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Valuation of mortgage products with stochastic prepayment-intensity models [Elektronische Ressource] / Andreas Kolbe

212 pages
Technische Universit¨at Mu¨nchenZentrum Mathematik - HVB-Stiftungsinstitut fu¨r FinanzmathematikValuation of mortgage products withstochastic prepayment-intensitymodelsAndreas KolbeVollst¨andiger Abdruck der bei der Fakult¨at fu¨r Mathematik der TechnischenUniversit¨at Mu¨nchen zur Erlangung des akademischen Grades einesDoktors der Naturwissenschaften (Dr.rer.nat.)genehmigten Dissertation.Vorsitzende: Univ.-Prof. Claudia Czado, Ph.D.Pru¨fer der Dissertation: 1. Univ.-Prof.Dr. Rudi Zagst2. Prof. Frank J. Fabozzi, Ph.D.(Yale University, USA),schriftliche Beurteilung3. Univ.-Prof.Dr. Ru¨diger Kiesel(Universit¨at Ulm)Die Dissertation wurde am 13. November 2007 bei der Technischen Univer-sit¨at eingereicht und durch die Fakult¨at fu¨r Mathematik am 30. Januar 2008angenommen.2iAbstractThis thesis is concerned with the valuation of mortgage products with un-certain time of termination. In particular, we develop new valuation modelsfor agency mortgage-backed securities (MBS) as they are traded in the USmarket. Standard US mortgages feature a prepayment option which is oftennot exercised optimally. This causes uncertainty with respect to the time oftermination of a mortgage contract and makes the valuation of mortgage-backed securities a mathematically challenging task.
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Technische Universit¨at Mu¨nchen
Zentrum Mathematik - HVB-Stiftungsinstitut fu¨r Finanzmathematik
Valuation of mortgage products with
stochastic prepayment-intensity
models
Andreas Kolbe
Vollst¨andiger Abdruck der bei der Fakult¨at fu¨r Mathematik der Technischen
Universit¨at Mu¨nchen zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften (Dr.rer.nat.)
genehmigten Dissertation.
Vorsitzende: Univ.-Prof. Claudia Czado, Ph.D.
Pru¨fer der Dissertation: 1. Univ.-Prof.Dr. Rudi Zagst
2. Prof. Frank J. Fabozzi, Ph.D.
(Yale University, USA),
schriftliche Beurteilung
3. Univ.-Prof.Dr. Ru¨diger Kiesel
(Universit¨at Ulm)
Die Dissertation wurde am 13. November 2007 bei der Technischen Univer-
sit¨at eingereicht und durch die Fakult¨at fu¨r Mathematik am 30. Januar 2008
angenommen.2i
Abstract
This thesis is concerned with the valuation of mortgage products with un-
certain time of termination. In particular, we develop new valuation models
for agency mortgage-backed securities (MBS) as they are traded in the US
market. Standard US mortgages feature a prepayment option which is often
not exercised optimally. This causes uncertainty with respect to the time of
termination of a mortgage contract and makes the valuation of mortgage-
backed securities a mathematically challenging task. Building on recently
introduced stochastic prepayment-intensity models for individual mortgage
contracts, we develop new mathematically consistent valuation models for
mortgage-backed securities. This modelling approach can also be considered
as an extension of the more traditional, purely econometric MBS valuation
models which are very popular in practice.
The intensity-based modelling framework also allows us to develop a
closed-form approximation formula for the value of agency MBS. Compared
to existing MBS valuation approaches in the academic and practitioner-
oriented literature, which usually rely on Monte-Carlo simulations or costly
numerical methods to solve multidimensional partial differential equations,
our closed-form approximation approach offers a computationally highly effi-
cient alternative. We apply this approach to some selected portfoliomanage-
ment applications with MBS, which require frequent product revaluations
under different scenarios and thus computationally efficient valuation rou-
tines.
