Foreword
Intherecentyears,therehasbeensigni ficantresearchinterestinmodellingand controlof financialsystemsthroughstate-spacerepresentationoftheirdynamics. Thisisbecausesuchapproacheseliminatetheuseofheuristicsin financial decision-makingwhileassuringstabilityandinseveralcasesoptimalityinthe functioningofeconomicsystems.Asonedelvesintothecomplexityofthe fi nancial dynamics,heperceivesthatdeterministicmodellingisunlikelytoworkandthat variability,parametricuncertaintyandstochasticityarefactorsthatshouldbe seriouslytakenintoaccountfortheefficientmanagementofeconomicsystems. Throughasynergismofsystemstheoryandmachinelearningmethods,this monographdevelopsmodellingandcontrolapproacheswhich fi nallyassurethatthe monitored financialsystemswillevolveaccordingtothespeci ficationsandoptimalityobjectives,whiletheriskofwrongdecision-makinginthemanagement ofthesesystemswillbealsominimized.
Theuseofstate-spacemodelsin financialengineeringallowstoeliminate heuristicsandempiricalmethodscurrentlyinuseindecision-makingproceduresfor finance.Ontheotherside,itpermitstoestablishmethodsoffault-freeperformance andoptimalityinthemanagementofassetsandcapitalsandmethodsassuring stabilityinthefunctioningof fi nancialsystems(e.g.ofseveral financialinstitutions andofthebankingsector).Thesystemstheory-basedandmachinelearning methodsdevelopedbythemonographstandforagenuineandsignifi cantcontributiontothe fi eldof financialengineering.First,themonographsolvesinaconclusivemannerproblemsassociatedwiththecontrolandstabilizationofnonlinear andchaoticdynamicsin financialsystems,whenthesearedescribedintheformof nonlinearordinarydifferentialequations.Next,itsolvesinaconclusivemanner problemsassociatedwiththecontrolandstabilizationof financialsystemsgoverned byspatiotemporaldynamics,thatissystemsdescribedbypartialdifferential equations(e.g.theBlack–ScholesPDEanditsvariants).Moreover,themonograph solvestheproblemof filteringfortheaforementionedtypesof fi nancialmodels,that isofestimationoftheentiredynamicsofthe financialsystemswhenusinglimited information(partialobservations)obtainedfromthem.Finally,themonograph solvesinaconclusiveandoptimalmannertheproblemofstatisticalvalidationof
computationalmodelsandtoolsusedtosupport financialengineersindecision taking.Throughthemethodsitdevelops,themonographenablestoidentify inconsistentandinappropriatelyparameterized financialmodelsandtotakenecessaryactionsfortheirupdate.
Thetopicsstudiedinthemonographareofprimaryimportancefor financial engineeringandfortheprofitablemanagementof financialsystems.Itisacommon sensethatdecision-makingin financeshouldstopbeingbasedonintuition, heuristicsandempiricalrulesandshouldmoveprogressivelytosystematicmethods ofassuredperformance.Tothisend,themonographdemonstrates fi rstthatitis possibletoidentifythecompletedynamicsof financialsystemsusinglimited informationoutofthem,andnext,itshowsthattheestimateddynamicscanbeused forthecontrolandstabilizationofsuchsystems.Themonograph’sestimation, forecastingandcontrolmethodsarenotonlyaddressedto financialsystemsdescribedbynonlinearordinarydifferentialequations,butalsoextendedto financialsystemsexhibitingspatiotemporaldynamics,asinthecaseofthe Black–ScholesPDE.OfferingsolutiontoestimationandcontrolproblemsinPDE modelsmetoftenin financeisoneofthemaincontributionsofthemonograph,and thiscanbeusefulbothfortheacademiccommunityandfor financialengineers workinginpractice.Anothermajorcontributionofthemonographisinstatistical validationofdecision-makingtoolsusedin financialengineering.Takinginto accounttheneedforreliablefunctioningofsoftwaredevelopedfordecisionsupport in finance,onecaneasilyunderstandthesignificanceofthemonograph'sresults aboutvalidationofthecomputationalmodelsof financialsystemsandofthe associatedforecastingtools.Themonographoffersto fi nancialengineersoptimal statisticalmethodsfordeterminingwhetherthemodelsusedinestimationofthe stateof financialsystemsareaccurateorwhethertheycontaininconsistent parameterswhichresultinforecastingoflowprecision.
Thecontentsofthemonographcoverthefollowingkeyareasfor financial engineering:(i)controlandstabilizationof financialsystemsdynamics,(ii)state estimationandforecastingand(iii)statisticalvalidationofdecision-makingtools. Themonographisprimarilyaddressedtotheacademiccommunity.Thecontent ofthemonographcanbeusedforteachingundergraduateorpostgraduatecourses in financialengineering.Therefore,itcanbeusedbybothacademictutorsand studentsasareferencebookforsuchacourse.Asignificantpartofthemonographs readershipisalsoexpectedtocomefromtheengineeringandcomputerscience community,aswellasfromthe financeandeconomicscommunity.Thenonlinear PDEcontrolandestimationmethodsanalysedintheproposedmonographcanbea powerfultoolandusefulcompanionforpeopleworkingonapplied financial engineering.
Athens,GreeceGerasimosG.Rigatos March2017
Preface
Thepresentmonographcontainsnewresultsand findingsoncontrolandestimation problemsfor financialsystemsandforstatisticalvalidationofcomputationaltools usedfor financialdecision-making.Theuseofstate-spacemodelsin financial engineeringwillallowtoeliminateheuristicsandempiricalmethodscurrentlyin useindecision-makingproceduresfor finance.Ontheotherside,itwillpermitto establishmethodsoffault-freeperformanceandoptimalityinthemanagementof assetsandcapitalsandmethodsassuringstabilityinthefunctioningof financial systems(e.g.ofseveral financialinstitutionsandofthebankingsector).Asitcanbe confirmedfromanoverviewoftherelevantbibliography,thesystemstheory-based andmachinelearningmethodsdevelopedbythemonographstandforagenuine andsigni ficantcontributiontothe fi eldof fi nancialengineering.First,themonographsolvesinaconclusivemannerproblemsassociatedwiththecontroland stabilizationofnonlinearandchaoticdynamicsin financialsystems,whentheseare describedintheformofnonlinearordinarydifferentialequations.Next,itsolvesin aconclusivemannerproblemsassociatedwiththecontrolandstabilizationof financialsystemsgovernedbyspatiotemporaldynamics,thatissystemsdescribed bypartialdifferentialequations(e.g.theBlack–ScholesPDEanditsvariants). Moreover,themonographsolvestheproblemof filteringfortheaforementioned typesof financialmodels,thatisofestimationoftheentiredynamicsofthe fi nancial systemswhenusinglimitedinformation(partialobservations)obtainedfromthem. Finally,themonographsolvesinaconclusiveandoptimalmannertheproblemof statisticalvalidationofcomputationalmodelsandtoolsusedtosupport financial engineersindecisiontaking.Throughthemethodsitdevelops,themonograph enablestoidentifyinconsistentandinappropriatelyparameterized financialmodels andtotakenecessaryactionsfortheirupdate.
Themonographcomestoaddresstheneedaboutdecision-makingin fi nancethat willbenolongerbasedonheuristicsandintuitionbutwillmakeuseofcomputationalmethodsandtoolscharacterizedbyfault-freeperformanceandoptimality. Throughthesynergismofsystemstheoryandmachinelearningmethods,the monographofferssolutions,inaconclusivemanner,tothefollowingkeyproblems metin financialengineering:(i)controlandstabilizationof financialsystems
exhibitingnonlinearandchaoticdynamics;(ii)controlandstabilizationof fi nancial systemsexhibitingspatiotemporaldynamicsdescribedbypartialdifferential equations;(iii)solutiontotheassociated filteringproblems,thatisestimationofthe completedynamicsoftheaforementionedcomplextypesof financialmodelswith theuseoflimitedinformationextractedoutofthem;(iv)elaboratedcomputational toolsfortheassessmentofriskin fi nancialsystemsandfortheoptimizedmanagementofcapitalsandassets;and(v)statisticalvalidationofdecisionsupport toolsusedin finance,suchasforecastingmodelsandmodelsof financialsystems dynamics.Themonographisprimarilyaddressedtotheacademicandresearch communityof financialengineeringaswellastotutorsofrelevantuniversity courses.Itcanalsobeausefulreferenceforstudentsof financialengineering,at bothundergraduateandpostgraduatelevel,helpingthemtogetacquaintedwith establishedapproachesforcontrol,estimationandforecastingin financeaswellas withmethodsforvalidatingtheprecisionofcomputationaltoolsusedindecision support.Finally,itisaddressedto financialengineersworkingonpracticalproblemsofrisk-freedecision-makingandaimingatprofitablemanagementoffunds, commoditiesand financialresources.
Themanagementof financialsystemshastoaddressthefollowingissues: (i)stability,(ii)modellingandforecastingand(iii)validationandupdateof decision-makingtools.About(i),itisnotedthatalthoughthedynamicsof fi nancial systemshasbeendescribedefficientlybytheBlack–ScholesPDEanditsvariants, littlehasbeendoneaboutitsstabilization.Theproblemofcontrolandstabilization ofdiffusionPDEsofthistypeisanon-trivialoneandhastobeimplementedusing ascontrolinputsonlythePDEsboundaryconditions.Themonographoffers solutionofassuredconvergenceandperformanceforthisdifficultcontrolproblem. Additionally,thereareseveraltypesof financialsystemsdescribedbynonlinear ODEswhichexhibitchaoticdynamics.Themonographprovidesstabilizingcontrol methodsforsuchsystemstoo.About(ii),itiseasytounderstandthatforecastingin financialsystemsissignificantforriskassessmentandsuccessfuldecision-making. Bybeinginpositiontopredictfuturestatesofthe financialsystem,earlywarning indicationsarehandledandprofitableactionsaretakenforassetandcapitalmanagement.Themonograph’smethodcontributestothisdirection.About(iii),itis apparentthattheeffectivenessofalldecision-makingprocessesin financeis dependentonthesufficiencyoftheinformationcollectedfromthe fi nancialsystem andontheaccuracyandcredibilityofdecisionsupporttools.Thestatisticalvalidationofdecision-makingsoftwareandofthemodelsusedbyitisimportantforthe maximizationofprofitsin financialsystemsmanagementandfortheminimization ofrisks.Clearly,themonographsolvesthestatisticalvalidationproblemsina conclusivemanner.
