Deep learning for CVA computations of large portfolios of financial derivatives

Delft University of Technology

Deep learning for CVA computations of large portfolios of financial derivatives

Andersson, Kristoffer ; Oosterlee, Cornelis W.

DOI

10.1016/j.amc.2021.126399

Publication date

2021

Document Version

Final published version

Published in

Applied Mathematics and Computation

Citation (APA)

Andersson, K., & Oosterlee, C. W. (2021). Deep learning for CVA computations of large portfolios of

financial derivatives. Applied Mathematics and Computation, 409, 1-21. [126399].

https://doi.org/10.1016/j.amc.2021.126399

Important note

To cite this publication, please use the final published version (if applicable).

Please check the document version above.

Copyright

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy

Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim.

This work is downloaded from Delft University of Technology.

ContentslistsavailableatScienceDirect

Applied

Mathematics

and

Computation

journalhomepage:www.elsevier.com/locate/amc

Deep

learning

for

CVA

computations

of

large

portfolios

of

financial

derivatives

Kristoffer

Andersson

a,∗

_,

_{Cornelis W.}

_Oosterlee

a,b

a CWI - National Research Institute for Mathematics and Computer Science, Amsterdam, the Netherlands b Delft Institute of Applied Mathematics, Delft University of Technology, Delft, the Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 25 February 2021 Revised 18 May 2021 Accepted 19 May 2021 Keywords: Portfolio CVA Expected shortfall WWR Bermudan options Deep learning

a

b

s

t

r

a

c

t

Inthispaper,weproposeaneuralnetwork-basedmethodforCVAcomputationsofa port-folio ofderivatives. In particular, wefocus onportfoliosconsisting ofacombinationof derivatives,withandwithouttrueoptionality,e.g.,aportfolioofamixofEuropean-and Bermudan-typederivatives.CVAiscomputed,withandwithoutnetting,fordifferentlevels ofWWRandfordifferentlevelsofcreditquality ofthecounterparty.Weshowthatthe CVAisoverestimatedwithupto25%byusingthestandardprocedureofnotadjustingthe exercisestrategyforthedefault-riskofthecounterparty.FortheExpectedShortfallofthe CVAdynamics,theoverestimationwasfoundtobemorethan100%insomenon-extreme cases.

© 2021 The Author(s). Published by Elsevier Inc. ThisisanopenaccessarticleundertheCCBYlicense

(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Inthispaper,weconsiderasetoffinancialcontracts,whichwerefertoastheportfolioofderivatives,orjusttheportfolio, writtenbetweentwoparties.Thefirstpartyisreferredtoasthebankandisconsideredtobedefault-free.Thesecondparty, whichmaydefault,isreferredtoasthecounterparty.Wetaketheperspectiveofthedefault-freebankinordertoinvestigate someoftherisksassociatedwithadefaultablecounterparty.Itisstraightforwardtoextendthemethodologiesusedinthis papertoadefaultablebankaswellastomultiplecounterparties.

1.1. Risk-freevaluation

We consider the problemof finding the value of a portfolio of derivatives with early-exercise features. In particular, we are focusing onportfolios withmultiplederivativeswithtrue optionality,e.g., AmericanorBermudan derivatives.We constructaportfolioofJ derivatives,wheretheindividual derivativesdependond1,d2,...,dJrisk factors.Thismeansthat wecouldfacehigh-dimensionalityintwoways:

1. Derivative j coulddependonalargenumberofriskfactors,i.e.,djcouldbelarge; 2. Wecouldhavemanyderivativesintheportfolio,i.e.,J couldbelarge.

∗_{Corresponding author. National Research Institute for Mathematics and Computer Science, P.O. Box 94079, 1090 GB, Amsterdam, the Netherlands.}

E-mail address: kristoffer.andersson@cwi.nl (K. Andersson).

https://doi.org/10.1016/j.amc.2021.126399

0 096-30 03/© 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

Table 1

Exposure given different exercise decisions at t, with and without a netting agreement. Without netting With netting

Exposure - no exercise 20 10

Exposure - exercise one of the American options 10 0 Exposure - exercise both American options 0 0

In [1],a neural network-based methodfor valuation ofa singleBermudan derivative wasproposed and provedto be highlyaccurateforderivativeswithupto100riskfactors.Later,thealgorithmwasextendedin[2]toalsoincludepathwise valuationsofthederivative(incontrasttoonlyfindingthevalueattheinitialtime).Inthispaper,weextend[1]and[2]to theportfoliocase,i.e.,findingthevalueofalargeportfolioof,possiblyhigh-dimensional,derivativeswithtrueoptionality, withouthavingtocomputethevalueofeachindividualderivative.

In a traditionalsetting, theso-called continuation value iscomputed, andsubsequently, thevalue of thederivative is givenbythemaximumofthecontinuationvalueandtheimmediatepay-off.Forasinglederivative,thisisstraightforward. For instance,thecontinuation value can be computedby solving an associatedPDE,which isdone in e.g.,[3–6] and[7], orthe continuationvalue canbe approximatedby aFouriertransformmethodology,whichis doneine.g., [8,9]and[10]. Furthermore, classical tree-basedmethods such as[11,12] and[13],can be used. Thesetypes ofmethods are, ingeneral, highlyaccurate butthey sufferseverelyfromthecurse ofdimensionality,meaning thatthey are computationallyfeasible onlyinlowdimensions(sayupto4riskfactors),see[14].Inhigherdimensions,Monte-Carlo-basedmethodsareoftenused, seee.g.,[15–18]and[19].Monte-Carlo-basedmethodscangeneratehighlyaccuratederivativevaluesattheinitialtime,but oftenlessaccuratevaluesbetweentheinitialtimeandmaturityofthecontract.