Furthermore, we consider the valuation of reverse mortgages in this the-
sis. Reverse mortgages also feature uncertainty with respect to the time of
termination of the contract and their mathematical valuation is thus non-
trivial. We develop a consistent valuation model, again based on a stochastic
termination-intensity, and illustrate our approach with some examples di-
rected towards the German market, where reverse mortgages are not yet
available.iiiii
Zusammenfassung
Die im amerikanischen Markt ublichen Hypothekenkredite beinhalten eine¨
Option, die es dem Kreditnehmer erlaubt, den Kredit jederzeit vorzeitig und
ohne Vorfa¨lligkeitsentsch¨adigung zu tilgen (prepayment). Die Existenz der
prepayment-Option und die Tatsache, dass viele Kreditnehmer die Option
suboptimal ausu¨ben, erzeugen Unsicherheit hinsichtlich des Terminierungs-
zeitpunktes von Hypothekenkontrakten und machen die finanzmathemati-
sche Bewertung von Hypothekendarlehen (mortgages)und Mortgage-Backed
Securities (MBS) zu einem anspruchsvollen Problem. Aufbauend auf inten-
sit¨atsbasierten Modellen fu¨r individuelle Hypthekenkredite, werden in dieser
Dissertation Bewertungsmodelle fur MBS entwickelt, die auch als Erweite-¨
rung der in der Praxis gebr¨auchlichen, rein ¨okonometrischen Modelle inter-
pretiert werden konnen.¨
Der intensitatsbasierte Ansatz ermoglicht es zudem, eine approximative,¨ ¨
geschlossene Bewertungsformel fu¨r Mortgage-Backed Securities mit festem
Zinssatzherzuleiten.ImVergleichzubestehendenMBS-Bewertungsroutinen,
die u¨blicherweise eine Monte-Carlo Simulation oder aufwa¨ndige numerische
Verfahren zur Losung mehrdimensionaler partieller Differentialgleichungen¨
erfordern, bietet die entwickelte geschlossene Approximationsformel eine nu-
merischsehreffiziente Bewertungsalternative. Dieseermoglichtesauch,MBS¨
im Rahmen einiger ausgewa¨hlter Anwendungen im Portfoliomanagement zu
betrachten, die eine wiederholte Produktbewertung unter verschiedenen Sze-
narien erfordern.
Abschließend werden in dieser Dissertation Reverse Mortgages betrach-
tet. Die mathematische Bewertung von Reverse Mortgages ist nicht-trivial,
da deren Terminierungszeitpunkt ebenfalls zuf¨allig ist. Der in dieser Arbeit
entwickelte mathematisch konsistente Bewertungsansatz basiert, wie bereits
die Bewertung von MBS, auf einer stochastischen Terminierungsintensit¨at.
Das Bewertungsmodell wird schließlich mit einigen Beispielen fur den deut-¨
schen Markt illustriert, in dem Reverse Mortgages bisher nicht erh¨altlich
sind.ivv
Acknowledgements
First of all, I would like to thank my supervisor Prof. Dr. Rudi Zagst. He
offeredmethepossibility todoadissertationattheHVB-InstituteforMath-
ematical Finance and significantly contributed to the success of this research
project through his valuable ideas, advice, feedback and encouragement in
numerousdiscussions. Heprovidedtheacademicallyproductiveenvironment
and also gave me the opportunity to present my work at various conferences.
Furthermore, I am grateful to Prof. Frank J. Fabozzi, Ph.D. and to Prof.
Dr. Ru¨diger Kiesel for agreeing to serve as referees for this thesis and to
Prof. Dr. Claudia Czado for agreeing to chair the examination board of my
dissertation.
I would alsolike tothankthe Market Risk ControlDivision atBayerische
Landesbank (BayernLB), headed by Dr. Stefan Peiss, for the financial sup-
portwhich madethisresearch cooperationbetween BayernLB andtheHVB-
Institute for Mathematical Finance possible. I am particularly grateful to
Kai-Uwe Radde, former head of the Market Risk team at BayernLB Munich
(nowAllianzS.E.), whoinitiatedtheresearch cooperationandsupported my
application. He also contributed to the success of this dissertation through
his ongoing interest, advice and encouragement during the last three years.
I would also like to thank all colleagues in the Market Risk and Quantita-
tive Analysis teams at BayernLB Munich and New York for the interesting
projects we jointly worked on and for the many discussions on prepayment
andmortgage-backed securities inparticular, which greatlyhelped me toun-
derstand the problems related to these topics from a practitioner’s point of
view.