Themonographcomprisesthefollowingchapters:
InChap. 1,systemstheoryandstabilityconceptsareoverviewed.Thischapter analysesthebasicsofsystemstheorywhichcanbeusedinthemodellingof nonlineardynamics.Tounderstandtheoscillatorybehaviourofnonlinearsystems thatcanexhibitsuchdynamics,benchmarkexamplesofoscillatorsaregiven. Moreover,usingexamplesfromstate-spacemodels,thefollowingpropertiesare
analysed:phasediagram,isoclines,attractors,localstability,bifurcationsof fixed pointsandchaosproperties.
InChap. 2,mainapproachestononlinearcontrolwithpotentialapplicationto financialsystemsareanalysed.Incontrolandstabilizationofthedynamicsof financialsystems,onecandistinguishthreemainresearchaxes:(i)methodsbased ongloballinearization,(ii)methodsbasedonasymptoticlinearizationand (iii)Lyapunovmethods.Asfarasapproach(i)isconcerned,thesearemethodsfor thetransformationofthenonlineardynamicsofthesystemtoequivalentlinear state-spacedescriptionsforwhichonecandesigncontrollersusingstatefeedback andcanalsosolvetheassociatedstateestimation(filtering)problem.Onecan classifyheremethodsbasedonthetheoryofdifferentially flatsystemsandmethods basedonLiealgebra.Asfarasapproach(ii)isconcerned,solutionsarepursuedto theproblemofnonlinearcontrolwiththeuseoflocallinearmodels(obtainedat localequilibria).Forsuchlocallinearmodels,feedbackcontrollersofprovenstabilitycanbedeveloped.Onecanselecttheparametersofsuchlocalcontrollersina mannerthatassurestherobustnessofthecontrollooptobothexternalperturbations andmodelparametricuncertainty.Asfarasapproach(iii)isconcerned,thatis methodsofnonlinearcontroloftheLyapunovtype,onecomesagainsttheproblems ofminimizationofLyapunovfunctionssoastoassuretheasymptoticstability ofthecontrolloop.ForthedevelopmentofLyapunov-typecontrollers,onecan eitherexploitamodelaboutthe fi nancialsystemsdynamicsorproceedina model-freemanner,asinthecaseofindirectadaptivecontrol.
InChap. 3,mainapproachestononlinearestimationwithpotentialapplicationto financialsystemsareanalysed.Totreatthe fi lteringproblemfornonlineardynamics in financialsystems,theExtendedKalmanFilterisanestablishedapproach. However,sincethisisbasedonapproximatelinearizationofthesystem’s state-spacedescriptionandinthetruncationofhigherordertermsintheassociated Taylorseriesexpansion,theUnscentedKalmanFilterisfrequentlyusedinitsplace. Thelatter filterperformsstateestimationbyaveragingonstatevectorsthatare selectedateachiterationofthe filteringalgorithmandbeingde finedbythecolumns oftheestimationerrorvectorcovariancematrix.Additionally,tohandlethecaseof non-Gaussiannoisesinthe filteringprocedure,theparticle filterhasbeenproposed. Anumberofpotentialstatevectorvalues(particles)areupdatedintimethrough elitismcriteria,andoutofthisset,theestimateofthestatevectoriscomputed.The topicofnonlinearestimationiscompletedbyanewnonlinear filteringapproach underthenameDerivative-freenonlinearKalmanFilter.This filterbasedonlinearizingtransformationofthemonitored financialsystemisproventoconditionally maintaintheoptimalityfeaturesofthestandardKalmanFilterandtobecomputationallyfasterthanothernonlinearestimationmethods.Moreover,totreatthe distributed filteringandstateestimationin financialsystems,onecanapply establishedmethodsfordecentralizedstateestimation,suchastheExtended InformationFilter(EIF)andtheUnscentedInformationFilter(UIF).EIFstandsfor thedistributedimplementationoftheExtendedKalmanFilter,whileUIFstandsfor thedistributedimplementationoftheUnscentedKalmanFilter.Additionally,to obtainadistributed filteringschemeinthismonograph,theDerivative-free
ExtendedInformationFilter(DEIF)isimplemented.Thisstandsforthedistributed implementationofadifferential flatnesstheory-based filteringmethodunderthe nameDerivative-freedistributednonlinearKalmanFilter.TheimprovedperformanceofDEIFcomparedtotheEIFandUIFisconfirmedbothintermsof improvedestimationaccuracyandintermsofimprovedspeedofcomputation. Finally,onecannotedistributed filteringwiththeuseofthedistributedparticle filter.Thisconsistsofmultipleparticle filtersrunningatdistributedcomputation units,whileaconsensuscriterionisusedtofusethelocalstateestimates.
InChap. 4,linearizingcontroland fi lteringfornonlineardynamicsin financial systemsisexplained.A flatness-basedadaptivefuzzycontrolisappliedtothe problemofstabilizationofthedynamicsofachaotic financesystem,describing interactionbetweentheinterestrate,theinvestmentdemandandthepriceexponent. First,itisproventhatthesystemisdifferentially flat.Thisimpliesthatallitsstate variablesanditscontrolinputscanbeexpressedasdifferentialfunctionsofa speci ficstatevariable,whichisaso-called flatoutput.Italsoimpliesthatthe flat outputanditsderivativesaredifferentiallyindependentwhichmeansthattheyare notconnectedtoeachotherthroughanordinarydifferentialequation.Byproving thatthe financesystemisdifferentially flatandbyapplyingdifferential flatness diffeomorphisms,itstransformationtothelinearcanonical(Brunovsky)isperformed.Forthelatterdescriptionofthesystem,thedesignofastabilizingstate feedbackcontrollerbecomespossible.A fi rstprobleminthedesignofsucha controlleristhatthedynamicmodelofthe fi nancesystemisunknown,andthus,it hastobeidentifiedwiththeuseneurofuzzyapproximators.Theestimateddynamics providedbytheapproximatorsisusedinthecomputationofthecontrolinput,thus establishinganindirectadaptivecontrolscheme.Thelearningrateoftheapproximatorsischosenfromtherequirementthesystem’sLyapunovfunctiontohave alwaysanegative first-orderderivative.Anotherproblemthathastobedealtwithis thatthecontrolloopisimplementedonlywiththeuseofoutputfeedback.To estimatethenon-measurablestatevectorelementsofthe financesystem,astate observerisimplementedinthecontrolloop.Thecomputationofthefeedback controlsignalrequiresthesolutionoftwoalgebraicRiccatiequationsateach iterationofthecontrolalgorithm.Lyapunovstabilityanalysisdemonstrates firstthat anH-in finitytrackingperformancecriterionissatis fi ed.Thissignifieselevated robustnessagainstmodellingerrorsandexternalperturbations.Moreover,global asymptoticstabilityisprovenforthecontrolloop.
InChap. 5,nonlinearoptimalcontroland filteringfor fi nancialsystemsis explained.Anewnonlinearoptimalcontrolapproachisproposedforthestabilizationofthedynamicsofachaotic financemodel.Thedynamicmodelofthe financialsystem,whichexpressesinteractionbetweentheinterestrate,theinvestmentdemand,thepriceexponentandtheprofitmargin,undergoesapproximate linearizationroundlocaloperatingpoints.Theselocalequilibriaaredefinedateach iterationofthecontrolalgorithmandconsistofthepresentvalueofthesystem’s statevectorandthelastvalueofthecontrolinputsvectorthatwasexertedonit.The approximatelinearizationmakesuseofTaylorseriesexpansionandofthecomputationoftheassociatedJacobianmatrices.Thetruncationofhigherordertermsin
theTaylorseriesexpansionisconsideredtobeamodellingerrorthatiscompensatedbytherobustnessofthecontrolloop.Asthecontrolalgorithmruns,the temporaryequilibriumisshiftedtowardsthereferencetrajectoryand finallyconvergestoit.ThecontrolmethodneedstocomputeanH-in finityfeedbackcontrol lawateachiterationandrequirestherepetitivesolutionofanalgebraicRiccati equation.ThroughLyapunovstabilityanalysis,itisshownthatanH-in fi nity trackingperformancecriterionholdsforthecontrolloop.Thisimplieselevated robustnessagainstmodelapproximationsandexternalperturbations.Moreover, undermoderateconditions,theglobalasymptoticstabilityofthe financesystem's feedbackcontrolisproven.
InChap. 6,aKalmanFilteringapproachforthedetectionofoptionmispricingin theBlack–ScholesPDEisintroduced.Financialderivativesandoptionpricing modelsareusuallydescribedwiththeuseofstochasticdifferentialequationsand diffusion-typepartialdifferentialequations(e.g.Black–Scholesmodels). Consideringthelattercaseinthischapter,anew filteringmethodfordistributed parametersystemsisdevelopedforestimatingoptionpricevariationswithoutthe knowledgeofinitialconditions.Theproposed filteringmethodistheso-called Derivative-freenonlinearKalmanFilterandisbasedonadecompositionofthe nonlinearpartialdifferentialequationmodelintoasetofordinarydifferential equationswithrespecttotime.Next,eachoneofthelocalmodelsassociatedwith theordinarydifferentialequationsistransformedintoamodelofthelinear canonical(Brunovsky)formthroughachangeofcoordinates(diffeomorphism) whichisbasedondifferential flatnesstheory.Thistransformationprovidesan extendedmodelofthenonlineardynamicsoftheoptionpricingmodelforwhich stateestimationispossiblebyapplyingthestandardKalmanFilterrecursion.Based ontheprovidedstateestimate,validationoftheBlack–ScholesPDEmodelcanbe performedandtheexistenceofinconsistentparametersintheBlack–ScholesPDE modelcanbeconcluded.