In contrast to the single derivative case, it is not enough to knowthe continuation value of a portfolio (with more than one derivative) in order to decide optimally which derivatives should be exercised. Therefore, it is commonto do the valuationatthe levelofeach derivative,andthenadd theindividual valuesofeach derivative toobtainthe portfolio value.Thisbecomescumbersomeforlargeportfolios.Asmentionedabove,themethodologyusedinthispaper,generalizes

[1]and[2],inwhichthe optimalexercisepolicy isapproximatedby maximizing expecteddiscountedcash-flows,i.e.,the continuation value is not computed. By not relying on computations of the continuation value, the algorithm isable to computetheportfoliovaluewithouthavingtocomputetheindividualvaluesforeachderivative.

1.2. RiskyvaluationandCVA

The Credit Valuation Adjustment (CVA) is the difference betweenthe risk-free portfolio value and the riskyportfolio value, where therisky portfoliovalue is definedasthe portfoliovalue when takingdefaultrisk of thecounterparty into account.Whilethereisnoambiguityoftherisk-freeportfoliovalue,itisnotcompletelyclearhowtheriskyportfoliovalue shouldbecomputed.Thequestioniswhethertheexercisepolicyshouldbeadjustedforthefactthatthecounterpartymay default. Forinstance,ifthecounterpartyends upinfinancial distress,it isreasonabletoassume thatthe bank(which in thispaperisassumedtobetherisk-freeparty)wouldbemorewillingtoexercisethecallablederivatives,inordertolower its exposure to thecounterparty. Eventhough it seemscommon toignore the effectofa defaultablecounterpartywhen computingriskyderivativevalues,ithasbeendiscussedintheliterature,seee.g.,[20–23].Inthecaseofasinglederivative

[20]states that the exercise regionfor a risk-free derivativeisalwaysasubsetoftheexerciseregionforariskycounterpart. However,inthecaseofaportfolio,thesituationismorecomplex,anddependsoncontractualdetailssuchastheclose-out and nettingagreements. One consequenceis that, inthe presence ofa netting agreement,the exercise decisions canno longerbemadeindividually.Toexplainthis,wegiveasimpleexample.

Example1.1. Assumethatwehaveaportfolio,consistingofthreederivatives,oneEuropeanfutureandtwoAmerican op-tions. Allcontractsareinitialized attime 0,matureattimeT anddependonthesamerisk-factor

(

Xt

)

t∈[0,T].Assume that, attimet∈

(

0,T

)

,andgivenXt=x,theintrinsicvaluesare

Vfuture

(

t,x

)

=−10, V1Am

(

t,x

)

=10, V2Am

(

t,x

)

=10, andtheimmediatepay-off fortheAmericanoptionssatisfy

gAm

1

(

t,x

)

<10, gAm2

(

t,x

)

<10.

In a risk free environment (no-defaultablecounterparty), itis sub-optimalto exercise the Americanoptions. However, in caseofadefaultablecounterparty,thesituationislesstrivial. InTable1,theexposuretothecounterparty,givendifferent exercisedecisionsatt,isgivenwithandwithoutanettingagreement.

Ifthecounterpartyisinseverefinancialdistress,thenitislikelyoptimalforthebanktoexercisebothAmericanoptions inthe caseofnonettingagreement,andoneoftheminthecaseofanettingagreement.From thissimple example,two things becomeclear;1)TheexercisedecisionsfortheAmericanoptionsareaffectednot onlybyariskycounterparty,but alsobywhetherornotanettingagreementexists.2)inthepresenceofnetting,exercisedecisionscannotbemadeforone derivativeinisolation,butonlyforalltheAmericanoptionssimultaneously.

In general, for a risky portfolio, it is not possible to describe the value of a single derivative, but only the value of the entireportfolio.This isaninteresting problemsincealmost allexisting algorithmsrely onexercise decisionsmadein isolationandriskyderivativevaluesthatcanbeaddeduptoobtaintheriskyportfoliovalue.

Ifthisisnottakenintoaccountwe wouldobtaina biasedlow valuationfortheriskyportfoliobyusinga sub-optimal exercise strategy. Since the CVA is the difference betweenthe risk-free and risky portfolio values, we would obtain an overestimationoftheCVA.Furthermore,thiseffectislikelytoincrease withdecreasingcreditquality ofthecounterparty. Inpractice,thismeansthatthecounterpartyispayingaCVAwhichisbasedonasub-optimalexercisestrategyusedbythe bank,whichisout ofcontrolforthecounterparty.EvenmoreproblematicisthattheoverestimationoftheCVAishigher forcounterpartiesthatalreadyareunderfinancialdistress.

Onecouldarguethatitisreasonableforthebanktocharge thecounterpartythehigherCVA,sincethebankwill prob-ablynot followthetheoretically optimalriskyexercise strategy.However,thereisanotherlevelofcomplexitynot yet dis-cussed.Whenthemark-to-market(MtM)CVAmovesintimeagainstthebank,thebankcouldfacelosses,notbecausethe counterparty actually defaults, butbecause disadvantageous changes in the MtMCVA. For instance, in BaselIII [24] the followingisstated:

“Under BaselII,theriskofcounterparty defaultandcreditmitigationriskwereaddressedbutmark-to-marketlosses dueto creditvaluationadjustments(CVA)werenot.Duringtheglobalfinancialcrisis,however,roughlytwo-thirdsoflossesattributedto counterpartycreditriskwereduetoCVAlossesandonlyaboutone-thirdwereduetoactualdefaults.”

Thisisfurtherdiscussedin[25],inwhichtheauthorsalsorecommendcomputationsofdifferentrisk measuresforthe futuredistributionofCVA.Twoexamplesofsuch measuresarethe ValueatRisk oftheCVA (VaR-CVA)andtheExpected ShortfalloftheCVA(ES-CVA).TheadvantageoftheES-CVAisthatitisacoherentrisk-measure,andwethereforefocuson ES-CVAinthispaper.

