Interest rate models: an infinite dimensional stochastic analysis perspective

239 Pages·2006·2.029 MB·English

by  Rene A. Carmona, M.R. Tehranchi

Springer Finance
EditorialBoard
M.Avellaneda
G.Barone-Adesi
M.Broadie
M.H.A.Davis
E.Derman
C.Klüppelberg
E.Kopp
W.Schachermayer
Springer Finance
SpringerFinanceisaprogrammeofbooksaimedatstudents,academicsand
practitionersworkingonincreasinglytechnicalapproachestotheanalysisof
financialmarkets.Itaimstocoveravarietyoftopics,notonlymathematicalfinance
butforeignexchanges,termstructure,riskmanagement,portfoliotheory,equity
derivatives,andfinancialeconomics.
AmmannM.,CreditRiskValuation:Methods,Models,andApplication(2001)
BackK.,ACourseinDerivativeSecurities:IntroductiontoTheoryandComputation(2005)
BarucciE.,FinancialMarketsTheory.Equilibrium,EfficiencyandInformation(2003)
BieleckiT.R.andRutkowskiM.,CreditRisk:Modeling,ValuationandHedging(2002)
BinghamN.H.andKieselR.,Risk-NeutralValuation:PricingandHedgingofFinancial
Derivatives(1998,2nded.2004)
BrigoD.andMercurioF.,InterestRateModels:TheoryandPractice(2001)
BuffR.,UncertainVolatilityModels-TheoryandApplication(2002)
CarmonaR.A.andTehranchiM.R.,InterestRateModels:anInfiniteDimensionalStochastic
AnalysisPerspective(2006)
DanaR.A.andJeanblancM.,FinancialMarketsinContinuousTime(2002)
DeboeckG.andKohonenT.(Editors),VisualExplorationsinFinancewithSelf-Organizing
Maps(1998)
DelbaenF.andSchachermayerW.,TheMathematicsofArbitrage(2005)
ElliottR.J.andKoppP.E.,MathematicsofFinancialMarkets(1999,2nded.2005)
FenglerM.R.,SemiparametricModelingofImpliedVolatility(200)
GemanH.,MadanD.,PliskaS.R.andVorstT.(Editors),MathematicalFinance–Bachelier
Congress2000(2001)
GundlachM.,LehrbassF.(Editors),CreditRisk+intheBankingIndustry(2004)
KellerhalsB.P.,AssetPricing(2004)
KülpmannM.,IrrationalExuberanceReconsidered(2004)
KwokY.-K.,MathematicalModelsofFinancialDerivatives(1998)
MalliavinP.andThalmaierA.,StochasticCalculusofVariationsinMathematicalFinance
(2005)
MeucciA.,RiskandAssetAllocation(2005)
PelsserA.,EfficientMethodsforValuingInterestRateDerivatives(2000)
PrigentJ.-L.,WeakConvergenceofFinancialMarkets(2003)
SchmidB.,CreditRiskPricingModels(2004)
ShreveS.E.,StochasticCalculusforFinanceI(2004)
ShreveS.E.,StochasticCalculusforFinanceII(2004)
YorM.,ExponentialFunctionalsofBrownianMotionandRelatedProcesses(2001)
ZagstR.,Interest-RateManagement(2002)
ZhuY.-L.,WuX.,ChernI.-L.,DerivativeSecuritiesandDifferenceMethods(2004)
ZieglerA.,IncompleteInformationandHeterogeneousBeliefsinContinuous-timeFinance
(2003)
ZieglerA.,AGameTheoryAnalysisofOptions(2004)
René A. Carmona · Michael R. Tehranchi
Interest Rate Models:
an Infinite Dimensional
Stochastic Analysis
Perspective
With12Figures
123
RenéA.Carmona
BendheimCenterforFinance
DepartmentofOperationsResearch
andFinancialEngineering
PrincetonUniversity
Princeton,NJ08544
USA
E-mail:[email protected]
MichaelR.Tehranchi
StatisticalLaboratory
CentreforMathematicalSciences
UniversityofCambridge
WilberforceRoad
CambridgeCB30WB
UK
E-mail:[email protected]
MathematicsSubjectClassification(2000):46T05,46T12,60H07,60H15,91B28
JELClassification:E43,G12,G13
LibraryofCongressControlNumber:2006924286
ISBN-10 3-540-27065-5 SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-27065-2 SpringerBerlinHeidelbergNewYork
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations
areliabletoprosecutionundertheGermanCopyrightLaw.