InChap. 7,aKalmanFilteringapproachtothedetectionofoptionmispricingin electricpowermarketsisanalysed.Asmentionedinthepreviouschapter,option pricingmodelsareusuallydescribedwiththeuseofstochasticdifferentialequations anddiffusion-typepartialdifferentialequations(e.g.Black–Scholesmodels).In caseofelectricpowermarketsthesemodelsarecomplementedwithintegralterms whichdescribetheeffectsofjumpsandchangesinthediffusionprocessandwhich areassociatedwithvariationsintheproductionrates,intheconditionofthe transmissionanddistributionsystem,inthepay-offcapability,etc.Consideringthe lattercase,thatisapartialintegrodifferentialequationfortheoption’sprice,anew filteringmethod,isdevelopedforestimatingoptionpricevariationswithout knowledgeofinitialconditions.Theproposed filteringmethodistheso-called Derivative-freenonlinearKalmanFilterandisbasedonatransformationofthe initialoptionpricedynamicsintoastate-spacemodelofthelinearcanonicalform. Thetransformationisshowntobebasedondifferential flatnesstheoryand fi nally providesamodeloftheoptionpricedynamicsforwhichstateestimationispossible byapplyingthestandardKalmanFilterrecursion.Basedontheprovidedstate estimate,validationoftheBlack–Scholespartialintegrodifferentialequationcanbe
performedandtheexistenceofinconsistentparametersintheelectricitymarket pricingmodelcanbeconcluded.
InChap. 8,corporations’ defaultprobabilityforecastingusingthe Derivative-freenonlinearKalmanFilterisexplained.Thischapterproposesa systematicmethodforforecastingdefaultprobabilitiesfor financial firmswith particularinterestinelectricpowercorporations.Accordingtothecreditrisktheory,acompany’sclosenesstodefaultisdeterminedbythedistanceofitsassets’ valuefromitsdebts.Theassets’ valuedependsprimarilyonthecompany’smarket (option)valuethroughacomplexnonlinearrelation.Byforecastingwithaccuracy theenterprise’soptionvalue,itbecomesalsopossibletoestimatethefuturevalue oftheenterprise’sassetsandtheassociatedprobabilityofdefault.Thischapter proposesasystematicmethodforforecastingtheproximitytodefaultforcompanies (option/assetvalueforecastingmethods)usingthenewnonlinearKalmanFiltering methodunderthenameDerivative-freenonlinearKalmanFilter.The fi rm’soption valueisconsideredtobedescribedbytheBlack–Scholesnonlinearpartialdifferentialequation.Usingdifferential flatnesstheory,thepartialdifferentialequationis transformedintoanequivalentstate-spacemodelintheso-calledcanonicalform. UsingthelattermodelandbyredesigningtheDerivative-freenonlinearKalman Filterasam-stepaheadpredictor,estimatesareobtainedofthecompany’sfuture optionvalues.Byforecastingthecompany’smarket(option)values,itbecomes finallypossibletoforecasttheassociatedassetvalueandvolatilityandalsoto estimatethecompany'sfuturedefaultrisk.
InChap. 9,validationof fi nancialoptionsmodelsusingneuralnetworkswith invariancetoFouriertransformisexplained.Itisknownthatnumericalsolution oftheBlack–ScholesPDEenablestocomputewithprecisionthevaluesof financial options,withina finite-timehorizon.Itisalsoknownthatsolutionstotheoption pricingproblemcanbeobtainedinclosedformusingFouriermethods,suchasthe FastFourierTransform,theexpansioninFourier-cosineseriesortheexpansionin Fourier–Hermiteseries.Inthischapter,modellingof financialoptions’ dynamicsis performed,usinganeuralnetworkwith2DGauss-Hermitebasisfunctionsthat remaininvarianttoFouriertransform.KnowingthattheGauss-Hermitebasis functionssatisfytheorthogonalitypropertyandremainunchangedunderthe Fouriertransform,subjectedonlytoachangeofscale,onehasthattheconsidered neuralnetworkprovidesthespectralanalysisoftheoptions’ dynamicsmodel. Actually,thesquaresoftheweightsoftheoutputlayeroftheneuralnetworkdenote thespectralcomponentsforthemonitoredoptions’ dynamics.Byobserving changesintheamplitudeoftheaforementionedspectralcomponents,onecanhave alsoanindicationaboutdeviationsfromnominalvalues,forparametersthataffect theoptions’ dynamics,suchasinterestrate,dividendpaymentandvolatility. Moreover,sincespecifi cparametricchangesareassociatedwithamplitudechanges ofspecifi cspectralcomponentsoftheoptions’ model,isolationofthedistorted parameterscanbealsoperformed.
InChap. 10,statisticalvalidationof financialforecastingtoolswithgeneralized likelihoodratioapproachesisanalysed.Thelocalstatisticalapproachforfault detectionandisolationisappliedtotheproblemofvalidationofafuzzymodel
whichcanbeusedinforecasting.Themethoddetectstheinconsistenciesbetweena fuzzyrulebaseandthemodelledsystem.Itcanalsoidentifywhicharethefaulty parametersofthefuzzymodel.TheFisherinformationmatrixexplainsthe detectabilityofchangesintheparametersofthefuzzymodel.Simulationtests illustratethemethod'scredibility.Asacasestudy,statisticalvalidationofaneurofuzzymodelofchaotictimeseriesisconsidered.
InChap. 11,distributedKalmanFilteringforriskassessmentininterconnected financialmarketsisanalysed.In financialdecision-making,suchasinthetradingof options,itisimportanttoregularlyvalidatetheaccuracyandreliabilityofdecision supporttools.Inthiscontext,thischapterintroducesadistributedschemeforthe validationofoptionpriceforecastingmodelsenablingearlydiagnosisofoptions mispricing.Itisconsideredthat N independentagentsmonitorandforecastthe variationofoptionpricesthroughlocallyparameterizedKalmanFilters.Itisalso assumedthat finaldecisionabouttheoptions’ priceistakenthroughafuzzyconsensusscheme,thatistheindividualforecastsofthedistributedagents,providedby localKalmanFiltersarefusedwithafuzzyweightingprocess.Thus,forecastingis finallyperformedbyafuzzyKalmanFilter.Itislikely,though,thatsomeofthe distributedmodelsareimproperlyparametrizedandfailtodescribeaccuratelythe realdynamicsoftheoption’smarket.Tothisend,astatisticalmethodisdeveloped capableof(i)detectingiftheestimationabouttheoptions’spricethatisprovided bythemulti-agentsystemissufficientlypreciseornotand(ii)isolatingthe ithagent thatmakesuseofanimproperlyparameterizedmodel.Thischapterprovidesone ofthefewapproachesfortestingtheaccuracyofdistributedKalmanFiltersfor financialdecision-makingandtheonlyonethatpermitstodetectparametric changesthatareofmagnitudeoflessthan1%ofthenominalvalueofthemonitored financialsystem.
InChap. 12,stabilizationof financialsystemsdynamicsthroughfeedback controloftheBlack–ScholesPDEisanalysed.Theobjectiveofthischapterwasto developaboundarycontrolmethodfortheBlack–ScholesPDEwhichdescribes optiondynamics.ItisshownthattheprocedurefornumericalsolutionofBlack–ScholesPDEresultsinasetofnonlinearordinarydifferentialequations(ODEs)and anassociatedstateequationsmodel.Forthelocalsubsystems,intowhichaBlack–ScholesPDEisdecomposed,itbecomespossibletoapplyboundary-basedfeedbackcontrol.Thecontrollerdesignproceedsbyshowingthatthestate-spacemodel oftheBlack–ScholesPDEstandsforadifferentially flatsystem.Next,foreach subsystemwhichisrelatedtoanonlinearODE,avirtualcontrolinputiscomputed, whichcaninvertthesubsystem’sdynamicsandcaneliminatethesubsystem’s trackingerror.Fromthelastrowofthestate-spacedescription,thecontrolinput (boundarycondition)thatisactuallyappliedtotheBlack–ScholesPDEisfound. ThiscontrolinputcontainsrecursivelyallvirtualcontrolinputswhichwerecomputedfortheindividualODEsubsystemsassociatedwiththepreviousrowsofthe state-spaceequation.Thus,bytracingtherowsofthestate-spacemodelbackwards, ateachiterationofthecontrolalgorithm,onecan finallyobtainthecontrolinput thatshouldbeappliedtotheBlack–ScholesPDEsoastoassurethatallitsstate variableswillconvergetothedesirablesetpoints.
InChap. 13,stabilizationofthemulti-assetBlack–ScholesPDEusingdifferential flatnesstheoryisanalysed.Amethodforfeedbackcontrolofthemulti-asset Black–ScholesPDEisdeveloped.Byapplyingoncemoresemi-discretizationanda finitedifferencesscheme,themulti-assetBlack–ScholesPDEistransformedintoa state-spacemodelconsistingofordinarynonlineardifferentialequations.Forthis setofdifferentialequations,itisshownthatdifferential flatnesspropertieshold. Thisenablestosolvetheassociatedcontrolproblemandtosucceedstabilization oftheoptions’ dynamics.Itisshownthatthepreviousprocedureresultsinasetof nonlinearordinarydifferentialequations(ODEs)andtoanassociatedstateequationsmodel.Forthelocalsubsystems,intowhichamulti-assetBlack–ScholesPDE isdecomposed,itbecomespossibletoapplyboundary-basedfeedbackcontrol.The controllerdesignproceedsbyshowingthatthestate-spacemodelofthemulti-asset Black–ScholesPDEstandsforadifferentially flatsystem.Next,foreachsubsystem whichisrelatedtoanonlinearODE,avirtualcontrolinputiscomputed,whichcan invertthesubsystem'sdynamicsandcaneliminatethesubsystem'strackingerror. Fromthelastrowofthestate-spacedescription,thecontrolinput(boundarycondition)thatisactuallyappliedtothemulti-assetBlack–ScholesPDEsystemis found.Thiscontrolinputcontainsrecursivelyallvirtualcontrolinputswhichwere computedfortheindividualODEsubsystemsassociatedwiththepreviousrows ofthestate-spaceequation.Thus,bytracingtherowsofthestate-spacemodel backwards,ateachiterationofthecontrolalgorithm,onecan finallyobtainthe controlinputthatshouldbeappliedtothemulti-assetBlack–ScholesPDEsoasto assurethatallitsstatevariableswillconvergetothedesirablesetpoints.
