Advanced Financial Modelling

465 Pages·2009·3.391 MB·English

by  Albrecher H., Runggaldier W.J., Schachermayer W. (eds.)

Radon Series on Computational and Applied Mathematics 8
Managing Editor
Heinz W.Engl (Linz/Vienna)
Editors
Hansjörg Albrecher (Lausanne)
Ronald H.W.Hoppe (Augsburg/Houston)
Karl Kunisch (Graz)
Ulrich Langer (Linz)
Harald Niederreiter (Singapore)
Christian Schmeiser (Linz/Vienna)
Radon Series on Computational and Applied Mathematics
1 Lectures on Advanced Computational Methods in Mechanics
Johannes Kraus and Ulrich Langer (eds.), 2007
2 Gröbner Bases in Symbolic Analysis
Markus Rosenkranz and Dongming Wang (eds.), 2007
3 Gröbner Bases in Control Theory and Signal Processing
Hyungju Park and Georg Regensburger (eds.), 2007
4 A Posteriori Estimates for Partial Differential Equations
Sergey Repin, 2008
5 Robust Algebraic Multilevel Methods and Algorithms
Johannes Kraus and Svetozar Margenov, 2009
6 Iterative Regularization Methods for Nonlinear Ill-Posed Problems
Barbara Kaltenbacher, Andreas Neubauer and Otmar Scherzer, 2008
7 Robust Static Super-Replication of Barrier Options
Jan H. Maruhn, 2009
8 Advanced Financial Modelling
Hansjörg Albrecher, Wolfgang J. Runggaldier and Walter Schachermayer
(eds.), 2009
Advanced Financial Modelling
Edited by
Hansjörg Albrecher
Wolfgang J.Runggaldier
Walter Schachermayer
≥
Walter de Gruyter · Berlin · New York
Editors
HansjörgAlbrecher WolfgangJ.Runggaldier
Universite´ deLausanne DipartimentodiMatematicaPuraedApplicata
QuartierUNIL-Dorigny Universita` degliStudidiPadova
BaˆtimentExtranef ViaTrieste63
1015Lausanne,Switzerland 35121Padova,Italy
E-mail:[email protected] E-mail:[email protected]
WalterSchachermayer
FacultyofMathematics
UniversityofVienna
Nordbergstraße15
1090Vienna,Austria
E-Mail:[email protected]
Keywords
Mathematicalfinance,actuarialmathematics,stochasticdifferentialequations,optimization,
mathematicalmodelling,computationalmethods.
MathematicsSubjectClassification2000
91-02,60G35,60H35,60J60,62P05,65C05,91B16,91B28,91B70,93E20.
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ISBN 978-3-11-021313-3
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TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableintheInternetathttp://dnb.d-nb.de.
(cid:2)Copyright2009byWalterdeGruyterGmbH&Co.KG,10785Berlin,Germany.
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Preface
Thisbookisacollectionofstate-of-the-artsurveysonvarioustopicsinmathemati-
calfinance,withanemphasisonrecentmodelingandcomputational approaches. The
volumeisrelatedtoaSpecialSemesteronStochasticswithEmphasisonFinancethat
tookplacefromSeptembertoDecember2008attheJohannRadonInstituteforCom-
putational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences
inLinz,Austria.
TheSpecialSemesterwasbuiltaroundanumberofselectedtopicsandeachofthese
topicswasthethemeofaninternationalworkshopwithabout20invitedspeakers. Be-
sides a Tutorial, a Kick-Off Workshop focusing also on “Academics meeting Practi-
tioners”andaConcludingWorkshop,thethematicworkshopsconcernedthefollowing
topics:
Advanced Modelling in Finance and Insurance; Optimization and Optimal Con-
trol;InverseandPartialInformationProblems: MethodologyandApplications; Com-
putational Methods with Applications in Finance, Insurance and the Life Sciences;
StochasticMethodsinPartialDifferentialEquationsandApplicationsofDeterministic
andStochasticPDEs.
Inadditiontotheworkshops,theideaarosetocollectsurveysonimportantaspects
and recent developments related to the topics of the Special Semester and this forms
the contents of the present volume. The topics covered include the following (listed
alphabetically and grouped according to their relation with the topics of the Special
Semesterintheaboveorder):
• Affinediffusionprocessesinfinance
• DefaultandprepaymentmodelingusingLevyprocesses
• Volatilityinferenceinmodelsbeyondsemimartingales
• Optimalassetallocation
• Optimalconsumptionandinvestmentinilliquidmarketsandwithdownsiderisk
measures
• Multiperiodacceptabilityfunctionals
• Worst-caseportfoliooptimization
• Gooddealbounds
• Optimalinvestmentandhedgingunderpartialandinsideinformation
• Regularizationofinverseproblemsandcalibrationofoptionpricemodels
• Advancedsimulationtechniques
• ApplicationsofMalliavinCalculus
• ProbabilisticschemesforfullynonlinearPDE’s
Thecontributionsthemselvesarearrangedinalphabeticorderaccordingtothefirst
namedauthor.
vi Preface
Moredetails onthe Special Semester andthe fullworkshop programcan befound
attheRICAMSpecialSemesterwebpageat:
http://www.ricam.oeaw.ac.at/specsem/sef
Wewouldliketotakethisopportunitytothankallthosewhohavecontributedscientif-
icallytothisSpecialSemester,inparticulartheauthorsofthisvolumeandthespeakers
attheworkshopsaswellasthe(morethan250)participantsintheworkshops. Further