SpringerisapartofSpringerScience+BusinessMedia
springer.com
©Springer-VerlagBerlinHeidelberg2006
PrintedinGermany
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot
imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Coverdesign:design&production,Heidelberg
Typesettingandproduction:LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig
Printedonacid-freepaper 41/3100YL-543210
To Lenny Gross
Preface
The level of complexity of the bond market is higher than for the equity
markets: one simple reason is contained in the fact that the underlying in-
struments on which the derivatives are written are more sophisticated than
mere shares of stock. As a consequence, the mathematical models needed to
describe their time evolution will have to be more involved. Indeed on each
given day t, instead of being given by a single number S as the price of
t
one shareof a common stock,the term structure of interest rates is givenby
acurvedeterminedbyafinitediscretesetofvalues.Thiscurveisinterpreted
asthesamplingofthegraphofafunctionT֒ P(t,T)ofthedateofmaturity
→
oftheinstrument.In particular,wheneverwehavetodealwith stockmodels
involving ordinary or stochastic differential equations or finite dimensional
dynamical systems, we will have to deal with stochastic partial differential
equations or infinite dimensional systems!
The main goal of the book is to present, in a self-contained manner, the
empiricalfacts needed to understandthe sophisticatedmathematicalmodels
developed by the financial mathematics community over the last decade. So
after a very elementary introduction to the mechanics of the bond market,
and a thorough statistical analysis of the data available to any curious spec-
tator without any special inside track information, we gradually introduce
the mathematicaltools neededto analyzethe stochastic models mostwidely
used in the industry. Our point of view has been strongly influenced by re-
cent works of Cont and his collaborators and the Ph.D. of Filipovi´c. They
merge the originalproposalof Musiela inviting us to rewritethe HJM model
asastochasticpartialdifferentialequation,togetherwithBjo¨rk’sproposalto
recast the HJM model in the framework of stochastic differential equations
in a Banach space. The main thrust of the book is to present this approach
from scratch, in a rigorous and self-contained manner.
Quick Summary. The first part comprises two chapters. The first one is
very practical. Starting from scratch, it offers a lowbrow presentation of the
VIII Preface
bond markets. This chapter is of a descriptive nature, and it can be skipped
by readers familiar with the mechanics of these markets or those who are
more interested in mathematical models. The presentation is self-contained
and no statistical prerequisites are needed despite the detailed discussion of
PrincipalComponentAnalysis(PCAforshort)andcurveestimation.Onthe
otherhand,thediscussionofthefactormodelsfoundinthefollowingchapter
assumes some familiarity with Itˆo’s stochastic calculus and the rudiments
of the Black–Scholes pricing theory. This first part of the book constitutes
a good introduction to the fixed income markets at the level of a Master in
Quantitative Finance.
The second part is a course on infinite dimensional analysis. If it weren’t
for the fact that the choice of topics was motivated by the presentation of
thestochasticpartialdifferentialandrandomfieldapproachestofixedincome
models, this part could be viewed as stand-alone text. Prompted by issues
raised at the end of Part I, we introduce the theory of infinite dimensional
Itˆo processes, and we develop the tools of infinite dimensional stochastic
analysis (including Malliavin calculus) for the purpose of the general fixed
income models we study in Part III of the book.
The last part of the book resumes the analysis of fixed income market
models where it was left off at the end of Part I. The dynamics of the term
structurearerecastasastochasticsysteminafunctionspace,andtheresults
ofPartIIarebroughttobeartoanalyzetheseinfinitedimensionaldynamical
systemsbothfromthegeometricandprobabilisticpointofviews.Oldmodels
are revisited and new financial results are derived and explained in the light
of infinitely many sources of randomness.
Acknowledgments. The first version of the manuscript was prepared as
a setof lecturenotes for a graduateseminar givenby the first-namedauthor
during the summer of 2000 at Princeton University. Subsequently, the crash
course on the mechanics of the bond market was prepared in December of
2000 for the tutorial presented in Los Angeles at the IPAM on the 3rd,
4th and 5th of January 2001. Rough drafts of the following chapters were
added in preparation for short courses given in Paris in January 2001 and
Warwick in Marchof the same year.RCwould like to thank JaksaCvitanic,
Nizar Touzi, and David Elworthy respectively, for invitations to offer these
shortcourses.MTacknowledgessupportduringwritingofthisbookfromthe
NationalScienceFoundationintheformofaVIGREpostdoctoralfellowship.