Finally I would like to express my gratitude to my colleagues at the
HVB-Institute for Mathematical Finance for many helpful discussions and
the always pleasant working atmosphere and to my family and friends for
making these last three years a highly enjoyable time.viContents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objectives and structure . . . . . . . . . . . . . . . . . . . . . 3
2 Mortgage products and prepayment 5
2.1 Prepayment and prepayment risk: A definition . . . . . . . . . 5
2.2 Mortgage-backed securities (MBS) . . . . . . . . . . . . . . . . 7
2.2.1 Subtypes of MBS and trading mechanics . . . . . . . . 7
2.2.2 Prepayment . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.3 Basic MBS cash flow conventions . . . . . . . . . . . . 13
2.3 Reverse mortgages . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Mathematical preliminaries 19
3.1 The Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Interest-rate markets . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 General definitions . . . . . . . . . . . . . . . . . . . . 22
3.2.2 The Vasicek and Hull-White Models . . . . . . . . . . 25
3.2.3 The Cox-Ingersoll-Ross Model . . . . . . . . . . . . . . 29
3.3 Point processes and intensities . . . . . . . . . . . . . . . . . . 31
3.3.1 Theoretical overview . . . . . . . . . . . . . . . . . . . 31
3.3.2 Application to the pricing of contingent claims . . . . . 38
3.4 The Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 Mortgage and MBS valuation 45
4.1 The different model classes . . . . . . . . . . . . . . . . . . . . 46
4.1.1 Econometric models . . . . . . . . . . . . . . . . . . . 46
4.1.2 Option-theoretic models . . . . . . . . . . . . . . . . . 48
4.1.3 Intensity-based models . . . . . . . . . . . . . . . . . . 51
4.2 Current frontiers and further challenges . . . . . . . . . . . . . 52
viiviii CONTENTS
5 A new hybrid-form MBS valuation model 57
5.1 The model set-up for a fixed-rate MBS . . . . . . . . . . . . . 57
5.2 Application to market data . . . . . . . . . . . . . . . . . . . 62
5.2.1 Parameter estimation and model calibration . . . . . . 62
5.2.2 Prices and option-adjusted spreads . . . . . . . . . . . 74
5.2.3 Effective duration, convexity and
parameter sensitivities . . . . . . . . . . . . . . . . . . 78
5.3 Adjustable-Rate MBS . . . . . . . . . . . . . . . . . . . . . . 84
5.4 Collateralized Mortgage Obligations . . . . . . . . . . . . . . . 88
6 A closed-form approximation for fixed-rate MBS 93
6.1 The model set-up . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2 Application to market data . . . . . . . . . . . . . . . . . . . 110
6.2.1 Parameter estimation and model calibration . . . . . . 110
6.2.2 Model performance, prices & sensitivities . . . . . . . . 114
7 The contribution of our MBS pricing models 123
7.1 A comparative assessment . . . . . . . . . . . . . . . . . . . . 123
7.2 Implications for the use in practice . . . . . . . . . . . . . . . 128
8 Optimal portfolios with MBS 129
8.1 The set-up: assets and scenarios . . . . . . . . . . . . . . . . . 130
8.2 Scenario-based portfolio optimisation
with MBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.2.1 Expected utility approach . . . . . . . . . . . . . . . . 138
8.2.2 Portfolio optimisation with CVaR constraints . . . . . 143
9 Valuation and Pricing of Reverse Mortgages 151
9.1 The default-free modelling framework . . . . . . . . . . . . . . 152
9.1.1 Fixed-rate reverse mortgages . . . . . . . . . . . . . . . 157
9.1.2 Adjustable-rate reverse mortgages . . . . . . . . . . . . 158
9.2 Introducing default risk . . . . . . . . . . . . . . . . . . . . . . 162
9.3 Results and implications . . . . . . . . . . . . . . . . . . . . . 167
10 Summary and conclusion 177
A A Monte-Carlo algorithm 181
B The moving block bootstrap 185
C Discussion of approximation errors 189

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