InChap. 14,stabilizationofcommoditiespricingPDEusingdifferential flatness theoryisexplained.Pricingofcommodities(e.g.oil,carbon,miningproducts, electricpowerandagriculturalcrops)isvitalforthemajorityoftransactionstaking placein financialmarkets.Amethodforfeedbackcontrolofcommoditiespricing dynamicsisdeveloped.ThePDEmodelofthecommoditiespricedynamicsis showntobeequivalenttoamulti-assetBlack–ScholesPDE.Actually,itisa diffusionprocessevolvingina2Dassetsspace,wherethe fi rstassetisthecommodity'sspotpriceandthesecondassetistheconvenienceyield.Asintheprevious chapters,byapplyingsemi-discretizationanda finitedifferencesscheme,this multi-assetPDEistransformedintoastate-spacemodelconsistingofordinary nonlineardifferentialequations.Forthelocalsubsystems,intowhichthecommoditiesPDEisdecomposed,itbecomespossibletoapplyboundary-basedfeedbackcontrol.Thecontrollerdesignproceedsbyshowingthatthestate-spacemodel ofthecommoditiesPDEstandsforadifferentially flatsystem.Next,foreach subsystemwhichisrelatedtoanonlinearODE,avirtualcontrolinputiscomputed, whichcaninvertthesubsystem’sdynamicsandcaneliminatethesubsystem’s trackingerror.Fromthelastrowofthestate-spacedescription,thecontrolinput (boundarycondition)thatisactuallyappliedtothemulti-factorcommodities’ PDE systemisfound.Thiscontrolinputcontainsrecursivelyallvirtualcontrolinputs whichwerecomputedfortheindividualODEsubsystemsassociatedwiththe previousrowsofthestate-spaceequation.Thus,bytracingtherowsofthe state-spacemodelbackwards,ateachiterationofthecontrolalgorithm,onecan
finallyobtainthecontrolinputthatshouldbeappliedtothecommoditiesPDE systemsoastoassurethatallitsstatevariableswillconvergetothedesirable setpoints.Bydemonstratingthefeasibilityofsuchacontrolmethoditisalsoproven thatthroughselectedpurchaseandsalesduringthetradingprocedure,theprice ofthenegotiatedcommoditiescanbemadetoconvergeandstabilizeatspeci fic referencevalues.
InChap. 15,stabilizationofmortgagepricedynamicsusingdifferential flatness theoryisanalysed.Pricingofmortgages(loansforthepurchaseofresidences,land orfarms)isvitalforthemajorityoftransactionstakingplacein financialmarkets. Inthischapter,amethodforstabilizationofmortgagepricedynamicsisdeveloped. ItisconsideredthatmortgagepricesfollowaPDEmodelwhichisequivalenttoa multi-assetBlack–ScholesPDE.Actually,itisadiffusionprocessevolvingina2D assetsspace,wherethe firstassetistheresidencepriceandthesecondassetisthe interestrate.Byapplyingsemi-discretizationanda finitedifferencesscheme,this multi-assetPDEistransformedintoastate-spacemodelconsistingofordinary nonlineardifferentialequations.Forthelocalsubsystems,intowhichthemortgage PDEisdecomposed,itbecomespossibletoapplyboundary-basedfeedbackcontrol.Thecontrollerdesignproceedsbyshowingthatthestate-spacemodelofthe mortgagepricePDEstandsforadifferentially flatsystem.Next,foreachsubsystem whichisrelatedtoanonlinearODE,avirtualcontrolinputiscomputed,whichcan invertthesubsystem'sdynamicsandcaneliminatethesubsystem’strackingerror. Fromthelastrowofthestate-spacedescription,thecontrolinput(boundarycondition)thatisactuallyappliedtothemulti-factormortgagepricePDEsystemis found.Thiscontrolinputcontainsrecursivelyallvirtualcontrolinputswhichwere computedfortheindividualODEsubsystemsassociatedwiththepreviousrows ofthestate-spaceequation.Thus,bytracingtherowsofthestate-spacemodel backwards,ateachiterationofthecontrolalgorithm,onecan finallyobtainthe controlinputthatshouldbeappliedtothemortgagepricePDEsystemsoasto assurethatallitsstatevariableswillconvergetothedesirablesetpoints.By showingthefeasibilityofsuchacontrolmethod,itisalsoproventhatthrough selectedmodifi cationofthePDEboundaryconditions,thepriceofthemortgage canbemadetoconvergeandstabilizeatspeci ficreferencevalues.
Themainpurposeofthisbookwastodisseminatenew fi ndingsusefulfor academicteachingandresearchintheareaof financialengineeringandtodevelop systematicmethodsformanagementandriskminimizationin financialsystems. Methodsforsolvingcontrolandestimationproblemsin fi nancialsystemsbecome progressivelypartofthecurriculumofseveralacademicdepartmentsatundergraduatelevel.Thisisbecausethereisaneedtoacquaintfutureengineerswith technologiesthatenablethefunctioningof financialsystemsaccordingtothe desirablespeci fications,evenunderuncertaintyandpartialinformationabouttheir dynamicmodel.Thepresentbookcontainsteachingmaterialwhichcanbeusedfor independentcourseson financialengineering.Thisbookcanalsoserveperfectly theneedsofpostgraduatecourseson financialengineeringwheremoreemphasis canbegiventoadvancedcomputationalandthemathematicaltechniquesforthe profitableandrisk-freemanagementof financialsystems.Thetitleofthecoursecan
bethesameasthetitleofthebook,i.e.state-spaceapproachestomodellingand controlin financialengineering:systemstheoryandmachinelearningmethods. Startingfromtheanalysisofdynamicalsystemstheoryandofestablished approachesforcontrolandestimationinnonlineardynamicalsystems,themonographmovesprogressivelytothesolutionofkeyproblemsmetin financialengineering,suchas(i)nonlinearcontroland filteringfor financialsystemsexhibiting complexandchaoticdynamics,(ii)controlandestimationforthePDEdynamicsof financialsystems,and(iii)statisticalvalidationofdecisionsupporttoolsusedin financialengineering.Throughthebalancedinteractionbetweenthetheoreticaland theapplicationpart,studentscanassimilatethenewknowledgeandcanbecome efficientincontrolandestimationof financialsystemsandinmethodsforthe optimizedmanagementofcapitalsandassets.
However,thisbookandisnotonlyaddressedtotheacademiccommunitybut alsotargetspeopleworkinginpracticalproblemsandapplicationsof financial engineering.Thereiscontinuousdemandfordevelopingelaboratedsoftwaretools thatwillenableoptimaldecision-makingabout financialsystems.Tothisend,there isaneedtoeliminateheuristicsandintuition-basedapproachesin financialengineeringandtoestablishmethodsthatassurestabilizationandconvergenceof financialsystemstodesirableperformanceindexes.Themonograph'scontribution tothisdirectionisclear.
Athens,GreeceGerasimosG.RigatosPh.D. ElectricalandComputerEngineer March2017
Acknowledgements
Theauthorofthismonographwouldliketothankresearchersintheareaof financialsystems,aswellasintheareaofdynamicalsystemsmodellingandcontrol forcontributingtothedevelopmentandcompletionofthisresearchwork,through reviews,commentsandmeaningfulremarks.
1.6.2Saddle-NodeBifurcationsofFixedPoints
2.5.1TransformationofNonlinearSystemsintoa
3.2LinearStateObservers
3.3TheContinuous-TimeKalmanFilterforLinearModels
3.4TheDiscrete-TimeKalmanFilterforLinearSystems
3.5TheExtendedKalmanFilterforNonlinearSystems
3.7.1TheParticleApproximationofProbability
3.7.2ThePredictionStage
3.7.3TheCorrectionStage
3.7.5ApproachestotheImplementationofResampling
3.8TheDerivative-FreeNonlinearKalmanFilter
3.8.1Conditionsforsolvingtheestimationproblemin single-inputnonlinearsystems
3.8.2StateEstimationwiththeDerivative-FreeNonlinear KalmanFilter
3.8.3Derivative-FreeKalmanFilteringformultivariable NonlinearSystems .............................
3.9DistributedExtendedKalmanFiltering
3.9.1CalculationofLocalExtendedKalmanFilter Estimations
3.9.2ExtendedInformationFilteringforStateEstimates
3.10DistributedSigma-PointKalmanFiltering .................
3.10.1CalculationofLocalUnscentedKalmanFilter Estimations
3.10.2UnscentedInformationFilteringforStateEstimates
3.12.2FusingEstimationsfromLocalDistributedFilters
3.12.3CalculationoftheAggregateStateEstimation
4.3.1ConditionsforApplyingtheDifferentialFlatness
5.2.1DynamicModeloftheChaoticFinanceSystem
5.2.2State-SpaceModeloftheChaoticFinancialSystem
5.2.3ChaoticDynamicsoftheFinanceSystem
5.3DesignofanH-InfinityNonlinearFeedbackController .......
5.3.1ApproximateLinearizationoftheChaoticFinance System
5.3.2EquivalentLinearizedDynamicsoftheChaotic FinanceSystem
5.3.3TheNonlinearH-InfinityControl
5.3.4ComputationoftheFeedbackControlGains
5.3.5TheRoleofRiccatiEquationCoefficientsin H1 ControlRobustness
5.4LyapunovStabilityAnalysis
5.4.1StabilityProof
5.4.2RobustStateEstimationwiththeUseofthe H
6.2OptionPricingModelingwiththeUseoftheBlack–Scholes
6.2.1OptionPricingModelingwiththeUseofStochastic DifferentialEquations
6.2.2TheBlack–ScholesPDE
6.2.3SolutionoftheBlack–ScholesPDE
6.2.4SensitivitiesoftheEuropeanCallOption
6.2.5NonlinearitiesintheBlack–ScholesPDE
6.2.6DerivativePricing
6.3EstimationofNonlinearDiffusionDynamics
6.3.1FilteringinDistributedParameterSystems
6.4StateEstimationfortheBlack–ScholesPDE
6.4.1ModelinginCanonicalFormoftheNonlinear Black–ScholesEquation ........................
