Robust static super-replication of barrier options

209 Pages·2009·3.022 MB·English

by  Jan H Maruhn

Radon Series on Computational and Applied Mathematics 7
Managing Editor
Heinz W.Engl (Linz/Vienna)
Editors
Hansjörg Albrecher (Linz)
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
Jan H. Maruhn
Robust Static Super-Replication
of Barrier Options
≥
Walter de Gruyter · Berlin · New York
Keywords
Statichedging,barrieroptions,robustoptimization,stochasticvolatility,
semi-infiniteoptimization,semidefiniteprogramming.
MathematicsSubjectClassification2000
49-02,91-02,91B28,90C30,90C34,91B24.
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ToSylvia

Preface
Since the ground-breaking work of Black and Scholes it is common practice to
hedgeexoticoptions bydynamicallyadjustingportfoliopositionsbasedonthesensi-
tivities (Greeks) of the target option. However, while this works sufficiently well in
practice for instruments with well-behaved sensitivities, it may be impossible to dy-
namically hedge options with discontinuous payoff profile and hence wildly moving
Greeks.Motivatedbythisproblemaswellastheideatoavoidacontinuousadjustment
oftheportfoliopositions,theconceptofstatictradingstrategieshasbeendevelopedin
theliterature.
Duetothefactthatbarrieroptionsarecomponentsofavarietyofproductstradedin
themarket,variousauthorshavefocussedonthestatichedgingofbarrieroptionswith
discontinuous payoff profile like up-and-out calls. For these options, a static trading
strategyconsistsofbuyingaportfolioofstandardcallsandholdingthisportfoliocon-
stantuntil either the barrier option expires or the barrier is hit. In the latter caseof a
barrierhitatsometimeinthefuture,theportfolioofcallsisliquidatedandhopefully
providesapayoffequaltothevalueoftheup-and-outcall.
As the future value of the calls changes with the volatility surface, it is intuitively
clear that a static hedging strategy has to be robust against changes of this surface.
However,this keypropertyhas sofarnotbeenthroroughly adressedinthe literature,
becauseconsideringthedynamicsofthevolatilitysurfaceleadstoanalyticalandcom-
putationaldifficulties.
The book at hand closes this gap by developing an optimization approach which
combines the idea of static hedging with the concept of super-replication and finally
addsrobustnessagainstmovementsofthevolatilitysurfaceinthesenseofaworstcase
design. The resulting robust static hedging problem is analyzedin detailfrom a the-
oreticalandnumericalpoint of view. This analysisdrawsfrom the fields of financial
mathematics,stochasticandsemi-infiniteoptimization,convexanalysis,partialdiffer-
entialequations as wellas semidefinite optimization, to deriveappropriate existence,
dualityandconvergenceresults.
Detailedexamples show that the proposed statichedging frameworkis superior to
the approaches developed in the literature. In particular the computed robust static
hedgingstrategiespreventpotentiallyhugelossesduetochangesofthevolatilitysur-
faceand only consistof a smallnumber of liquidly tradedstandardoptions. Surpris-
ingly, the robustness can be gained by relatively little additional cost. Moreover, in
contrasttootherapproachesthecomputationsprovetobenumericallystableduetoan
implicitregularizationofthehedgingproblem.
Thebookdevelopsthestaticsuper-replicationapproachinseveralsteps. Chapter1
summarizes the theoretical tools needed throughout the book, including results from
viii Preface
thefieldsofmathematicalfinanceandoptimization. Afterwardsweintroducethecon-
ceptofstaticallyhedgingbarrieroptionsinChapter2andreviewthemainapproaches
thathavebeenconsideredintheliteraturesofar.
Chapter 3 presents the starting point of our analysis in the form of a stochastic
optimizationproblemcharacterizingcost-optimalstaticsuper-replicationstrategiesin
generalfinancial market models. As it turns out, the structure of the problem allows
to prove the existence of optimal static hedging strategies based on suitable results
from the fieldof convexanalysis. In addition we introduce a firstnaive Monte Carlo
discretization of the stochastic super-replication constraint which allows to compute
the strategyby solving a large scalelinear programming problem. Numericalresults
fortheBlack–ScholesandHeston’sstochasticvolatilitymodelshowthattheproposed
approachoutperformsallotherapproachesintheliterature. Inparticularthecomputed
portfolio matches the sensitivities of the target option and is only marginally more
expensivethanthebarrieroptionitself.
Although these results are promising, the Monte Carlo method has the significant
drawbackofslowconvergencewhichcanleadtolongcomputationtimes. Tocircum-
vent this problem, Chapter 4 reformulates the stochastic super-replication constraint
intoaninfinitenumberofdeterministicconstraints. Thestructureoftheresultinglin-
earsemi-infinite optimization problem canbe exploited to drasticallyreduce compu-
tationtimeincomparisontotheMonteCarlo-basedmethod. Moreover,theequivalent
formulation allows to derive the dual optimization problem with an interesting eco-
nomical interpretation. In analogy to the case of dynamic hedging the dual problem
maximizes the discounted expectation of the barrier option payoff. But this time the
expectationismaximizedoverthesetofmeasureswhichareprice-consistentwithre-
specttothestandardoptionsinthehedgeportfolio–afarbiggersetthantheequivalent
martingalemeasures. Thisalsotheoreticallyunderlinesthattheabilitytoapproximate