1.3. Structureofthepaper

InSection2themathematicalproblemformulationisgiven.Wedefinetherisk-freeandriskyportfolios,close-out agree-mentsbothwithandwithoutnettingagreementsandtheassociateCVA.Furthermore,theproblemsareformulatedinterms ofso-calleddecisionfunctions,whichcontroltheexercisestrategies.InSection3,thealgorithmsarepresented.Inthefirst part,thealgorithmforlearningoptimalexercisestrategiesisgivenandinthesecondpart,analgorithmforlearning path-wiseentitiessuchasthepathwiseportfolioexposureispresented.Finally,inSection4numericalexperimentsarepresented. Theexperimentsincludeafirstpart,inwhichrisk-freevaluesarecomputedandcomparedtoawell-establishedregression based method.Inthe second partwecompare CVAcomputedwiththe risk-freeandthe riskyexercise strategyto verify that,indeed,theCVAisoftenoverestimatedwithalgorithmsinusetoday.Wepresentcomparisonswithandwithout net-ting,fordifferentlevelsofWrongWayRisk(WWR),andfordifferentcreditqualityofthecounterparty.Asafinalexample, weanalysetheeffectofthedifferentexercisestrategiesonES-CVA.IntheAppendix,weprovidesomeadditionaldetailson thealgorithmsandthespecificchoiceofneuralnetworks.

2. Problemformulation

Let

(

,F,Q

)

beaprobabilityspacecompletedwiththeQ−null-setsofF.ForT∈

(

0,∞

)

,and1_{d∈N,letX:[0,T]×}

_

_→

Rd _andr:[0,T]×

_

_{→Rrepresentthe(market)risk-factorsoftheportfolioandthe shortrate, respectively.Furthermore,} we denoteby

τ

D _{thedefault eventof the counterparty, whichis a stopping time defined on}

_(

_,F,Q

₎

_{andwe let 1}D_: [0,T]×



→

{

0,1

}

,bethejump-to-defaultprocessgivenby

1D_t :=I_{t<τD_}. (2.1)

Theinformationstructureisgivenbythesub₋

σ

_−algebrasgeneratedbyX,r and1D,i.e.,_HX

t =

σ

(

Xs:s∈[0,t]

)

,Htr=

σ

(

rs:

s∈[0,t]

)

andGt=

σ

(

1Ds :s∈[0,t]

)

andwe define the enlarged filtrations Ht=HXt ∧Hrt and Ft=Ht∧Gt. In thispaper, we use either a constant short rate (risk-free rate), or we view the short rate as one of the risk factors. In the latter case, we model the shortrate asone ofthe d component processes ofX, whichimplies that, Ht=HtX. The motivation for introducinga separate notationfortheshort rateisto simplifythenotationwhen theshort rateisused todiscount cash-flows. For commonly used conditional expectations, we introduce the short-hand notations Et,x[· ]:=EQ[·

|

Xt=x], Et,x,ν[· ]:=EQ_[·

_|

_Xt=x,1D

t =

ν

]andEt[· ]:=EQ[·

|

Ht].

We usea numéraire, which,for t∈[0,T], is definedby Bt:= exp

(

₀trsds

)

,which should be interpreted as the value at timet of a savings-account, whichwas worth 1attime 0. Fort,u∈[0,T] witht≤ u,we use Dt,u:= BBtu to discount a cash-flowobtainedattime uback totimet.ThemeasureQistherisk-freemeasure,underwhichalltradeableassetsare martingalesrelativetothenuméraire,e.g.,ifcomponenti∈

{

1,2,...,d

}

ofXistradeable,then (Xt)i

B_t isaQ−martingale. Ifnotspecificallystatedotherwise,equalitiesandinequalitiesofrandomvariablesshouldbeinterpretedinaQ−almost suresense.

1 We use N = _{{ 1 , 2 , 3 . . . } and N}

2.1. Aportfolioofderivatives

WeassumeaportfolioofJ∈Nderivatives.Fort∈[0,T],andforderivative j∈

{

1,2,...,J

}

,wedenotethesetofexercise datesgreaterthanorequaltot byTj

(

t

)

⊆ [0,T],andsetT

(

t

)

=

{

T1

(

t

)

,T2

(

t

)

,...,TJ

(

t

)

}

.Inotherwords,Tj

(

t

)

represents the remaining exercise dates ofderivative j at timet and T

(

t

)

is thea listof theremaining exercise dates forall the J

derivatives.NotethatforaEuropean-typecontract,theonlyexercisedateisatthematurity,foraBermudan-typecontract therearemultipleexercisedates,andforanAmerican-typecontract,thereareinfinitelymanyexercisedates.Weemphasize that theexercisedatesare simplysubsetsofthetimeinterval [0,T],andprovidenoinformationonwhichexercisepolicy tofollow,exceptinsometrivialcasese.g.,whenthereisonlyoneexercisedate.

Sincewewanttobeabletotreatderivativeswithearly-exercisefeatures,weneedtointroduceaframeworkforstopping times.For j∈

{

1,2,...,J

}

,anX−stoppingtimewithrespecttoTj

(

0

)

,isarandomvariable,

τ

j,definedon

(

,F,Q

)

,taking onvaluesinTj

(

0

)

,suchthatforalls∈Tj

(

t

)

,itholdsthattheevent

{

τ

j=s

}

∈Hs.Furthermore,wedefineanXt,x−stopping timeasanX−stoppingtime,conditionalonXt=x,and

τ

≥ t.

Foreachderivative, j_∈

{

1_,2_,...,J

}

weuseindividualpay-off functions,g_j:[0_,T]_{× R}d_{→R,which,fort,s∈[0,T],with}

s≥ t,areassumedtosatisfy

E0[

|

Dt,sgj

(

s,Xs

)

|

2]<∞. (2.2)

Sincewearetreatingportfolioswheretheindividualderivativesmayhavedifferentmaturities,weseteachpay-off function to zero for all times larger than its maturity, i.e., for j∈

{

1,2,...,J

}

, x∈Rd _andfort>max

{

_T

j

(

0

)

}

,we set gj

(

t,x

)

≡ 0, wheremax

{

Tj

(

0

)

}

representsthelargestelementbelongingtothesetTj

(

0

)

.

