Table Of ContentInternational Series in
Operations Research & Management Science
Rogemar S. Mamon
Robert J. Elliott Editors
Hidden Markov
Models in
Finance
Further Developments and
Applications, Volume II
International Series in Operations Research
& Management Science
Volume 209
SeriesEditor
CamilleC.Price
StephenF.AustinStateUniversity,TX,USA
Forfurthervolumes:
http://www.springer.com/series/6161
Rogemar S. Mamon • Robert J. Elliott
Editors
Hidden Markov Models
in Finance
Further Developments and Applications,
Volume II
123
Editors
RogemarS.Mamon RobertJ.Elliott
DepartmentofStatistical&Actuarial SchoolofMathematics
Sciences UniversityofAdelaide
UniversityofWesternOntario Adelaide,Australia
London,ON,Canada
ISSN0884-8289 ISSN2214-7934(electronic)
ISBN978-1-4899-7441-9 ISBN978-1-4899-7442-6(eBook)
DOI10.1007/978-1-4899-7442-6
SpringerNewYorkHeidelbergDordrechtLondon
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Preface
Since the publicationof our coedited monographHidden Markov Models (HMM)
in FinancebySpringerin 2007,therehasbeensubstantialresearchinmanyareas
offinancewhichemployHMM.Itistheobjectiveofthiseditedvolumetopresent
someupdatesonthecurrentstateofthefield.Thebookbringstogetherseveralar-
ticleswhichexplorethemostrecentdevelopmentsofHMMsandtheirapplications
tofinancialmodelling.Themainthemesinthiscollection,combiningnewtheoret-
icaladvancesandapplications,areimprovementsoftheEMalgorithmforHMMs,
interest rates, foreign exchange, insurance, option pricing, trading and parameter
estimationwithinsomeextendedHMMframeworks.
Theleadpaper‘RobustificationofanOnlineEMAlgorithmforModellingAsset
Prices Within an HMM’, by C. Erlwein-Sayer and P. Ruckdeschel, considers the
robustification of the EM algorithm in conjunctionwith Elliott’s pioneeringwork
(1994)onexactadaptivefiltersforMarkovchainsobservedinGaussiannoise.This
innovation aims to tackle the presence of outliers in financial data and preserves
confidenceintheresultsoftheHMMestimationprocedure.
Thesucceedingthreearticlesshowcasevariousapproachesinmodellinginterest
rates underHMM-drivenregime-switchingframeworks.Intheir paper‘Stochastic
VolatilityorStochasticCentralTendency:evidencefromaHiddenMarkovModelof
theShort-TermInterestRate’,C.A.WilsonandR.J.Elliottshowtheimportanceof
stochasticvolatilityoverstochasticcentraltendencyincapturingtheevolutionofin-
terestrates.Itfeaturesaniterativeprocedureindeterminingthelikelihoodfunction;
numericalmaximisation was used to find maximum likelihoodestimates. Wu and
ZengputforwardamodelwithananalyticallysimplerepresentationoftheMarkov
regimeshiftsbutwhichiscapableofhandlingthestylisedfeaturesoftheyieldcurve
intheirpaper‘AnEconometricModeloftheTermStructureofInterestRatesUnder
Regime-SwitchingRisk’.Theefficientmethodofmomentswasutilisedintheirem-
piricalexaminationofUSdatawithinatwo-factorversionoftheirproposedmodel.
Continuing the theme of interest-rate modelling, L. Steinru¨cke, R. Zagst and A.
SwishchukextendedtheLIBORmarketmodeltoincorporatesuddenmarketshocks,
structuralbreaksandchangesineconomicclimate.Theirpaper‘TheLIBORmarket
v
vi Preface
model:aMarkov-switchingjumpdiffusionextension’includesmodelcalibrationto
realdataillustratingpotentialusabilityforpractitioners.
‘ExchangeRates and Net Portfolio Flows: a Markov-SwitchingApproach’,by
F. Menla,F. SpagnoloandN. Spagnolo,is apaperwhichdescribestheusefulness
ofMarkov-switchingtoolstoinvestigatethefinerstructureoftheforeignexchange
rate market. In particular, it probes the effects of bond and equity portfolio flows
toexchangeratedynamics.CoveringmajorcurrenciesagainsttheUSdollar,itisa
far-reachingempiricalstudythatexploresthe relationshipbetweenportfolioflows
andfluctuationsintheexchangeratechanges.
In recentyears,there hasbeen a proliferationof finsurancebusiness dueto the
significantgrowthinthe creationofderivativesthatblendthecharacteristicsoffi-
nancialandinsuranceproducts.Theseinstrumentsrequirenewmethodstodealwith
theircomplexity.P.Azimzadeh,P.A.ForsythandK.R.Vetzalexaminethevaluation
andhedgingofguaranteedlifelongwithdrawalanddeathbenefitscontractsintheir
paper‘HedgingCostsforVariableAnnuitiesUnderRegimeSwitching’.Ageneral
approachisconstructedwhichenablesutility-basedpricingandotherfactorstobe
takenintoaccountandyieldsasystemofpartialdifferentialequations(PDEs).The
systemissolvedusinganimplicitmethodforalargeclassofutilityfunctionsthat
governthewithdrawalbehaviourofthepolicyholders.
D.Nguyen,G.YinandQ.Zhanganalyseanoptimaltradingstrategyassuming
abull-bearregime-switchingmarket.Theirpaper‘AStochasticApproximationAp-
proachforTrend-FollowingTrading’delvesintoabuy–sellstrategythatmaximises
expectedreturn.Theoptimalityofsuchastrategyisachievedbydeterminingthresh-
oldlevelsthroughastochasticapproximationalgorithm.Thiscircumventstheneed
tosolveHamilton–Jacobi–Bellman-typeequations.