6.4.2StateEstimationwiththeDerivative-FreeNonlinear KalmanFilter ................................
6.4.3ConsistencyCheckingoftheOptionPricing Model ......................................
6.5SimulationTests .....................................
6.5.1EstimationwiththeUseofanAccurate Black–ScholesModel ..........................
6.5.2DetectionofMispricingintheBlack–Scholes Model
7KalmanFilteringApproachtotheDetectionofOption MispricinginElaboratedPDEFinanceModels
7.2OptionPricingintheEnergyMarket
7.2.1EnergyMarketandSwingOptions
7.2.2EnergyOptionsPricingModels
7.3ValidationoftheEnergyOptionsPricingModel
7.3.1StateEstimationwiththeDerivative-FreeNonlinear KalmanFilter
7.3.2ConsistencyCheckingoftheOptionPricing
8.2.1TheMerton-KMVCredit-RiskModel
8.3EstimationoftheMarketValueoftheCompany UsingtheBlack–ScholesPDE
8.3.1State-SpaceDescriptionoftheBlack–Scholes Equation
8.4ForecastingDefaultwiththeDerivative-FreeNonlinear KalmanFilter
8.4.1StateEstimationwiththeDerivative-FreeNonlinear KalmanFilter
8.4.2TheDerivative-FreeNonlinearKalmanFilteras Extrapolator
8.4.3ForecastingoftheMarketValueUsingthe Derivative-FreeNonlinearKalmanFilter
8.4.4AssessmentoftheAccuracyofForecasting withtheUseofStatisticalCriteria
9.3.1GeneralizedFourierSeries .......................
9.3.2TheGauss–HermiteSeriesExpansion ..............
9.3.3NeuralNetworksUsing2DHermiteActivation Functions ....................................
9.4SignalsPowerSpectrumandtheFourierTransform
9.4.1Parseval'sTheorem
9.4.2PowerSpectrumoftheSignalUsingthe Gauss–HermiteExpansion
10StatisticalValidationofFinancialForecastingTools withGeneralizedLikelihoodRatioApproaches
10.2.2DeterminationoftheNumberandType ofFuzzyRules ...............................
10.2.3StagesofFuzzyModelling ......................
10.2.4FuzzyModelValidationfortheAvoidanceof Overtraining .................................
10.3FuzzyModelValidationwiththeLocalStatistical Approach
10.3.1TheExactModel
10.3.4IsolationofParametricChangeswiththeMin-Max Test
10.3.5ModelValidationReducestheNeedforModel
10.5.1FuzzyRuleBaseinInputSpacePartitioning
10.5.2FuzzyModellingwiththeInputDimension
11DistributedValidationofOptionPriceForecastingToolsUsing aStatisticalFaultDiagnosisApproach
–ScholesPDE
11.2.1State-SpaceDescriptionoftheBlack–ScholesPDE
11.4ConsistencyoftheKalmanFilter ........................
11.5EquivalenceBetweenKalmanFiltersandRegressor Models ............................................
11.6ChangeDetectionoftheFuzzyKalmanFilterUsingtheLocal StatisticalApproach ..................................
11.6.1TheGlobal v2 TestforChangeDetection ...........
11.6.2IsolationofInconsistentKalmanFilterParameters withtheSensitivityTest ........................
11.6.3IsolationofInconsistentKalmanFilterParameters withtheMin–MaxTest
–Scholes
12StabilizationofFinancialSystemsDynamicsThroughFeedback ControloftheBlack-ScholesPDE
12.2TransformationoftheBlack-ScholesPDEintoNonlinear ODEs .............................................
12.2.1DecompositionofthePDEModelintoEquivalent
12.2.2ModelinginState-SpaceFormoftheBlack-Scholes PDE
12.4ComputationofaBoundaryConditions-BasedFeedback ControlLaw
13StabilizationoftheMulti-assetBlack–ScholesPDEUsing DifferentialFlatnessTheory
13.2BoundaryControloftheMulti-assetBlack–ScholesPDE
13.3Flatness-BasedControloftheMulti-assetBlack–Scholes PDE
13.4StabilityAnalysisoftheControlLoop
14StabilizationofCommoditiesPricingPDEUsingDifferential FlatnessTheory ..........................................
14.2.1ElaboratedSchemesforTradingElectricPower
14.4Flatness-BasedControloftheMulti-factorCommodities
15.4BoundaryControloftheMulti-factorMortgage
Chapter1 SystemsTheoryandStabilityConcepts
1.1Outline
Thechapteranalyzesthebasicsofsystemstheorywhichcanbeusedinthemodeling offinancialsystems.Financialsystemsmayexhibitcomplexdynamicscharacterized byoscillationsorchaos.Parametricvariationsinthemodelsoffinancialsystems mayalsoaffectandmodifytheirstabilityproperties.Thefollowingpropertiesare analyzed:phasediagram,isoclines,attractors,localstability,bifurcationsoffixed pointsandchaosproperties.
1.2CharacteristicsoftheDynamicsofNonlinearSystems
Mainfeaturescharacterizingthestabilityofnonlineardynamicalsystemsaredefined asfollows[121, 274]:
1. Finiteescapetime:Itisthefinitetimewithinwhichthestate-vectorofthenonlinear systemconvergestoinfinity.
2. Multipleisolatedequilibria:Alinearsystemcanhaveonlyoneequilibriumto whichconvergesthestatevectorofthesysteminsteady-state.Anonlinearsystem canhavemorethanoneisolatedequilibria(fixedpoints).Dependingontheinitial stateofthesystem,insteady-statethestatevectorofthesystemcanconvergetoone oftheseequilibria.
3. Limitcycles:Foralinearsystemtoexhibitoscillationsitmusthaveeigenvalueson theimaginaryaxis.Theamplitudeoftheoscillationsdependsoninitialconditions. Innonlinearsystemsonemayhaveoscillationsofconstantamplitudeandfrequency, whichdonotdependoninitialconditions.Thistypeofoscillationsisknownas limit cycles.
4. Sub-harmonic,harmonicandalmostperiodicoscillations:Astablelinearsystem underperiodicinputproducesanoutputofthesamefrequency.Anonlinearsystem,
©SpringerInternationalPublishingAG2017
G.G.Rigatos, State-SpaceApproachesforModellingandControl inFinancialEngineering,IntelligentSystemsReferenceLibrary125, DOI10.1007/978-3-319-52866-3_1
underperiodicexcitationcangenerateoscillationswithfrequencieswhichareseveral timessmaller(subharmonic)ormultiplesofthefrequencyoftheinput(harmonic). Itmayalsogeneratealmostperiodicoscillationswithfrequencieswhicharenot necessarilymultiplesofabasisfrequency(almostperiodicoscillations).
5. Chaos:Anonlinearsysteminsteady-statecanexhibitabehaviorwhichisnot characterizedasequilibrium,periodicoscillationoralmostperiodicoscillation.This behaviorischaracterizedaschaos.Astimeadvancesthebehaviorofthesystem changesinarandom-likemanner,andthisdependsontheinitialconditions.Although thedynamicsystemisdeterministicitexhibitsrandomnessinthewayitevolvesin time.
6. Multiplemodesofbehavior :Itispossiblethesamedynamicalsystemtoexhibit simultaneouslymorethanoneoftheaforementionedcharacteristics(1)–(5).Thus,a systemwithoutexternalexcitationmayexhibitsimultaneouslymorethanonelimit cycles.Asystemreceivingaperiodicexternalinputmayexhibitharmonicorsubharmonicoscillations,oranevenmorecomplexbehaviorinsteadystatewhichdepends ontheamplitudeandfrequencyoftheexcitation.
1.3ComputationofIsoclines
Anautonomoussecondordersystemisdescribedbytwodifferentialequationsof theform
Themethodoftheisoclinesconsistsofcomputingtheslope(ratio)between f2 and f1 foreverypointofthetrajectoryofthestatevector (x1 , x2 )
Thecase s(x ) = c describesacurveinthe x1 x2 planealongwhichthetrajectories ˙ x1 = f1 (x1 , x2 ) and ˙ x2 = f2 (x1 , x2 ) haveaconstantslope.
Thecurve s(x ) = c isdrawninthe x1 x2 planeandalongthiscurveonealso drawssmalllinearsegmentsoflength c.Thecurve s(x ) = c isknownasisocline. Thedirectionofthesesmalllinearsegmentsisaccordingtothesignoftheratio f2 (x1 , x2 )/f1 (x1 , x2 ).
Example1:
Thefollowingsimplifiednonlineardynamicalsystemisconsidered
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561 So Jesus says (p. 230, Copt.) of “the man who receives and accomplishes the Mystery of the Ineffable One”; “he is a man in the Cosmos, but he will reign with me in my kingdom; he is a man in the Cosmos, but he is a king in the light; he is a man in the Cosmos, but he is not of the Cosmos, and verily I say unto you, that man is I, and I am that man.”
562. p. 246, Copt.
563. See last note and n. 5, p. 147 supra.
564. Hatch, op. cit. p. 302 and note.
565. pp. 236, 237, Copt.
566. Loc. cit. Or they may cover a kind of allegory, as we might say that Agape or Love makes Faith, Hope, and Charity. But I believe it to be more likely that the “12 mysteries” are letters in a word. So in the Μέρος
it is said of the “Dragon of the Outer Darkness,” which is in fact the worst of all the hells described in that book: “And the Dragon of the Outer Darkness hath twelve true (αὐθέντη) names which are in his gates, a name according to each gate of the torturechambers. And these names differ one from the other, but they belong to each of the twelve, so that he who saith one name, saith all the names. And these I will tell you in the Emanation of the Universe”—(p. 323, Copt.). If this be thought too trivial an explanation, Irenaeus tells us that the 18 Aeons remaining after deducting the Decad or Dodecad (as the case may be) from the rest of the Pleroma were, according to the Valentinians, signified by the two first letters of the name of Jesus:
Irenaeus, Βk I. c. 1, § 5, p. 26, Harvey. Equally absurd according to modern ideas are the words of the Epistle of Barnabas (c. X., pp. 23, 24, Hilgenfeld), where after quoting a verse in Genesis about Abraham circumcising 318 of his slaves (cf. Gen. xiv. 14), the
author says “What then is the knowledge (γνῶσις) given therein? Learn that the 18 were first, and then after a pause, he says 300. (In) the 18, I = 10, H = 8, thou hast Jesus (Ἰησοῦν). And because the Cross was meant to have grace in the T, he says also 300. He expresses therefore Jesus by two letters and the Cross by one. He knows who has placed in us the ungrafted gift of teaching. None has learned from me a more genuine word. But I know that ye are worthy.”