2.2. Risk-freeandriskyportfoliovaluationwithoutnetting

The valueofa derivative(not)takingdefaultriskofthe counterpartyintoaccount isreferred toasthe risky(risk-free) value.Wedefinetherisk-freeandtheriskyvaluesofderivative j∈

{

1,2,...,J

}

,atmarketstate

(

t∈[0,T],Xt=x∈Rd

)

,and defaultstate1D_t =

ν

∈

{

0_,1

}

,by Vj

(

t,x

)

:= sup τ∈Tj(t) Et,x



Dt,τgj

(

τ

,Xτ

)



, (risk-freevalue), (2.3) Uj

(

t,x,

ν

)

:=

ν

sup τ∈Tj(t) Et,x,1



1D_τDt,τgj

(

τ

,Xτ

)

+

(

1−1D_τ

)

D_t,τD



RVj

(

τ

D,X_τD

)

++V_j

(

τ

D,X_τD

)

−



, (riskyvalue), (2.4)

where _Tj

(

t

)

is the set of all X−stopping times taking on values in Tj

(

t

)

and for x∈R,

(

x

)

+=max

{

0,x

}

and

(

x

)

−= min

{

0,x

}

. Inthe above, we assume a close-outagreement which uses the risk-free derivative values asreference valua-tion. At defaultof the counterparty, the bank receives only a fraction,R∈[0,1

)

, referred to as the recovery-rate, of the positive partofeach derivative. Ontheother hand,each derivativewitha negativerisk-freevalue atdefaultneeds tobe addedentirelytotheportfolio.

Notethatfortheriskyvalue weneedadditionalinformationofpriordefaultsofthecounterparty,whichiscapturedin the realization,

ν

_∈

{

0_,1

}

,ofthejump-to-default process, i.e.,

ν

₌1ifno defaulthasoccurredpriorto, oratt,and

ν

₌0 otherwise.ThenotationabovetriviallyholdsforEuropean-typederivativessincetheonlyexercisedateisatmaturityofthe contract.Furthermore,abarrier-typefeaturecouldbeaddedbyalsoincludingaspatialdimensiontoTj

(

0

)

.Thevalueofa portfolio,consistingofJ derivatives,atmarketstate

(

t,Xt=x

)

anddefaultstate1Dt =

ν

,withoutnetting,isgivenby



V

₍

_t,x

₎

_:= J 

j=1

Vj

(

t,x

)

, (risk-freeportfoliovalue),



U

₍

_t,x,

_ν

₎

_:= J  j=1 Uj

(

t,x,

ν

)

=

ν

J  j=1

Uj

(

t,x,1

)

, (risky portfoliovalue). Using(2.3)and(2.4),theabovecanbewrittenas



V

₍

_t,x

₎

₌ J  j=1 sup τj∈Tj(t) Et,x



Dt,τjgj

(

τ

j,Xτj

)



(2.5)



U

₍

_t,x,

ν

₎

₌

ν

J  j=1 sup τj∈Tj(t) Et,x,1



1D_τjDt,τjgj

(

τ

j,Xτj

)

+

(

1−1D_τj

)

Dt,τD



RVj

(

τ

D,X_τD

)

++V_j

(

τ

D,X_τD

)

−



. (2.6)

Sincetheaimistoapproximatetheoptimalexercisepolicywithneuralnetworks,wewishtore-formulatetheproblem intoanoptimizationproblem,inwhichthetargetfunctioncanberepresentedbyaneuralnetwork.Following[1]and[2]we useso-calleddecisionfunctions,todetermineforeachderivative andgivenamarketstate,whetherornottoexercise the derivative.Forj_∈

{

1_,2_,...,J

}

,decisionfunction j denotedby f_j,isoftheformf_j:[0_,T]_{× R}d_→

_{

_0,1

_}

_{.Inordertoguarantee} thatanexercisedecisioncanonlyoccuratanexercisedate,werequirefors∈/Tj

(

0

)

,that fj

(

s,·

)

≡ 0.

We nowrestrictourattentiontothecasewhenthere,foreachderivative, isafinitenumberofexercisedates,i.e.,for

j_∈

{

1_,2_,...,J

}

,itholdsthat

|

Tj

(

0

)

|

∈N.Froma theoreticalperspective,thisexcludesAmerican-typederivatives,butfrom apracticalperspective,aninfinitenumberofexercisedatesisoftenapproximatedbyalarge,butfinite,numberofexercise dates.ThisimpliesthatwecanstillconsiderAmerican-typederivativesbyincreasingthenumberofexercisedatesuntilthe derivativevalueconverges(untilthevaluedoesnotincreasewithadditionalexercisedates).WedenotebyT

(

t

)

thesetof dateswhichrepresentanexercisedateforatleastoneoftheJ derivatives.Mathematically,wedefinetheexercisedatesof theportfolioas T

(

t

)

:= J j=1 Tj

(

t

)

, (2.7)

andthenumberofuniqueexercisedatesintheportfolioisgivenbyN=

|

T

(

0

)

|

.WeassumethattheinitialtimeT0=0is notanexercisedateforanyofthederivatives.