thanks go to the Austrian Academy of Sciences and in particular the Johann Radon
InstituteofComputationalandAppliedMathematicsinLinzanditsdirectorHeinzW.
EnglformakingthisSpecial Semester possible. WealsothankRobertPlatofromthe
publishinghousedeGruyter fortheprofessionaleditorial supportduringtheprepara-
tionofthisvolume.
Lausanne,PaduaandVienna,June2009 HansjoergAlbrecher
WolfgangRunggaldier
WalterSchachermayer
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
O. E. BARNDORFF-NIELSEN, J. SCHMIEGEL
Browniansemistationaryprocessesandvolatility/intermittency . . . . . . . . . 1
D. BECHERER
Fromboundsonoptimalgrowthtowardsatheoryofgood-dealhedging . . . . 27
C. BLANCHET-SCALLIET, R. GIBSON BRANDON, B. DE SAPORTA,
D. TALAY, E. TANRE´
Viscositysolutionstooptimalportfolioallocationproblemsinmodelswith
randomtimechangesandtransactioncosts . . . . . . . . . . . . . . . . . . . . . 53
B. BOUCHARD, R. ELIE, N. TOUZI
Discrete-timeapproximationofBSDEsandprobabilisticschemesforfully
nonlinearPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
D. FILIPOVIC´, E. MAYERHOFER
Affinediffusionprocesses: theoryandapplications . . . . . . . . . . . . . . . . 125
M. B. GILES, B. J. WATERHOUSE
Multilevelquasi-MonteCarlopathsimulation . . . . . . . . . . . . . . . . . . . 165
H. JO¨NSSON, W. SCHOUTENS, G. VAN DAMME
ModellingdefaultandprepaymentusingLe´vyprocesses: anapplicationtoasset
backedsecurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
B. JOURDAIN
Adaptivevariancereductiontechniquesinfinance . . . . . . . . . . . . . . . . 205
S. KINDERMANN, H. K. PIKKARAINEN
Regularisationofinverseproblemsanditsapplicationtothecalibrationofoption
pricemodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
C. KLU¨PPELBERG, S. PERGAMENSHCHIKOV
Optimalconsumptionandinvestmentwithboundeddownsideriskmeasuresfor
logarithmicutilityfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
A. KOHATSU-HIGA, K. YASUDA
AreviewofsomerecentresultsonMalliavinCalculusanditsapplications . . 275
R. KORN, M. SCHA¨L
Thenumeraireportfolioindiscretetime: existence,relatedconceptsand
applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
viii Contents
R. KORN, F. SEIFRIED
Aworst-caseapproachtocontinuous-timeportfoliooptimisation . . . . . . . . 327
R. KOVACEVIC, G. CH. PFLUG
Timeconsistencyandinformationmonotonicityofmultiperiodacceptability
functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
M. MONOYIOS
Optimalinvestmentandhedgingunderpartialandinsideinformation . . . . . 371
H. PHAM
Investment/consumptionchoiceinilliquidmarketswithrandomtradingtimes 411
T. ZARIPHOPOULOU
Optimalassetallocationinastochasticfactormodel–anoverviewandopen
problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
RadonSeriesComp.Appl.Math8,1–25 (cid:2)c deGruyter2009
Brownian semistationary processes and
volatility/intermittency
OleE. Barndorff-Nielsen and Ju¨rgenSchmiegel
Abstract.Anewclassofstochasticprocesses,termedBrowniansemistationaryprocesses(BSS),is
introducedanddiscussed. ThisclasshassimilaritiestothatofBrowniansemimartingales(BSM),
but is mainly directed towards the study of stationary processes, and BSS processes are not in
general of thesemimartingale type. Wefocus onsemimartingale - nonsemimartingale issuesand
oninferenceproblemsconcerning theunderlying volatility/intermittencyprocess, inthenonsemi-
martingalecaseandbasedonnormalisedrealisedquadraticvariation.TheconceptofBSSprocesses
hasarisenoutofanongoingstudyofturbulentvelocityfieldsandisthepurelytemporalversionof
thegeneraltempo-spatial frameworkofambitprocesses. Thelatter, whichmayhaveapplications
alsotothefinanceofenergymarkets,isbrieflyconsideredattheendofthepaper,againwithrefer-
encetothequestionofinferenceonthevolatility/intermittency.
Key words. Ambit processes, intermittency, nonsemimartingales, stationary processes, realised
quadraticvariation,turbulence,volatility.
AMSclassification.60G10
1 Introduction
ThispaperdiscussesstochasticprocessesY ={Yt}t∈R oftheform
(cid:2) (cid:2)
t t
Yt =μ+ g(t−s)σsdBs+ q(t−s)asds (1.1)
−∞ −∞
where μ is a constant, B is Brownian motion, g and q are nonnegative deterministic
functions on R, with g(t) = q(t) = 0 for t ≤ 0, and σ and a are ca`dla`g processes.
Whenσ anda arestationarythensoisY. Accordingly we shallrefer toprocessesof
this typeas Browniansemistationary (BSS) processes. Itissometimes convenient to
indicatetheformulaforY as
Y =μ+g∗σ•B+q∗a•Leb, (1.2)
whereLebdenotesLebesguemeasure.
We consider the BSS processes to be the natural analogue, for stationarity related
processes,oftheclassBSMofBrowniansemimartingales
(cid:2) (cid:2)
t t
Yt = σsdBs+ asds. (1.3)
0 0
Inthepresentpapertheprocessesσ andawill,unlessotherwisestated,betakento
bestationary,andwethenrefertoσasthevolatilityorintermittencyprocess. Theterm