Table Of ContentLecture Notes in Economics
and Mathematical Systems 615
FoundingEditors:
M.Beckmann
H.P.Künzi
ManagingEditors:
Prof.Dr.G.Fandel
FachbereichWirtschaftswissenschaften
FernuniversitätHagen
Feithstr.140/AVZII,58084Hagen,Germany
Prof.Dr.W.Trockel
InstitutfürMathematischeWirtschaftsforschung(IMW)
UniversitätBielefeld
Universitätsstr.25,33615Bielefeld,Germany
EditorialBoard:
A.Basile,A.Drexl,H.Dawid,K.Inderfurth,W.Kürsten
Detlef Repplinger
Pricing of Bond Options
Unspanned Stochastic Volatility
and Random Field Models
Dr. Detlef Repplinger
Man Investments AG
Huobstrasse 3
8808 Pfäffikon SZ
Switzerland
[email protected]
ISBN978-3-540-70721-9 e-ISBN978-3-540-70729-5
DOI10.1007/978-3-540-70729-5
LectureNotesinEconomicsandMathematicalSystemsISSN0075-8442
LibraryofCongressControlNumber:2008931347
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Foreword
There is still a consistency problem if we want to price interest rate deriva-
tivesonzerobonds,likecapletsorfloorlets,andonswaps,likeswaptions,at
thesametimewithinonemodel.Thepopularmarketmodelsconcentrateei-
theronthevaluationofcapsandfloorsoronswaptions,respectively.Musiela
and Rutkowski (2005) put it this way: ”We conclude that lognormal market
models of forward LIBORs and forward swap rates are inherently incon-
sistent with each other. A challenging practical question of the choice of a
benchmarkmodelforsimultaneouspricingandhedgingofLIBORandswap
derivativesthusarises.”
Repplinger contributes to the research in this area with a new system-
atic approach. He develops a generalized Edgeworth expansion technique,
called Integrated Edgeworth Expansion (IEE), to overcome the aforemen-
tioned consistency problem. Together with a ’state of the art’ Fractional
Fourier Transform technique (FRFT) for the pricing of caps and floors this
method is applied to price swaptions within a set of ’up to date’ multidi-
mensionalstochasticinterestratemodels.Besidethetraditionalmulti-factor
Heath-Jarrow-Mortonmodels,termstructuremodelsdrivenbyrandomfields
and models with unspanned stochastic volatility are successfully covered.
Alongthewaysomenewclosedformsolutionsarepresented.
IamratherimpressedbytheresultsofthisthesisandIamsure,thatthis
monographwillbemostusefulforresearchersandpractitionersinthefield.
Tu¨bingen,May2008 RainerScho¨bel
v
Acknowledgements
This Ph.D. thesis has been prepared and accepted by the College of Eco-
nomicsandBusinessAdministrationattheEberhard-Karls-UniversityTu¨bin-
gen.There,Ihadthepleasuretoworkasaresearchassistantandlecturerat
theFinanceDepartmentofProf.Dr.-Ing.RainerScho¨bel.
First of all, I would like to thank my academic supervisor and teacher
Prof.Dr.-Ing.RainerScho¨bel,notonlyforhisguidanceandadviceworking
at the finance department, but also for being the promoter of my further ca-
reeratManInvestments.Moreover,IamgratefultothesecondrefereeProf.
Dr.JoachimGrammigandthefurthermemberofmythesiscommitteeProf.
Dr.WernerNeus.IwouldalsoliketothankProf.Dr.ManfredStadler,Pro-
fessorDr.GerdRonning,Prof.Dr.UweWalzandProf.Dr.WernerNeusfor
lettingmeparticipateinthePh.D.seminarasamemberofthepostgraduate
programme.
Finally, I owe many thanks to my colleagues and friends Bjo¨rn Lutz, Dr.
JochenVeith,Dr.RalfGampferandDr.FrankHenze,notonlyforthefruitful
andinspiringdiscussionsbutalsoforourtimetogether.
Last but not least, my gratitude goes to my wife Birgit, my sons Moritz
and Matthes and my parents for their support throughout the entire seven
yearsworkingonthatproject.