Tosimplify,weusethefollowingnotationfortheN exercisedates,therisk-factorsevaluatedattheN exercisedates,and thediscountingbetweenexercisedates

T

(

0

)

=

{

T1,T2,...,TN=T

}

, (2.8)

Xk:=XTk, andDk,:=DTk,T fork,=1,2,...,N. (2.9)

Furthermore,fort∈[0,T],therisk-factorprocess on[t,T],conditionalonXt=x,isdenotedby Xt,x=

(

Xs

)

s∈[t,T],wherewe alsonotethatX0,x0=X.TheabovenotationallowsustoexpressanXt,x_−stoppingtimeintermsofdecisionfunctions2

τ

n j[fj]

(

Xt,x

)

:= N  k=n Tkfj

(

Tk,Xk

)

k−1  m=n

(

1− fj

(

Tm,Xm

))

. (2.10)

The notationabove isusedtoemphasizethat, thedecisionfunction fj,controlstheexercise strategy,giventhestochastic processXt,x_.Moreover,Xt,x_{is not justa random value ata specific time,but the entire process,starting atXt=x and until} stoppingoccurs.Inlatersectionsthevaluationofaderivativeoraportfolioisformulatedasanoptimizationproblem,which isoptimizedbyvarying f_j.Althoughthenotationispracticalwhenoptimizationisdiscussed,itiscumbersometousewhen wedefinethevalueofaderivative.Wethereforeusethefollowingshort-handnotation

¯

τ

n, j:=

τ

jn[fj]

(

Xt,x

)

, (2.11)

and keep in mind, that the strategy is controlled by a decision function, fj, and for u≥ t, the event I{τ¯_{n, j}≤u} is

σ

(

Xt,x

₎

_{−measurable.Wecannowdefinethevalueoftherisk-freeandriskyderivatives,givenanexercisestrategyexpressed} intermsofdecisionfunctions.Forderivative j∈

{

1,2,...,J

}

,marketstate

(

t,x

)

∈

(

Tn−1,Tn]× Rd,wedefinetheparametrized valuationfunctions Vj

(

t,x

|

fj

)

:=Et,x



Dt,τn, j¯ gj



¯

τ

n, j,Xτn, j¯



, (2.12) Uj

(

t,x

|

fj

)

:=Et,x,1



1D_{τn, j¯} D_t,τ¯n, jgj



¯

τ

n, j,Xτ¯n, j



+



1−1D_{τn, j¯}



Dt,τD



RVj

(

τ

D,XτD

)

++V_j

(

τ

D,XτD

)

−



. (2.13)

Similarly,we definetheportfoliovalueswithrespecttotheexercise strategygivenby f,astheparametrizedfunctions

ϒ

V

₍

_t,x

_|

_f

₎

_:= J  j=1 Vj

(

t,x

|

fj

)

,

ϒ

U

(

t,x,

ν |

f

)

:=

ν

J  j=1 Uj

(

t,x

|

fj

)

, (2.14)

wherethevalueoftheriskyportfolioalsodependsonthedefaultstateofthecounterparty,1D_t =

ν

∈

{

0,1

}

.Wenowwantto finddecisionfunctionssuchthat,wheninsertedin(2.12)and(2.13),weobtain(2.3)and(2.4).Withthisinmind,wedefine for j∈

{

1,2,...,J

}

, att∈[0,T],the (optimal)exercise regions, EZ

j

(

t

)

,inwhich itisoptimalto exercise,andthe (optimal) continuationregions,CZ

j

(

t

)

,inwhichitisoptimaltoholdon,by

EZ j

(

t

)

:=



x∈Rd

|

Zj

(

t,x

)

=gj

(

t,x

)

and t∈Tj

(

0

)

,

CZ j

(

t

)

:=



x∈Rd

|

Zj

(

t,x

)

>gj

(

t,x

)

or t∈/Tj

(

0

)

, forZ∈

{

V,U

}

.

Theabove statesthatthederivativeshouldbeexercisedifitsvalue equalstheimmediateexercisevalue,andweareatan exercisedateandthederivativeshouldnotbeexercisedifitsvalueisgreaterthantheimmediateexercisevalueorifweare notatanexercisedate.NotethatEZ

j

(

t

)

∪CZj

(

t

)

=RdandEZj

(

t

)

∩CZj

(

t

)

=∅.Fort∈[0,T],adecisionfunction, j∈

{

1,2,...,J

}

, canthenbedefinedas

fZ

j

(

t,x

)

:=I{x∈EZ

j(t)}, forZ∈

{

V,U

}

. (2.15)

Furthermore,wedenoteby fZ,thevectorconsistingoftheindividualdecisionfunctions fZ

(

t,x

)

:=

(

fZ

1

(

t,x

)

,f2Z

(

t,x

)

,...,fJZ

(

t,x

))

T, forZ∈

{

V,U

}

. (2.16) Formarketstates

(

t,x

)

∈[0,T]× Rd _{andforaderivative j∈}

_{

_1,2,...,J

_}

_,itholdsthat

Vj



t,x

|

fV j



=Vj

(

t,x

)

, Uj



t,x

|

fU j



=Uj

(

t,x,1

)

.

ThevalidityoftheaboveisadirectconsequenceofProposition4in[1].Inturn,thisimpliesthatbyinsertingtheoptimal decisionfunctionsinthefunctionalsinequations(2.14),weobtaintheriskyandrisk-freeportfoliovalues,i.e.,

ϒ

V



_t,x

_|

_fV



=



V

₍

_t,x

₎

_,

_ϒ

U



_t,x,

_{ν |}

_fU



=



U

₍

_t,x,

_ν

₎

_.

Insubsequentsectionstheoptimaldecisionfunction fZ,forZ_∈

{

V_,U

}

,isapproximatedwithaseriesofneuralnetworks. The reasonforusing therathercomplicated notation,(2.10),is that thisstructure allowsusto view thevaluation of the derivativesasanoptimizationproblemoverthesetofdecisionfunctions,whichweapproximateonsomefinite-dimensional functionspace.Oneexampleofsuchfunctionspaceisthefunctionsgeneratedbyaseriesofneuralnetworkswithafixed numberofparameters.Whenweuseaspecificstrategy,e.g., fV or fU,thisisspecifiedbyaddingasuperscriptreferringto theparticularstrategy.ForZ∈

{

V,U

}

,wedefinetheshort-handnotation

¯

τ

Z

n, j:=

τ

jn[fjZ]

(

Xt,x

)

, (2.17)

whereitisassumedthatt∈

(

Tn−1,Tn].