567. “The True Word” or the Word of the Place of Truth. The latter expression is constantly used in other parts of the book, and seems to refer to the χώρημα or “receptacle,” that is the heaven, of the Aeon Ἀλήθεια, that is the Decad. Cf. especially the Μέρος
(pp. 377, 378, Copt.), where it is said that certain baptisms and a “spiritual chrism” will lead the souls of the disciples “into the Places of Truth and Goodness, to the Place of the Holy of all Holies, to the Place in which there is neither female, nor male, nor shape in that Place, but there is Light, everlasting, ineffable.”
568. These ἀποτάγματα are set out in detail in the Μέρος
Σωτῆρος (pp. 255 sqq. Copt.), where the disciples are ordered to “preach to the whole world ... renounce (ἀποτασσετε) the whole world and all the matter which is therein, and all its cares and all its sins, and in a word all its conversation (ὁμιλιαι) which is therein, that ye may be worthy of the mysteries of the Light, that ye may be preserved from all the punishments which are in the judgments” and so on. It should be noted that these are only required of the psychics or animal men.
569. No doubt in the Greek original the actual seal was here figured. For examples, see the Bruce Papyrus, passim. The idea is typically Egyptian. As M. Maspero says in his essay on “La Table d’Offrandes,” R.H.R. t. xxxv No. 3 (1897), p. 325: no spell was in the view of the ancient Egyptians efficacious
unless accompanied by a talisman or amulet which acted as a material support to it, as the body to the soul.
570. p. 238, Copt.
571. Hatch, op. cit. p. 296, n. 1, for references.
572. 1 Cor. xv. 29. The practice of “baptizing for the dead,” as the A. V. has it, evidently continued into Tertullian’s time. See Tertull. de Resurrectione Carnis, c. XLVIII. p. 530, Oehler.
573. Döllinger, First Age, p. 327.
574. Hatch, op. cit. p. 307. The Emperor Constantine, who was baptized on his deathbed, was a case in point. The same story was told later about the Cathars or Manichaeans of Languedoc. The motive seems in all these cases to have been the same: as baptism washed away all sin, it was as well to delay it until the recipient could sin no more.
575. Hatch, op. cit. p. 295 and note, for references.
576. p. 236, Copt.
577 See n. 2, p. 166 supra
578. Döllinger, First Age, pp. 234, 235.
579. Ibid. p. 235. Rom. vi. 4; Gal. iii. 27, 29, are quoted in support.
580. Ibid. p. 235. Rom. vii. 22; 1 Cor. vi. 14; Eph. iii. 16 and v. 30 are quoted in support.
581. Hatch, op. cit. p. 342.
582. p. 228, Copt.
583. pp. 230, 231, Copt.
584 The Pistis Sophia proper comes to an end twenty pages later
585. Döllinger, First Age, p. 239. 1 Cor. x. 16 sqq.; Eph. v. 30, quoted in support.
586. Justin Martyr was probably born 114, and martyred 165 A.D. For the passage quoted in text, see his First Apology, c. LXVI., where he mentions among other things that the devils set on the worshippers of Mithras to imitate the Christian Eucharist by celebrating a ceremony with bread and a cup of water.
587. Hatch, op. cit. p. 308. This visible change of the contents of the cup of water to the semblance of blood is described in the Μέρος τευχῶν Σωτῆρος (p. 377, Copt.), and with more detail in the Bruce Papyrus. Cf. p. 183 infra.
588. Μέρος
p. 354, Copt.
589. Whether the author of the Pistis Sophia really intended to describe them may be doubted; but it is to be noted that the sacraments which Jesus is represented as celebrating in the Μέρος τευχῶν Σωτῆρος can hardly be they, although Jesus calls them in one place (p. 374, Copt.), “the mysteries of the light which remit sins, which themselves are appellations and names of light.” These are administered to the twelve disciples without distinction, and it is evident that the author of these books is quite unacquainted with any division into pneumatic and psychic, and knows nothing of the higher mysteries called in the Pistis Sophia proper “the mysteries of the Ineffable One” and “the mysteries of the First Mystery.” We should get over many difficulties if we supposed the two later books to be Marcosian in origin, but in any event they are later than the Pistis Sophia.
590. p. 246, Copt. So in the Manichaean text described in Chapter XIII, Jesus is Himself called “the Tree of Knowledge.”
591 So Irenaeus, Bk I. c. 1, § 11, pp. 58, 54, Harvey:
“For the psychic (animal) men are taught psychic things, they being made safe by works and by mere faith, and not having perfect knowledge. And they say that we of the Church are these people. Wherefore they declare that good deeds are necessary for us: for otherwise we could not be saved. But they decree that they themselves are entirely and in every thing saved, not by works, but because they are pneumatic (spiritual) by nature.”
592. p. 249, Copt.
593. p. 250, Copt. It is to be observed that these “cleansing mysteries” will only admit their recipients to the light of the Kingdom of Jesus—not to that of the First Mystery or of the Ineffable One.
594. As did perhaps the Manichaeans afterwards. See J.R.A.S. for January, 1913, and Chap. XIII infra
595. So Charles Kingsley in Hypatia. Gibbon, Decline and Fall, vol. IV c. 60, n. 15, quotes a statement of Rufinus that there were nearly as many monks living in the deserts as citizens in the towns.
596. Mallet, Le Culte de Neit à Saïs, p. 200, points out that the God Nu described in the 18th Chapter of the Book of the Dead is “the infinite abyss, the Βυθός, the πατὴρ ἄγνωστος of the Gnostics.” So Maspero in Rev. Critique, 30 Sept. 1909, p. 13, who declares that the author of the Pistis Sophia was influenced directly or indirectly by Osirian beliefs.
597 Moret, Le verbe créateur et révélateur, p. 286, for references.
598. Maspero, Ét. Égyptol. t. II. p. 187: “L’ogdoade est une conception hermopolitaine qui s’est répandue plus tard sur toute l’Égypte à côté de l’ennéade d’Heliopolis. Les théologiens d’Hermopolis avaient adopté le concept de la neuvaine, seulement ils avaient amoindri les huit dieux qui formaient le corps du dieu principal. Ils les avaient reduits à n’être plus que des êtres presque abstraits nommés d’après la fonction qu’on leur assignait, en agissant en masse sur l’ordre et d’après l’impulsion du dieu chef. Leur ennéade se composait donc d’un dieu tout-puissant et d’une ogdoade.”
599. “Son origine (l’ogdoade hermopolitaine subordonné à un corps monade) est fort ancienne: on trouve quelques-unes des divinités qui la composent mentionnées déjà dans les textes des Pyramides.” Maspero, op. cit. t. II. p. 383. As he says later the actual number of gods in the Ennead or Ogdoad was a matter of indifference to the ancient Egyptian: “les dieux comptaient toujours pour neuf, quand même ils étaient treize ou quinze,” ibid. p. 387. Cf. Amélineau, Gnost. Ég. pp. 294, 295.
600. See n. 5, p. 175 supra, and Maspero, “Hypogées Royaux,” Ét. Égyptol. II. p. 130, n. 2.
601. See n. 2, p. 153 supra.
602. Maspero, “Hypogées Royaux,” t. II. p. 121.
603. Maspero, Rev. Crit. 30 Sept. 1909, p. 13.
604. Maspero, “Hypogées Royaux,” t. II. p. 118. Cf. Pistis Sophia, p. 84, Copt. and elsewhere.
605. Maspero, “La Table d’Offrandes,” R.H.R. t. XXXV (1897) p. 325. As has been said, in the Ascensio Isaiae, anyone
passing from one heaven to another has to give a password, but not to exhibit a seal.
606. Amélineau, Gnost. Ég. p. 196; Schmidt, Koptisch-Gnostische Schriften, Bd I. p. xiii.
607 It is so used in the Excerpta Theodoti, and in the Papyrus Bruce. See p. 190, infra.
608. Jean Reville, Le Quatrième Évangile, Paris, 1901, p. 321. Mgr Duchesne, Early Christian Church, pp. 102, 192, says in effect that St John’s Gospel appeared after the Apostle’s death and was not accepted without opposition. He thinks Tatian and Irenaeus the first writers who quoted from it with acknowledgement of its authorship. If we put the date of Tatian’s birth at 120 (see Dict. Christian Biog. s.h.n.) and allow a sufficient period for the initiation into heathen mysteries which he mentions, for his conversion and for his becoming a teacher, we do not get a much earlier date than 170 for his acceptance of the Fourth Gospel. Irenaeus was, of course, later in date than Tatian.
609. Tertullian, Adv. Valentinianos, c. 2.
610. Amélineau, Gnost. Ég. p. 180.
611. Tertullian, de Carne Christi, c. 20.
612 E.g. p. 47, Copt. Cf. also ibid. pp. 147, 170, 176.
613. Tertullian, adv. Val. c. v.
614. Op. cit. c. 9.
615. Op. cit. c. 18.
616. Op. cit. c. 20.
617. Op. cit. c. 25.
618 Tertullian, de Carne Christi, c. 9. Irenaeus, Bk II. c. 7, § 1, p. 270, Harvey, seems to have known both of Barbelo and of the Virgin of Light, since he speaks of corpora sursum ... spiritalia et lucida, “spiritual and translucent bodies on high” casting a shadow below in quam Matrem suam descendisse dicunt “into which they allege their Mother descended.”
619. ⲞⲨ ⲘⲈⲢⲞⲤ ⲎⲦⲈ Ⲏ ⲦⲈⲨⲬⲞⲤ
, or (in Greek)
620. “This I say to you in paradigm, and likeness and similitude, but not in truth of shape, nor have I revealed the word in truth,” p. 253, Copt. So in the next page (p. 254, Copt.), Jesus says of the perfect initiate that “He also has found the words of the Mysteries, those which I have written to you according to similitude—the same are the members of the Ineffable One.” From His mention of “writing,” one would imagine that the reference here is to documents such as the Bruce Papyrus which gives the pictures of “seals” together with cryptographically written words.
