexpected Risk Modeling Engine
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PART II
INTEREST RATE MARKET FUNDAMENTALSAND PROPRIETARY TRADING STRATEGIES 387
Chapter 9
Simple Interest Rate Products
9.19.1.I9.I,29.29.2.19.2.29.2.39.39.49.4.19.4.29.4.39.4.49.4.59.59.5.1
Treasury IssuesTreasury BillsTreasury Notes and BondsFutures ContractsEuro-dollars and LIBOREuro-dollar FuturesNote and Bond FuturesInterest Rate DerivativesInterest Rate SwapsPlain Vanilla Interest Rate SwapForward SwapBasis SwapConstant Maturity SwapSwaptionBond OptionsOTC Options
389
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Chapter 10 Yield Curve Modeling
397
10.110.210.310.3.110.3.210.3.310.3.410.410.4.I10.4.2
IntroductionThe Bootstrap MethodOrthogonal Exponential Spline ModelExponential Basis FunctionsMaximum Likelihood Estimatesfor Spline CoeficientsImplementation of the Spline ModelSummarySwap CurveConstructing Euro-dollar Strip CurveConvexity Adjustment
Chapter 11 Two-Factor Risk Model11.111.211.311.411.511.5.111S.211.5.311.611.6.111.6.211.7
PCA and TFRM MethodologiesPrincipal Components AnalysisTwo-factor Risk Model SpecificationEmpirical ValidationApplicationsLevel-hedged Bullet/Barbell TradesTwo-factor Portfolio Hedging StrategyBond Indices with Level and Curve Risk ProfileAdjusted DurationsP-Adjusted DurationHedging the Extremely Long EndFuture Directions
397398399400403405406406407408
41141141341842 1423423423426427430432433
Chapter 12 The Holy Grail - Two-Factor Interest RateArbitrage
434
12.