2.3. Riskyportfoliovaluationwithnetting

When considering the risky portfolio value with netting, the problembecomes nonlinear in the sense that the risky portfolio valueisnolonger thesumoftheindividualrisky derivativevalues.Infact,there nolongerexists”a riskyvalue forasinglederivative”,sincethevaluationneedstobecarriedoutonaportfoliolevel.Beforewedefinetheriskyvalueof anettedportfolio,weneedtodefinetheprocess3_A:[0,T]×

_

_→

_{

_0,1

_}

J_{,whichfort∈[0,T]andj∈}

_{

_1,2,...,J

_}

_satisfies

(

At

)

j=



0, ifderivative jhasbeen exercisedpriortot,

1, else.

Similarto(2.9),weusetheshort-handnotationforAatinitialdate,T0,theexercisedates,T1,...,TN,

Ak:=ATk, fork=0,1,...,N. (2.18)

The process A is Ht−measurable but it isnot enough to knowXt=x inorder to determine At.The reason fordefining

A is that the exercise decisions forthe nettedrisky portfolio are definedby a J−dimensional

(

Xt,x,At,α

)

−stopping times vector, where At,α₌

₍

_As

₎

s∈[t,T] conditional on At=

α

. This means that, at each exercise date, in addition to the current marketstate,weneedtoknowwhichderivativesintheportfoliohavebeenexercisedpriortothecurrenttime,inorderto makeoptimalexercisedecisions.Wedenote,byT

₍

_t

₎

_,thespaceof

₍

_Xt,x_,At,α

₎

_{−stoppingtimesvectors,takingonvaluesin}

{

T1

(

t

)

,T2

(

t

)

,...,TJ

(

t

)

}

.Furthermore,fort∈

(

Tn−1,Tn],we denoteby

τ

_jn element j ofastoppingtimesvector

τ

n∈T

(

t

)

. Thenettedportfoliovaluewithariskycounterparty,givenmarketstateXt=x,defaultstateofthecounterparty1Dt =

ν

and portfoliostate(exercisestateofthederivativesintheportfolio)At=

α

,isgivenby



A

₍

_t,x,

ν

_,

α

₎

_:=

ν

_sup τ∈T_(t)Et,x,1,a



_J j=1

α

j1D_τjDt,τjgj

(

τ

j,Xτj

)

+D_t,τD



R



_J k=1

α

k

(

1−1D_τk

)

Vk

(

τ

D,X_τD

)



+₊



_J =1

α



(

1−1D_τ

)

V

(

τ

D,XτD

)



−



, (2.19) where Et,x,ν,α[· ]=EQ_[·

_|

_Xt=x,1D

t =

ν

,At=

α

]. To emphasize the importance, we put the following observations about

(2.19)intotworemarks.

Remark2.1. Theoptimalstoppingstrategiesoftheindividualderivativesintheportfolioarenolongerindependentofeach other,asin(2.5) and(2.6).Furthermore,theoptimalstrategy dependsonearlierexercisedecisions,meaningthatinorder to make theexercise decisions Markovian,we needto includeinformation aboutearlierdecisions.The reasonforthis is the non-linearityin thetwo sums inside theexpectation in(2.19).Therefore, theoptimal stoppingstrategies needto be computedfortheentireportfoliosimultaneously.Tothebestofourknowledge,thishasnotbeendoneinanordinaryleast squaressettingbefore.However,itisdiscussedinaPDEframeworkinthecaseofaportfolioofAmericanswaptionsin[23].

Remark2.2. Thevalueoftheriskyportfoliowithnetting,dependsontheJ risk-freederivativevaluesandwearetherefore required to approximate the risk-free derivative values.The reasonfor this is that, at default, the risk-free value of the portfolioisusedasreferencevalueintheclose-outagreement(seeEquation(2.19)).Ifwerestrictourportfoliotoderivatives withpositivepay-off functions,thenV_jcanbe replacedbyg_j(bythedefinitionofV_jandthelawofiteratedexpectations). Furthermore,intherestrictedportfolio,thevalueswithandwithoutnettingcoincide.

An

(

Xt,x,At,α

)

−stoppingtimesvectorcanbedefinedby

τ

n_[f]

₍

_Xt,x_,At,x

₎

_:= N  k=n Tkf

(

Tk,Xk,Ak

)

 k−1  M=n

(

1 J− f

(

Tm,Xm,Am

))

, (2.20)

where iselement-wisemultiplicationand1JistheJ−dimensionalvectorwithonlyones,

(

1,1,...,1

)

T.Wedenoteelement

j of thestopping timesvector by

τ

n

j[f]

(

Xt,x,At,α

)

=

(

τ

n[f]

(

Xt,x,At,x

))

j.We emphasize thateach element ofthe stopping time vector dependson f andnot onlyan element j whichisthe casewithout netting.Similar to(2.11),we introducea short-handnotation,whichsimplifiesthevaluationfunction

ˆ

τ

n:=

τ

n[f]

(

Xt,x,At,x

)

,

τ

ˆn, j:=

(

τ

ˆn

)

j. (2.21)

Wehereuse” ˆ

τ

”,insteadof” ¯

τ

” asin(2.11),toemphasizethatthestoppingtimealsotakesAt,α_{asanargument.Thenetted} riskyportfoliovalue,giventheexercisestrategyobtainedbydecisionfunction f,isthengivenby

ϒ

A



_t,x,

_ν

_,

_{α |}

_f



_:=

_ν

_E t,x,1,α



_J j=1

α

j1Dτˆn, jDt,τn, jˆ gj



ˆ

τ

n, j,Xτn, jˆ



+Dt,τD



R



_J j=1

α

j

(

1−1Dτn, jˆ

)

Vj

(

τ

D,XτD

)



+₊



_J j=1

α

j

(

1−1Dτn, jˆ

)