Laupen,Switzerland,May2008 DetlefRepplinger
vii
Contents
1 Introduction ............................................. 1
2 Theoptionpricingframework ............................. 7
2.1 Zero-couponbondoptions.............................. 8
2.2 Couponbondoptions .................................. 10
3 TheEdgeworthExpansion................................. 15
3.1 ThegeneralizedEEscheme............................. 16
3.2 Theapproximationofaχ2-pdf .......................... 21
v
3.3 Theapproximationofalognormal-pdf.................... 24
4 TheIntegratedEdgeworthExpansion ...................... 29
4.1 ThegeneralizedIEEscheme ............................ 30
4.2 Anapproximationoftheχ2-cdf ......................... 32
v
4.3 Anapproximationofthelognormal-cdf................... 34
5 Multi-FactorHJMmodels................................. 39
5.1 Thechangeofmeasure................................. 43
5.2 Pricingofzero-couponbondoptions ..................... 44
5.2.1 Theclosed-formsolution ......................... 45
5.2.2 Theclosed-formsolutionperformingaFRFT ........ 49
5.3 Pricingofcouponbondoptions.......................... 53
5.3.1 Aspecialclosed-formsolution .................... 55
5.3.2 ThespecialsolutionperforminganIEE ............. 57
5.3.3 Theone-factorsolutionperforminganIEE .......... 62
5.3.4 Themulti-factorsolutionperforminganEEE ........ 65
6 Multiple-RandomFieldstermstructuremodels ............. 71
6.1 RandomFields ....................................... 71
ix
x Contents
6.2 Multiple-RandomFieldHJM-framework.................. 75
6.3 Changeofmeasure .................................... 80
6.4 Pricingofzerobondoptions ............................ 81
6.4.1 Aclosed-formRandomFieldsolution .............. 81
6.5 Pricingofcouponbondoptions.......................... 86
6.5.1 Thesingle-RandomFieldsolutionperforminganIEE . 86
6.5.2 Themultiple-RandomFieldsolutionrunninganIEE .. 89
7 Multi-factorUSVtermstructuremodel .................... 93
7.1 Thechangeofmeasure................................. 97
7.2 Pricingofzero-couponbondoptions ..................... 98
7.2.1 TheindependentsolutionperformingaFRFT........ 99
7.2.2 ThedependentsolutionperformingaFRFT .........102
7.3 Pricingofcouponbondoptions..........................106
7.3.1 Theone-factorsolutionperforminganIEE ..........107
7.3.2 Themulti-factorsolutionperforminganIEE.........110
8 Conclusions..............................................113
9 Appendix ................................................117
9.1 IndependentBrownianmotions..........................117
9.2 DependentBrownianmotions ...........................119
9.2.1 Case1.........................................120
9.2.2 Case2.........................................121
9.2.3 Case3.........................................122
10 MatlabcodesfortheEEandIEE...........................125
10.1 Integerequation ......................................125
10.2 Computationofthecumulantsgiventhemoments ..........126
10.3 ComputationoftheHermitepolynomial ..................127
10.4 TheEE ..............................................128
10.5 TheIEE .............................................129
References...................................................131
Listoffigures ................................................135
Listoftables .................................................137
Chapter 1
Introduction
BeforetheworkofHoandLee[39]andHeath,JarrowandMorton[35]the
point of view in the literature was explaining the term structure of interest
ratesorrespectivelythecrosssectionofbondprices.ThenewHeath,Jarrow
and Morton (HJM) models perfectly fit to an observed initial term struc-
ture by focussing on the arbitrage-free pricing of related derivatives. Given
a specification of the volatility for the forward rates or bond prices together
with the initial term structure completely determines the risk-neutral bond
price dynamics or equivalently the short rate process (see e.g. de Jong and
SantaClara[24],Casassus,Collin-DufresneandGoldstein[14]).Thevolatil-
itystructureingeneralcanbecomputedbyinvertingtheoptionpricessimilar
tothecalculationofimpliedvolatilitiesthatareextractedfromstockoption
prices. One drawback of these models lies in the non-Markovian structure
of the short rate dynamics resulting in a computationally low tractability.
Hence, most of the HJM-models in the literature are restricted to a deter-
ministic volatility structure leading to a Markovian short rate process. It is
wellknownthatadeterministicvolatilityfunctionalwaysleadstoGaussian
interestratesandthereforewehavetodealwithnegativeinterestrates.
Unlike these models the traditional models, such as Cox, Ingersoll, and
Ross[22]andVasicek[73]arebuiltonstatevariablesstartingfromagiven
short rate process. Hence, they directly causes a Markovian structure. On
the other hand it is well known that they can fit the initial term structure
only by making the model parameters time dependent. In contrary, coming
fromtheHJM-frameworkthesocalledextendedformmodels(e.g.Hulland
White [41]) are a result of the arbitrage-free HJM setting, where the short
ratedynamicsaredefinedendogenously.
Cox,IngersollandRoss[22]andJamshidian[42]demonstratethatclosed-
formsolutionsforzero-couponbondoptionscanbederivedforsingle-factor
squarerootandGaussianmodels.Moregenerally,Duffie,PanandSingleton
1
2 1 Introduction
[28]demonstratethattheentireclassofaffinemodelspossessesclosed-form
solutions for zero-coupon bond options, which can be derived by applying
standard Fourier inversion techniques. The option pricing formula for zero-
couponbondoptions(caplets/floorlets)arediscussede.g.inChenandScott
[16],Duffie,PanandSingleton[28],BakshiandMadan[6]andChackoand
Das[15].Unfortunately,thesepaperssayonlylittleaboutthepricingofop-
tionsoncouponbonds(swaptions).