621. p. 357, Copt. This opening sentence could not have been written by one of the Valentinians of Hadrian’s time, who, as has been said above, “did not choose to call Jesus, Lord,” Irenaeus, Bk I. c. 1, I. p. 12, Harvey
622. In the address of Jesus beginning “O my Father, Father of every Fatherhood, boundless light” with which this part of the Μ. τ. σ. opens, we can, with a little good will, identify nearly every word of the “galimatias” which at first sight seems mere gibberish. Thus, the whole invocation reads: αεηιουω, ϊαω,
The seven vowels to which many mystical interpretations have been assigned, and which have even been taken for a primitive system of musical notation (C. E. Ruelle, “Le Chant des Sept Voyelles
Grecques,” Rev des Ét. Grecques, Paris, 1889, t. II. p. 43, and pp. 393-395), probably express the sound to Greek ears of the Jewish pronunciation of Yahweh or Jehovah. The word Iao we have before met with many times both as a name of Dionysos and otherwise, and is here written anagrammatically from the difficulty which the Greeks found in dealing with Semitic languages written the reverse way to their own. The word ψινωθερ which follows and is also written as an anagram is evidently an attempt to transcribe in Greek letters the Egyptian words P, Shai, neter (P = Def. article, Shai = the Egyptian God of Fate whose name Revillout, Rev. Égyptol. Paris, 1892, pp. 29-38, thinks means “The Highest,” and neter or nuter the determinative for “god”), the whole reading “Most High God.” The words ζαγουρη παγουρη (better, πατουρη) are from the Hebrew roots רטפ רגס and seem to be the “he that openeth and no man shutteth; and shutteth and no man openeth” of Rev. iii. 7. Νεθμομαωθ, which is often found in the Magic Papyri, is reminiscent of the Egyptian neb maat “Lord of Truth,” the following νεψιομαωθ being probably a variant by a scribe who was uncertain of the orthography. Μαραχαχθα I can make nothing of, although as the phrase νεφθομαωθ μαραχαχθα appears in the Magic Papyrus of Leyden generally called W (Leemans, Papyri Graeci, etc. t. II. p. 154) in a spell there said to be written by “Thphe the Hierogrammateus” for “Ochus the king,” it is evidently intended for Egyptian. In the same spell appear the words θαρνμαχαχ ζοροκοθορα and θωβαρραβαυ which are evidently the same as those in the Μ. τ. σ., and of which I will only say that, while Mr King supposes ζοροκοθορα to mean “lightgatherer,” θωβαρραβαυ is in the leaden tabula devotionis of Carthage (Molinier, “Imprecation gravée sur plomb,” Mem. de la Soc. Nat. des Antiquaires de France, série VI. t. VIII. Paris, 1897, pp. 212-216) described as τον θεὸν του τῆς παλινγενεσιας “the god of rebirth.” The concluding words are of course merely “Yahweh of Hosts.”
623 The description of the moon-chariot drawn by two white oxen is found in Claudian’s Proserpine. According to Cumont (Textes et Monuments relatifs aux Mystères de Mithra, t. I. p. 126 and note) it was not until Hadrian’s time that this conception, which seems to have been Persian in origin, became fixed in the West.
624. This “Middle Way” has nothing to do with the τόπος or “place” of the middle, where are set in the Pistis Sophia proper the powers who preside over incarnation. It is below the visible sphere (p. 364, Copt.) and is met with in Rabbinic lore. See Kohler, op. cit. p. 587.
625. This division of the Twelve Aeons into two halves seems at first sight inconsistent with the description in the Pistis Sophia proper which always speaks of them as Twelve. Yet it is plain that the author of the Pistis Sophia knew the legend here given, as he makes John the Divine speak (p. 12, Copt.) of “the rulers who belong to the Aeon of Jabraoth” and had made peace with the mysteries of the light. These “rulers who repented” are again mentioned on p. 195, Copt. In the other part of the Μέρος
(p. 356, Copt.), it is also said that the souls of Abraham, Isaac, and Jacob are to be placed in “the Place of Jabraoth and of all the rulers who repented” until Jesus can take them with Him to the light. So the Papyrus Bruce (Amélineau, p. 239).
626. There are seven pages missing between the descriptions of the tortures of the Middle Way and those of Amenti and Chaos, the gap occurring at p. 379, Copt. It is possible that what follows after this is not from the Μέρος
but an extract from yet another document.
627. In the text of the Μ. τ. σ. (p. 377, Copt.), Jesus simply asks His father for a sign, and “the sign is made which Jesus had said.” In the Papyrus Bruce where the same ceremony is described in almost identical words, it is said that the wine of the offering was turned into water which leaped forth of the
vase which contained it so as to serve for baptism. Cf. Amélineau, Gnost. Ég. p. 253. That Marcus the magician by juggling produced similar prodigies, see Irenaeus, Bk I. c. 7, II. pp. 116, 117, Harvey.
628. The name of Jaldabaoth, which in the whole of the rest of the MS. is spelt ⲒⲀⲖⲆⲀⲂⲀⲰⲐ, appears on p. 380 immediately after the lacuna of seven pages as ⲒⲀⲖⲦⲀⲂⲀⲰⲐ, Ialtabaoth, which supports the theory of another author
629. This is also briefly mentioned in the part of the Μέρος
Σωτῆρος just described. See pp. 386, sqq., Copt.
630. This appears to contradict the Pistis Sophia proper, where it is said that the Virgin of Light gives the soul, and the Great Iao the Good the power.
631. Cf. the speech of the crocodile in the tale of the Predestined Prince: “Ah, moi, je suis ton destin qui te poursuit; quoi que tu fasses, tu seras ramené sur mon chemin.” Maspero, Contes Populaires de l’Égypte Ancienne, 3rd ed. Paris, n. d. p. 175.
632. Evidently the Egyptian ka or double. Cf. the “Heart Amulet” described by Erman, Handbook of Egyptian Religion, pp. 142, 143, where the dead says to his heart: “Oh heart that I have from my mother! Oh heart that belongs to my spirit, do not appear against me as witness, provide no opposition against me before the judges, do not contradict me before him who governs the balance, thou art my spirit that is in my body....” This seems to be a transcription of the 30th Chapter of the Book of the Dead, of which there are several variants, none of which however directly suggest that the heart is the accuser to be dreaded. See Budge, Book of the Dead, 1909, vol. II. pp. 146-152.
633. Thus the Μ. τ. σ. says (p. 355, Copt.) “For this I despoiled myself (i.e. laid aside my heavenly nature) to bring the mysteries into the Cosmos, for all are under [the yoke of] sin,
and all lack the gifts of the mysteries.... Verily, verily I say unto you: until I came into the Cosmos, no soul entered into the light.” Contrast this with the words of the Pistis Sophia proper (p. 250, Copt.): “Those who are of the light have no need of the mysteries, because they are pure light,” which are made the “interpretation” of the text: “They that are whole have no need of a physician, but they that are sick.” See also the Pistis Sophia, p. 246, Copt., where it is said of the mysteries promised by Jesus that “they lead every race of men inwards into the highest places according to the χωρημα of the inheritance, so that ye have no need of the rest of the lower mysteries, but you will find them in the two books of Jeû which Enoch wrote etc.”
634. p. 280, Copt.
635. Μ. τ. σ. p. 388, Copt., where it is said that the soul of the righteous but uninitiated man is after death taken into Amenti and afterwards into the Middle Way, being shown the tortures in each place, “but the breath of the flame of the punishments shall only afflict him a little.” Afterwards he is taken to the Virgin of Light, who sets him before the Little Sabaoth the Good until the Sphere be turned round so that Zeus (♃) and Aphrodite (♀) come into aspect with the Virgin of Light and Kronos (♄) and Ares (♂) come after them. She then puts the soul into a righteous body, which she plainly could not do unless under the favourable influence of the “benefics” ♃ and ♀. This seems also to be the dominant idea of the Excerpta Theodoti, q.v. Compare this, however, with the words of the Pistis Sophia proper (pp. 27, 28, Copt.) where Mary Magdalene explains that the alteration made by Jesus in the course of the stars was effected in order to baffle those skilled in the mysteries taught by the angels “who came down” (as in the Book of Enoch), from predicting the future by astrology and magic arts learned from the sinning angels.
636. p. 361, Copt.
637 That is the Sphere of Destiny acting through its emissary the Moira or Fate described above, p. 184 supra.
638. It is a curious example of the fossilizing, so to speak, of ancient names in magic that Shakespeare should preserve for us in the Tempest and Macbeth the names of Ariel and Hecate which we find in the Μ. τ. σ. No doubt both were taken by him from mediaeval grimoires which themselves copied directly from the Graeco-Egyptian Magic Papyri mentioned in Chap. III supra. Cf. the use of Greek “names of God” like ischiros (sic!) athanatos, etc. in Reginald Scot’s Discovery of Witchcraft, passim.
639. So that it could not profit by the knowledge of the awful punishments prepared for sinners. I do not know that this idea occurs elsewhere.
640. p. 380, Copt.
641. The Marcosian authorship of the whole MS. is asserted by Bunsen, Hippolytus and his Age, vol. I. p. 47. Köstlin, Über das gnostische System des Buch Pistis Sophia in the Theologische Jahrbücher of Baur and Zeller, Tübingen, 1854, will have none of it, and declares the Pistis Sophia to be an Ophite work. In this, the first commentator on the book is followed by Grüber, Der Ophiten, Würzburg, 1864, p. 5, §§ 3, 4.