Vj

(

τ

D,XτD

)



−



, (2.22)

where we, again, remind ourselves that the exercise strategy is controlled by f, and for u≥ t, the event I{τˆ_{t, j}≤u} is

σ

(

Xt,x_,At,α

₎

_{−measurable.}

Foranettedportfolio,theoptimalexerciseregions,describedinSection2.2,arelesstrivial.Firstly,theybecome depen-dent onthe stateofearlierexercise decisions,At=

α

t∈

{

0,1

}

J.Secondly,theexercise regionforderivative j∈

{

1,2,...,J

}

isexpressed undertheconditionthat anoptimalexercise strategyfortheother J− 1derivativesisapplied.Therefore,we only describetheoptimaldecisionfunction asbelongingtothe supremumoverthespace, D,ofallmeasurablefunctions,

f:[0_,T]_{× R}d_×

_{

_0,1

_}

J_→

_{

_0,1

_}

J_, fA∈argmax f∈D

ϒ

A



_0,x 0,

ν

0,

α

0

|

f



, (2.23)

where

ν

0=1(nodefaultpriortooratt=0)anda0=

(

1,1,...,1

)

T (noderivativeshavebeenexercisedpriortot=0).We thenassumethat,giventhestate

(

t,Xt=x,1tD=

ν

,At=

α

)

,thefollowingholds

ϒ

A

₍

_t,x,

_ν

_,

_{α |}

_fA

₎

₌

_

A

₍

_t,x,

_ν

_,

_α

₎

_{. (2.24)}

Similarto(2.17),whenwewanttoemphasizetheparticularchoiceofdecisionfunction, fA,weusetheshort-handnotation ˆ

τ

A n=

τ

n[f A ]

(

Xt,x,At,α

)

and

τ

ˆnA, j=



τ

n_[fA ]

(

Xt,x,At,α

)



_j, (2.25)

whereitisassumedthatt∈

(

Tn−1,Tn].

2.4. Creditvaluationadjustmentofaderivativeportfolio

The formaldefinitionofCVAisthe differencebetweentherisk-freeandtheriskyportfoliovalue. Givenmodels ofthe underlyingmarketanddefaulteventsofourcounterparty,theabovedefinitionofCVAisstraightforwardforaportfolio con-sistingofderivativeswithoutoptionality e.g.,Europeanoptions,barrieroptionsetc.When itcomes toportfoliosconsisting ofderivativeswithtrueoptionality, e.g.,theBermudanoptions,American optionsetc.thestandard procedureisnot clear. Inthissection,wedefinetheCVAforportfoliosofderivativeswithtrueoptionalityaswellassomeapproximations,which simplifythecomputations.InthedefinitionsofCVA,weusetheportfoliovaluationsintermsofoptimallychosendecision

functionsgiveninequations(2.14)and(2.24).TheCVAat

(

t₌0_,X0=x0

)

,withandwithoutnetting,respectively,aregiven by CVA:=

ϒ

V



_0,x 0

|

fV



−

ϒ

U



_0,x 0,1

|

fU



,

(

withoutnetting

)

, (2.26) CVANet:=

ϒ

V



_0,x 0

|

fV



−

ϒ

A



_0,x 0,1,1 J

|

fA



,

(

withnetting

)

. (2.27)

Acommonlyusedapproximationistoapply thesameexercisestrategy totherisk-freeandrisky portfolios.Onesuch ap-proximationisdefinedas CVA:=

ϒ

V



_0,x 0

|

fV



−

ϒ

U



_0,x 0,1

|

fV



,

(

Risk-freestrategy,withoutnetting

)

, (2.28)

CVANet:=

ϒ

V



_0,x 0

|

fV



−

ϒ

A



_0,x 0,1,1 J

|

fV



,

(

Risk-free strategy,withnetting

)

. (2.29)

Theonlydifferencebetween(2.26)-(2.27)and(2.28)-(2.29)isthatinthelattertherisk-freestrategyisusedalsofortherisky portfolios.Onecouldalsothinkofotherdefinitions,e.g.,usingtheriskystrategiesforbothportfolios.Thisparticularchoice is motivatedbythefact that fU and fA are,ingeneral, dependenton fV through theclose-outagreements in(2.13)and

(2.22).Moreover, fV is a sub-optimal strategy for both risky portfolios leading to CVA≤ CVA,andCVANet≤ CVANet,which is beneficial for the bank (but certainly not forthe counterparty). If we instead use only the risky descision functions,

i.e., replacing fV with fU in (2.28) and fA in (2.29),we wouldobtain an underestimation of theCVAs, which wouldbe unacceptableforthebank.

AsmentionedintheIntroduction,thebankisexposedtotheriskofCVAlosses, asaconsequenceoftheMtMvalueof theCVA movingagainst thebank.WethereforewanttofollowtheevolutionoftheCVAovertime,to gaininsightsinits distribution.OfparticularinterestisthetaildistributionoftheCVA,fortimesbetweeninitialtimeandthematurityofthe portfolio.Toexplorethis,wedefinethedynamicversionsof(2.26)-(2.29),whicharestochasticprocessesdependingonthe market andportfoliostate processesX andA.Fort∈[0,T],thedynamicversionsoftheCVAs (andtheir approximations) aregivenbythefollowingrandomvariables

CVA

(

t,Xt,At

)

:= J  j=1



Vj



t,Xt

|

fVj



− Uj



t,Xt

|

fUj



(

At

)

j, CVAnet



t,Xt,At

)

:= J  j=1 Vj

(

t,Xt

|

fVj



(

At

)

j−

ϒ

A



t,Xt,1,At

|

fA



, CVA

(

t,Xt,At

)

:= J  j=1



Vj



t,Xt

|

fVj



− Uj



t,Xt

|

fVj



(

At

)

j, CVAnet

(

t,Xt,At

)

:= J  j=1 Vj

(

t,Xt

|

fVj



(

At

)

j−

ϒ

A



t,Xt,1,At

|

fV



.