callpricesisakeypropertyforthederivationofasuccessfulstatichedge.
Based on the semi-infinite reformulation of the hedging problem Chapter 5 intro-
ducesarobustificationofthehedgingstrategyagainstchangesofthevolatilitysurface.
From an optimization point of view, the robustness is described by model parameter
uncertaintysetswhichcorrespondtoaninfinitenumberoffuturevolatilitysurfacesce-
narios. Theexistenceofsolutionsofthisproblemisprovenbyapplyingthemaximum
principle and Mizohata’s uniqueness theorem for parabolic partial differential equa-
tions. Furthermore,weprovetheconvergenceofsuitablealgorithmsforthenumerical
solutionoftherobusthedgingproblem. Thesemethodsforthefirsttimequantifyand
eliminate the model parameter sensitivity of hedge portfolios for barrier options by
addressingthenonlinearityofcalloptionpriceswithrespecttochangesofthevolatil-
itysurface. NumericalresultsfortheBlack–ScholesandHeston’sstochasticvolatility
model show that robustness against model parameter uncertainty can be gained by
surprisinglylowcost.
Chapter 6 shows that the previous results can in analogy be transfered to a sub-
replication setting. Combined with the super-replication prices this leads to robust
Preface ix
staticbounds forthe priceof barrieroptions whicharesurprisingly tightif compared
topurelymodel-independentboundsintheliterature. Moreover,wealsorobustifythe
static hedging problem against jumps by introducing an additional model parameter
uncertainty set. However, based on the maximum principle, we can eliminate this
uncertaintysetandtransformtheproblemtoaformulationwhichcanbeinterpretedas
moving the barrier. The chapter concludes with a section whichshows how to apply
therobusthedgingapproachtoalargevarietyofbarrieroptioncontracts.
Chapter 7 considers an additional robustification against model errors in the form
of ellipsoidaluncertaintysetsinthe pricespace. Fortunately, theseellipsoidal uncer-
taintysetscanbereplacedbyaninfinitenumberofsecondorderconeorsemidefinite
programmingconstraints. Tocomputeasolutionoftheproblem,weproposeanalgo-
rithm successively solving linear and nonlinear semidefinite programming problems
andproveitsconvergenceundermildconditions.
Finally, Chapter 8 analyzes the real world performance of the static superhedge
based on a seven year dataset. As it turns out, the theoretical robustness against
changes of the volatility surface (including the skew) is also confirmed empirically.
Compared to a static strike spread hedge and a dynamic local volatility hedge, the
robuststaticsuperhedgeistheportfolioleadingtothesmallesthedgeerrordispersion.
To summarize, the book develops a static hedging approach which allows to ro-
bustifytradingstrategiesagainstmodelparameteruncertaintyintheformofvolatility
shocks, skew and jump risk as well as model errors. The practical applicability of
theframeworkaswellastheperformanceofthetradingstrategiesareillustratedwith
numerousexamplesthroughoutthebook.
SomeWordsofGratitude
This book is based on the author’s dissertation accepted on March 5th, 2007, at the
Universityof Trier. Mostoftheworkevolvedduringmytimeasaresearchassociate
at the Department of Mathematics, in the researchgroup Numerical Analysis led by
Prof.Dr.EkkehardW.Sachs. AtthispointIwouldliketothankthepeoplewhohave
contributedtothisresearchandsupportedmeduringthepastyears.
FirstofallIwouldliketothankmyadvisorProf.Dr.EkkehardW.Sachsfornumer-
ous discussions aswellasthe jointworkandpublications on statichedging. Besides
his contributions to this work, I am especially grateful that he encouraged me to ex-
perience all aspects of the life of a researcher including the freedom to choose my
academicfocus,theopportunitytoparticipateinvariousinternationalconferencesand
that he always supported my ambitions to work on projects with external industry
partners. Itwasaveryinspiringandpleasanttimeintheresearchgroup.
Furthermore,IwishtothankDr.HansjörgAlbrecherforfruitfulcommentsandfor
actingasarefereeofmyPhDthesis.
In addition I would like to thank the Financial Engineering team (Equities, Com-
moditiesandFunds) ofUniCreditMarkets&InvestmentBanking, BayerischeHypo-
x Preface
undVereinsbankAG,forsupportingandinitiatingmyworkonstatichedgingduringa
jointresearchproject. IamparticularlyindebtedtoAlexanderGieseforverydeepand
enduringdiscussionsonthisworkandthepracticeofmathematicalfinanceingeneral.
Ourinitialpaperonstatichedgingwasthestartingpointfortheresearchpresentedin
thisbook.
Moreover, I wish to express my gratitude to Dr. Friedemann Leibfritz for detailed
discussions on the theory and practice of optimization as well as the joint work on
the robustification against model errors. My thanks also go to Morten Nalholm and
Matthias Fengler for the collaboration regarding the empirical performance of the
robust static hedge. I also thank my colleagues Ewgenij Hübner, Ilia Gherman and
ChristinaJagerattheUniversityofTrierforagreattimeinthedepartmentofmathe-
matics.
SpecialthanksgotoPDDr.RobertPlatofornumeroussuggestionsandforbeingso
patientwithrespecttonecessaryrevisionsofthemanuscript. Moreover,Iamgrateful
forthedetailedcommentsandsuggestionsofananonymousreferee.
Finally,IwouldliketothankmywifeSylviaforconstantlysupportingmeoverthe
pastyears. Withoutherthecompletionofthisresearchwouldnothavebeenpossible.
Iamalsoindebtedtomyparentsfortheirmoralandfinancialsupportofmyacademic
career.
JanMaruhn
Munich,April2009