642. Clem. Alex. Strom. Bk I. c. 19.
643. Thus, according to Marcus (Irenaeus, Bk I. c. 8, § 11, pp. 145, 146, Harvey), “that name of the Saviour which may be pronounced, i.e. Jesus, is composed of six letters, but His ineffable name of 24.” The cryptogram in the Pistis Sophia is in these words (p. 125, Copt.): “These are the names which I will give thee from the Boundless One downwards. Write them with a sign that the sons of God may show them forth of this place. This is the name of the Deathless One ααα ωωω, and
this is the name of the word by which the Perfect Man is moved: ιιι. These are the interpretations of the names of the mysteries. The first is ααα, the interpretation of which is φφφ. The second which is μμμ or which is ωωω, its interpretation is ααα. The third is ψψψ, its interpretation is οοο. The fourth is φφφ, its interpretation is ννν. The fifth is δδδ, its interpretation is ααα, which above the throne is ααα. This is the interpretation of the second αααα, αααα, αααα, which is the interpretation of the whole name.” The line drawn above the three Alphas and Omegas is used in the body of the text to denote words in a foreign (i.e. non-Egyptian) language such as Hebrew; but in the Papyrus Bruce about to be described, the same letters without any line above are given as the name of “the Father of the Pleroma.” See Amélineau’s text, p. 113. The “moving” of the image (πλάσμα) of the Perfect Man is referred to in Hippolytus (op. cit. p. 144, Cruice). That the Tetragrammaton was sometimes written by Jewish magicians with three Jods or i.i.i. see Gaster, The Oldest Version of Midrash Megillah, in Kohut’s Semitic Studies, Berlin, 1897, p. 172. So on a magic cup in the Berlin Museum, conjuration is made “in the name of Jahve the God of Israel who is enthroned upon the cherubim ... and in the name A A A A” (Stübe, Judisch-Babylonische Zaubertexte, Halle, 1895, pp. 23-27). For the meaning of the words “above the throne,” see Franck, La Kabbale, p. 45, n. 2.
644. The opening words of the invocation βασεμὰ
which Irenaeus (Bk I. c. 14, § 2, pp. 183, 184, Harvey) quotes in this connection from Marcus certainly read, as Renan (L’Église Chrétienne, p. 154, n. 3) points out, “In the name of Achamoth” (i.e. Sophia).
645. See n. 3, p. 180, supra. In the Pistis Sophia proper Jesus is never spoken of save as “the Saviour” or as “the First Mystery.”
646 Cf. Maspero, Hypogées Royaux, passim, esp. pp. 157 and 163.
647. Schmidt’s study of the Bruce Papyrus with a full text and translation was published in the Texte und Untersuchungen of von Gebhardt and Harnack under the title Gnostische Schriften in Koptischer Sprache aus dem Codex Brucianus, Leipzig, 1892. He republished the translation of this together with one of the Pistis Sophia in the series of early Greek Christian literature undertaken by the Patristic Committee of the Royal Prussian Academy of Sciences under the title Koptisch-Gnostische Schriften, Bd I. Leipzig, 1905. His arrangement of the papyrus leaves makes much better sense than that of Amélineau, but it is only arrived at by eliminating all passages which seem to be inconsequent and attributing them to separate works. The fragments which he distinguishes as A and B and describes as “gnostischen Gebetes,” certainly appear to form part of those which he describes as the two “books of Jeû.”
648. Amélineau, “Notice sur le Papyrus gnostique Bruce,” Notices et Extraits des MSS. de la Bibl. Nat. etc. Paris, 1891, p. 106. This would seem to make matter the creation of God, but the author gets out of the dilemma by affirming (op. cit. p. 126) that “that which was not was the evil which is manifested in matter” and that while that which exists is called αἰώνιος, “everlasting,” that which does not exist is called ὕλη, “matter.”
649. Amélineau, op. cit. p. 231.
650. This word arrangement (οἰκονομία) occurs constantly in the Pistis Sophia, as when we read (p. 193, Copt.) that the last παραστάτης by the command of the First Mystery placed Jeû, Melchisedek, and four other powers in the τόπος of those who belong to the right hand πρός οἰκονομίας of the Assembly of the Light. There, as here, it doubtless means that they were arranged in the same order as the powers above them in pursuance of the principle that “that which is above is like that
which is below,” or, in other words, of the doctrine of correspondences. From the Gnostics the word found its way into Catholic theology, as when Tertullian (adv. Praxean, c. 3) says that the majority of simple-minded Christians “not understanding that though God be one, he must yet be believed to exist with his οἰκονομία, were frightened.” Cf. Hatch, H.L. p. 324.
651. Perhaps the House or Place of Ἀλήθεια or Truth many times alluded to in the Μ. τ. σ.
652. Aerôdios is shortly after spoken of as a person or power, so that here, as elsewhere, in this literature, the place is called by the name of its ruler
653. This word constantly occurs in the Magic Papyri, generally with another word prefixed, as σεσενγεν βαρφαραγγης (Papyrus Mimaut, l. 12, Wessely’s Griechische Zauberpapyri, p. 116), which C. W. King (Gnostics and their Remains, 2nd ed. p. 289) would translate “they who stand before the mount of Paradise” or in other words the Angels of the Presence. Amélineau (Notices, etc. p. 144, n. 2) will have Barpharanges to be “a hybrid word, part Chaldean and part Greek” meaning “Son of the Abyss”—which is as unlikely as the other interpretation.
654. p. 143, Amélineau (Notices, etc.); p. 361, Schmidt, K.-G.S.
655. According to Amélineau, op. cit., “The Book of the Great Word in Every Mystery.”
656. pp. 188-199, Amélineau, op. cit.; Schmidt, K.-G.S. pp. 308314.
657. pp. 219, 220, Amélineau, op. cit.; Schmidt, K.-G.S. p. 226. She seems to be here called “the Great Virgin of the Spirit.” Cf. the
“For [some of them] suppose a certain indestructible Aeon continuing in a Virgin spirit whom they call Barbelo” of Irenaeus, Bk I. c. 27, § 1, p. 222, Harvey.
658. The powers named are thus called in both the Pistis Sophia and the Bruce Papyrus. See Pistis Sophia, pp. 248, 252 Copt.; Amélineau, op. cit. p. 177.
659 According to the Pistis Sophia (p. 1, Copt.), 11 years elapsed between the Crucifixion and the descent of the “Vestures” upon Jesus on the Mount of Olives. We may imagine another year to have been consumed by the revelations made in the book.
660. If the “Books of Jeû” were ever written we should expect them to bear the name of Enoch, who is said to have taken them down in Paradise at the dictation of Jesus. See p. 147, n. 5, supra. Very possibly the expression really does refer to some of the mass of literature once passing under the name of Enoch and now lost to us.
661. Amélineau, op. cit. p. 72.
662. Schmidt, K.-G.S. p. 26.
663. Amélineau, op. cit. p. 211; Schmidt, K.-G.S. p. 322. The West or Amenti is the Egyptian name for Hades.
664. Maspero, “Egyptian Souls and their Worlds,” Ét. Égyptol. t. I. p. 395.
665. Maspero, “Hypogées Royaux,” Ét. Égyptol. t. II. pp. 148, 165.
666. Ibid. pp. 178, 179.
667. Ibid. p. 31.
668. Ibid. pp. 14-15.
669. Ibid. p. 166. To make things more difficult, the guardian sometimes had a different name for every hour. Cf. ibid. p. 168.
670. Ibid. pp. 124, n. 2, 163. For the talismans or amulets, see Maspero, “La Table d’Offrandes,” R.H.R. t. XXXV (1897), p. 325.
671 Maspero, “Hyp. Roy.” pp. 113, 118.
672. Ibid. pp. 162, 163.
673. Ibid. pp. 41, 163.
674. Ibid. p. 178.
675. Ibid. p. 179.
676. The kings, according to a belief which was evidently very old in the time of the Pyramid-Builders, were supposed to possess immortality as being gods even in their lifetime. Later, the gift was extended to rulers of nomes and other rich men, and finally to all those who could purchase the spells that would assure it. In Maspero’s words “La vie d’au delà n’était pas un droit pour l’Égyptien: il pouvait la gagner par la vertu des formules et des pratiques, mais il pouvait aussi bien la perdre, et s’il était pauvre ou isolé, les chances étaient qu’il la perdit à bref délai” (op. cit. p. 174).
677. p. 254, Copt.
678. de Faye (Intro. etc. p. 110) shows clearly, not only that the aims and methods of the school of Valentinus changed materially after its founder’s death, but that it was only then that the Catholic Church perceived the danger of them, and set to work to combat them systematically.
679. To thinkers like Dean Inge (Christian Mysticism, 1899, p. 82) this was the natural and appointed end of Gnosticism, which according to him was “rotten before it was ripe.” “It presents,” he says, “all the features which we shall find to be characteristic of degenerate mysticism. Not to speak of its oscillations between fanatical austerities and scandalous licence, and its belief in magic and other absurdities, we seem, when we read Irenaeus’ description of a Valentinian heretic, to hear the voice of Luther venting his contempt upon
some Geisterer of the sixteenth century.” It may be so; yet, after all, Gnosticism in its later developments lasted for a longer time than the doctrines of Luther have done, particularly in the land of their birth.
680. Cf. Maspero, Life in Ancient Egypt and Assyria, Eng. ed. 1892, pp. 90-92, for the distaste of the Egyptians of Ramesside times for the life of a soldier and their delight in that of a scribe.
681. All these, especially alchemy, are illustrated in the Magic Papyrus of Leyden known as W. See Leemans, Pap. Gr. t. II. pp. 83 sqq.
682. Gibbon, Decline and Fall, vol. III. p. 214, Bury’s ed.
683. Renan, L’Église Chrétienne, pp. 154, 155, and authorities there quoted. Cf. Hatch, H. L. pp. 129, 130, 293, 307-309.
684. Harnack, What is Christianity? p. 210; Duchesne, Early Christian Church, p. 32.
685. Chap. IX. p. 118 supra.
686. Renan, Marc Aurèle, p. 49. Cf. Dill, Nero to Marcus, pp. 473477.
687. Renan, L’Église Chrétienne, pp. 31-33, and Hadrian’s letter there quoted.
688. Of the defences mentioned in the text the Apology of Quadratus is the only one still lost to us. Justin Martyr’s two Apologies are among the best known of patristic works. That of Aristides was found by Dr Rendel Harris in a Syriac MS. in 1889. For the identification of this by Dean Armitage Robinson with the story of Barlaam and Josaphat, see Cambridge Texts and Studies, vol. 1. No. 1.