He would also like to thank Thaleia Zariphopoulou for inviting him to give
a courseonthis materialat theUniversity ofTexas atAustin during the fall
of 2003.
Finally, we would like to dedicate this book to Leonard Gross. The foot-
prints of his seminal work on infinite dimensional stochastic analysis can
be found all over the text: the depth of his contribution to this corner of
mathematics cannot be emphasized enough. And to make matter even more
Preface IX
personal, RC would like to acknowledge an unrepayable personal debt to
L.Grossforbeinganenlighteningteacher,anenjoyableadvisor,arolemodel
for his humorous perspective on academia and life in general, and for being
a trustworthy friend.
Princeton, NJ, October 2005 Ren´e A. Carmona
Cambridge, UK, October 2005 Michael R. Tehranchi
Contents
Part I The Term Structure of Interest Rates
1 Data and Instruments of the Term Structure
of Interest Rates.......................................... 3
1.1 Time Value of Money and Zero Coupon Bonds ............. 3
1.1.1 Treasury Bills.................................... 4
1.1.2 Discount Factors and Interest Rates ................ 5
1.2 Coupon Bearing Bonds.................................. 6
1.2.1 Treasury Notes and Treasury Bonds ................ 7
1.2.2 The STRIPS Program ............................ 9
1.2.3 Clean Prices ..................................... 10
1.3 Term Structure as Given by Curves....................... 11
1.3.1 The Spot (Zero Coupon) Yield Curve ............... 11
1.3.2 The Forward Rate Curve and Duration ............. 13
1.3.3 Swap Rate Curves................................ 14
1.4 Continuous Compounding and Market Conventions ......... 17
1.4.1 Day Count Conventions ........................... 17
1.4.2 Compounding Conventions ........................ 19
1.4.3 Summary........................................ 20
1.5 Related Markets........................................ 21
1.5.1 Municipal Bonds ................................. 22
1.5.2 Index Linked Bonds .............................. 22
1.5.3 Corporate Bonds and Credit Markets ............... 23
1.5.4 Tax Issues....................................... 25
1.5.5 Asset Backed Securities ........................... 25
1.6 Statistical Estimation of the Term Structure ............... 25
1.6.1 Yield Curve Estimation ........................... 26
1.6.2 Parametric Estimation Procedures.................. 27
1.6.3 Nonparametric Estimation Procedures .............. 30
XII Contents
1.7 Principal Component Analysis ........................... 33
1.7.1 Principal Components of a Random Vector .......... 33
1.7.2 Multivariate Data PCA ........................... 34
1.7.3 PCA of the Yield Curve........................... 36
1.7.4 PCA of the Swap Rate Curve...................... 39
Notes & Complements....................................... 41
2 Term Structure Factor Models............................ 43
2.1 Factor Models for the Term Structure..................... 43
2.2 Affine Models .......................................... 46
2.3 Short Rate Models as One-Factor Models.................. 49
2.3.1 Incompleteness and Pricing........................ 50
2.3.2 Specific Models .................................. 51
2.3.3 A PDE for Numerical Purposes .................... 55
2.3.4 Explicit Pricing Formulae ......................... 57
2.3.5 Rigid Term Structures for Calibration............... 59
2.4 Term Structure Dynamics ............................... 60
2.4.1 The Heath–Jarrow–MortonFramework.............. 60
2.4.2 Hedging Contingent Claims........................ 62
2.4.3 A Shortcoming of the Finite-Rank Models ........... 63
2.4.4 The Musiela Notation............................. 64
2.4.5 Random Field Formulation ........................ 66
2.5 Appendices ............................................ 67
Notes & Complements....................................... 71
Part II Infinite Dimensional Stochastic Analysis
3 Infinite Dimensional Integration Theory .................. 75
3.1 Introduction ........................................... 75
3.1.1 The Setting...................................... 77
3.1.2 Distributions of Gaussian Processes................. 78
3.2 Gaussian Measures in Banach Spaces and Examples ........ 80
3.2.1 Integrability Properties ........................... 82
3.2.2 IsonormalProcesses .............................. 83
3.3 Reproducing Kernel Hilbert Space........................ 84
3.3.1 RKHS of Gaussian Processes ...................... 86
3.3.2 The RKHS of the Classical Wiener Measure ......... 87
3.4 Topological Supports, Carriers, Equivalence and Singularity.. 88
3.4.1 Topological Supports of Gaussian Measures.......... 88
3.4.2 Equivalence and Singularity of Gaussian Measures.... 89
3.5 Series Expansions ...................................... 91