Somecontributionstotheubiquitousandpopularproblemofoptionpricingun-
der an HMM setting are containedin two papersincludedin this monograph.TK
Siu’sarticle‘AHiddenMarkov-ModulatedJumpDiffusionModelforEuropeanOp-
tionPricing’isconcernedwithatwo-stepvaluationprocedure.Itconsistsofusing
filteringmethodsandtheEsschertransformtoderiveanintegro-partialdifferential
equationsatisfiedbythepriceofaEuropeanoption.Thepaper‘AnExactFormula
forPricingAmericanExchangeOptionswithRegimeSwitching’byL.Chanpro-
vides the valuation of exchange options of American style when the parameters
of the underlyingvariable are governedby an HMM. A homotopyanalysis-based
methodisutilisedtocalculatethevalueofanAmericanoptioninthisframework.
The last two papershighlightthe estimation of modelparametersunder HMM
frameworkswithenhancedflexibilityofmodellingotherfeaturesofobservedmar-
ket data. The emphasisis on their potentialimplementationin financialderivative
valuation, risk management and asset allocation. X. Xi and R. Mamon’s paper
‘Parameter Estimation in a Weak Hidden Markov Model with Independent Drift
and Volatility’ gives the estimation procedure for the drift and volatility compo-
nentsina modelwheretheyareindependentlyinfluencedbytwo separatehigher-
order HMMs with different state spaces. Such a procedure enables the incorpo-
ration of memory in historical data in addition to occasional structural changes.
Finally, another way to augmentthe capability of the HMM setting is to build an
Preface vii
estimation scheme assuming thatthe signal modelhas a non-normalperturbation.
Thisisdiscussedinthepaper‘ParameterEstimationinaRegime-SwitchingModel
with Non-normalNoise’byL. JalenandR. Mamon.Forthissetting,thedynamic
estimationoftransitionprobabilitiescanstillrelyonrecursivelyfilteredestimatesof
quantitiesthatarefunctionsoftheMarkovchain.Concentratingonnoisehavinga
Student’st-distribution,itisdemonstratedthattheestimationoftheotherremaining
parametersamountstosolvingasystemofnonlinearequationswhichcanbereadily
accomplishedbyusingmoderncomputingsoftware.
Acknowledgements RogemarS.Mamonwouldliketoexpresshisappreciationforthefinancial
supportreceivedfromtheNaturalSciencesandEngineeringResearchCouncil(grantno.341780-
2012).RobertJ.ElliottthankstheAustralianResearchCouncilandtheNaturalSciencesandEn-
gineeringResearchCouncilforthesupportofhisresearch.ParticularthanksgotoPhDcandidate
Ivy Zang (University of Western Ontario) for her technical help in producing the draft of this
monographconsistentwithSpringer’sstyle.
Botheditorsofthisvolumegratefullyacknowledgethefollowingdependablecolleagueswhose
invaluableinputswereessentialinthereviewprocessofeachcontribution:
LeunglungChan(UniversityofNewSouthWales,Australia)
BujarGhasi(UniversityofLiverpool,UK)
CodyHyndman(ConcordiaUniversity,Montre´al,Canada)
KennethJackson(UniversityofToronto,Canada)
MichaelLudkovski(UniversityofCaliforniaatSantaBarbara,USA)
NigelMeade(ImperialCollege,London,UK)
MarianitoRodrigo(UniversityofWollonggong,Australia)
KenSiu(CassBusinessSchool,CityUniversity,London,UK)
AnatoliySwishchuk(UniversityofCalgary,Canada)
ShuWu(UniversityofKansas,USA)
Finally, we wish to thank Fred Hillier (Springer’s international series in OR editor) and
Matthew Amboy (associate editor of OR & MS Springer Science + Business Media) for their
promptandreliableassistanceinmakingthisprojectasuccess.
London,ON,Canada RogemarS.Mamon
Adelaide,Australia RobertJ.Elliott
November2013
Contents
1 RobustificationofanOn-lineEMAlgorithmforModellingAsset
PricesWithinanHMM ...................................... 1
ChristinaErlwein-SayerandPeterRuckdeschel
1.1 Introduction.............................................. 1
1.2 HiddenMarkovModelFrameworkforAssetReturns ........... 3
1.3 EssentialStepsinElliott’sAlgorithm......................... 4
1.3.1 ChangeofMeasure ................................ 4
1.3.2 FilteringforGeneralAdaptedProcesses............... 4
1.3.3 Filter-BasedEMAlgorithm ......................... 7
1.3.4 SummaryofAlgorithm............................. 8
1.4 OutliersinAssetAllocationProblem......................... 8
1.4.1 OutliersinGeneral ................................ 8
1.4.2 Time-DependentContext:ExogenousandEndogenous
Outliers.......................................... 9
1.4.3 EvidenceforRobustnessIssueinAssetAllocation...... 10
1.5 RobustStatistics .......................................... 12
1.5.1 ConceptsofRobustStatistics........................ 12
1.5.2 OurRobustificationoftheHMM:GeneralStrategy ..... 14
1.5.3 RobustificationofStep(0) .......................... 15
1.5.4 RobustificationoftheE-Step........................ 15
1.5.5 Robustificationofthe(M)-Step...................... 18
1.6 ImplementationandSimulation ............................. 23
1.7 Conclusion............................................... 24
1.7.1 ContributionofThisPaper.......................... 24
1.7.2 Outlook.......................................... 26
Appendix...................................................... 27
References..................................................... 29
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