Givenasingle-factorGaussianinterestratemodelJamshidian[42]derives
a closed-form solution for the price of an option on coupon bonds. This so-
lutionstemsfromthefactthattheoptimalexercisedecisionatmaturityisa
onedimensionalboundaryandacouponbondcanbewrittenasaportfolioof
zero-coupon bonds. Unfortunately, the closed-form solution for options on
coupon bonds and zero-coupon bond options cannot be extended to multi-
factor models. Then the exercise boundary becomes a non-linear function
of the multiple state variables and cannot be computed in closed-form. Lit-
terman and Scheinkman [55] shows that typically multi-factor models are
needed to capture the dynamics of the term structure of interest rates. One
approachtohandlethenon-linearexerciseboundaryinamulti-factorsetting
is applied by Singleton and Umantsev [68]. They approximate the exercise
boundarywithalinearfunctionofthemultiplestate-variables.Ontheother
hand,therearetwonewdrawbacksfromthisapproach.Firstly,aseparateap-
proximation has to be performed for every single strike price coming along
with a low efficiency and tractability. Secondly, and even more restrictive
their approach becomes completely intractable for a large number of state
variables.Hence,weneedanewmethodforthecomputationofbondoption
pricesinageneralizedmulti-factorHJMsetup.
Miltersen, Sandmann, and Sondermann [60] and Brace, Gatarek, and
Musiela[9]derivethesocalledLIBORmarketmodel.Intheirapproachthe
lognormal distributed interest rates are given and closed-form solutions can
be derived to compute the prices of interest rate caps/floors and swaptions.
Their formulae are very tractable and easy to handle. On the other hand,
thereoccursamodelinconsistencybetweentheswaptionandcap/floormar-
kets coming from the fact that a lognormal LIBOR rate cannot coexist with
alognormaldistributedswaprate.
Weovercomethisinconsistency,byderivingaunifiedframeworkthatdi-
rectlyleadstoconsistentcap/floorandswaptionprices.Thus,ingeneralwe
start from a HJM-like framework. This framework includes the traditional
HJM model as well as an extended approach, where the forward rates are
drivenbymultipleRandomFields.Furthermore,eveninthecaseofamulti-
factor unspanned stochastic volatility (USV) model we are able to compute
thebondoptionpricesveryaccurately.First,wemakeanexponentialaffine
guess for the solution of an expectation, which is comparable to the solu-
1 Introduction 3
tionofaspecialcharacteristicfunction.Then,giventhissolutionweareable
to compute the prices of zero-coupon bond options by applying standard
Fourier inversion techniques. In limited cases this method can also be ap-
pliedforthepricingofcouponbearingbondoptions(seee.g.Singletonand
Umantsev [68]), but completely fails assuming a multi-factor framework.
In order to overcome this drawback, we use the solution of our exponen-
tial affine guess to compute the moments of the underlying random vari-
able.Giventhesemomentswearethenabletocomputethepricesofcoupon
bond options (swaptions) by performing an integrated approach of a gener-
alizedEdgeworthExpansion(IEE)technique(seechapter(4)).Thisisanew
method for the computation of the probability that an option matures in the
money.
In chapter (2), we derive a unified framework for the computation of the
priceofanoptiononazero-couponbondandacouponbondbyapplyingthe
wellknownFourierinversionscheme.Therefore,weintroducethetransform
Θ (z),whichlateroncanbeseenasacharacteristicfunction.Incaseofzero-
t
couponbondoptionsweareabletofindaclosed-formsolutionforthetrans-
formΘ (z) and apply standard Fourier inversion techniques. Unfortunately,
t
assuming a multi-factor framework there exists no closed-form solution of
thecharacteristicfunctionΞ (z)givenacouponbondoption.Hence,inthis
t
caseFourierinversiontechniquesfail.
Chapter (3) focuses on the derivation of a generalized approach of the
Edgeworth Expansion (EE) technique. This approach extends the series ex-
pansion technique of Jarrow and Rudd [44], Turnbull and Wakeman [72],
Collin-Dufresne and Goldstein [19] and Ju [47] to a generalized approxi-
mation scheme for the computation of the exercise probabilities that an op-
tionendsupinthemoney.Themainadvantageofthisnewtechniquestems
fromthefactthatthepricingschemeisstrictlyseparatedfromtheunderlying
model structure. Thus, the structure of the underlying dynamics enter only
inthecomputationofthemoments.Inotherwords,wederiveageneralized
algorithm to approximate the exercise probabilities, by using only the mo-
ments of the underlying random variable, which either can be computed in
closed-formorevennumerically1.
Inchapter(4),wederiveanewintegratedversionofthegeneralizedEE2.
Thisintegratedversioncanbeappliedtocomputetheexerciseprobabilities
directly,insteadofcomputinganintegrationovertheapproximatedpdf3.Fi-
nally, we obtain a series expansion of the exercise probabilities in terms of
Hermitepolynomialsandcumulants.Thisapproachisatechniquetoapprox-
1AMatlabprogramisavailableintheappendixsection(10.4).
2Therefore,wetermthisgeneralizedseriesexpansiontheIntegratedEdgeworthExpansion(IEE).
3TheEEoriginallyisderivedtoapproximatedensityfunctionsinsteadofprobabilities.