Intheabove,theCVAisconditionalonthatthecounterpartyhasnotdefaultedpriorto,orat,t (itdoesnotmakesenseto calculatetheCVAifthecounterpartyhasalreadydefaulted).From theabove wecandefinetheExpectedvalueoftheCVA (E-CVA),andfor

α

_∈

(

0_,1

)

,the

α

_−levelofValueatRiskoftheCVA(VaR-CVA)andExpectedShortfalloftheCVA(ES-CVA),

E-CVA

(

t

)

:=E[CVA

(

t,Xt,At

)

|

1t=1], (2.30) VaR-CVA_α

(

t

)

:=inf



P∈R



Q



CVA

(

t,Xt,At

)

≤ P



≥

α



, (2.31)

ES-CVA_α

(

t

)

:=E



CVA

(

t,Xt,At

)



1t=1,CVA

(

t,Xt,At

)

≥ VaR-CVAα

(

t

)



. (2.32)

InasimilarwayE-CVA

(

t

)

,ES-CVA_α

(

t

)

,E-CVAnet

(

t

)

,ES-CVAnet_α

(

t

)

,E-CVAnet

(

t

)

and

ES-CVAnet_α

(

t

)

aredefined.TheexpressionfortheES-CVAlookscomplicatedbutisbasicallyjusttheexpectedvalueofthe

α

−tailoftheCVAdistribution.WefocusonES-CVAinsteadofVaR-CVAbecauseitisacoherentriskmeasureandVaR-CVA isnot.

Remark2.3. SinceES_{− CVA}isanon-tradedriskmeasure,itshouldideallybecomputedundertherealworldmeasureP, seee.g.,[25]foradetaileddiscussion.Tobeprecise,

(

Xt,At

)

shouldbegeneratedundertheP−measureand,theCVA,which isatradeableasset,shouldbecomputedundertheQ−measure.Itisstraightforwardtoadjustthealgorithmsinthispaper beabletocomputeES− CVAundertheP−measure,see[2]fordetailsinthespecialcaseJ=1.

2.5. Exposureprofiles

In thissubsectionwe discusstheconceptof exposureprofiles foraportfolio ofderivatives.Thefinancial exposure(of thebank)isdefinedasthemaximumamountthebankstandstolooseifthecounterpartydefaults.Theexposureprofileis looselydefinedasthedistributionoftheexposureovertime.Theexposures,withandwithoutnetting,aredefinedas

ENet_t :=max



J  j=1 Vj

(

t,Xt

)(

At

)

j,0



, Et:= J  j=1 max



Vj

(

t,Xt

)(

At

)

j,0

,

where we recall that

(

At

)

j=I_{τj>t} with

τ

j beingthe exercise datefor derivative j. Furthermore,fora portfolio without netting,theexpectedexposure(EE),andfor

α

_∈

(

0_,1

)

,thepotentialfutureexposure(PFE)aredefinedas

EE

(

t

)

:=E0



D0,tEt



, (2.33) PFE_α

(

t

)

:=inf



P∈R



Q



D0,tEt ≤ P



≥

α

. (2.34)

Boththeexpectation andtheprobability in(2.33) and(2.34) shouldbeinterpreted asconditional onX0=x0∈Rd.TheEE andPFEinthepresenceofnetting,denotedbyEENet

(

·

)

andPFENet_α

(

·

)

,andareobtainedbyinsteadusingthenettedexposure in(2.33)and(2.34).

IfweassumeaconstantrecoveryrateR∈[0,1

)

,andthatX and1D areindependent,i.e., thedefaulteventofthe coun-terpartyisindependentoftheriskfactors,then(2.28)and(2.29)canbewrittenas

CVA=

(

1− R

)

T 0 EE

(

t

)

Q



τ

D_∈[t+dt

₎



_{, CVA}Net₌

₍

_{1− R}

₎

T 0 EENet

(

t

)

Q



τ

D_∈[t+dt

₎



_, whichcanbeapproximatedby

CVA≈

(

1− R

)

M m=1 EE

(

tm

)

Q



τ

D_∈

₍

_t m−1,tm]



, CVANet≈

(

1− R

)

M  m=1 EENet

(

tm

)

Q



τ

D_∈

₍

_t m−1,tm]



,

forsomepartitionof[0,T],witht0=0andtM=T.Theaboveformulationsrequireaccesstothedensityofdefaultevents, but maybe moreaccurate, especially for large M and a small probability of default(with a simulation based approach, problemswithalowprobabilityofdefaultcanoftenbetackledwithvariancereductiontechniques).

3. Algorithms

In thefirst partofthissection, we presentaneural network-based methodto approximatethe decisionfunctions in-troducedintheprevioussection.ThemethodgeneralizestheDeepOptimalStoppingproposedin[1]andextendedin[2], whichapproximatesstoppingdecisionsforasinglederivative,tobeapplicablealsoforportfoliosofderivativeswith early-exercise features. Furthermore,forthe risky portfolios, the algorithmis extended to be able to deal withdefaultrisk of thecounterparty.The algorithmisbasedonaseriesofneuralnetworks, whichareoptimizedbackwardsintimewiththe objectivetomaximizetheexpecteddiscountedcash-flows.

Inthesecond partofthissection,theexercisepolicyobtainedfromtheapproximatedecisionfunctionsisapplied path-wiseonrealizationsoftheriskfactorsofeachderivativeintheportfoliotogeneratepathwisecash-flows.Thesecash-flows areusedinaneuralnetworkbasedregressionalgorithmtoapproximatepathwisederivativevalues.Thesepathwise deriva-tivevaluescanthenbeusedtocomputeimportantriskmanagementmeasures.

Due to the problemformulation which giveshighdimensionality both in theunderlying assetand inthe number of derivativesintheportfolio,thenotationmaybesomewhatdifficulttofollow.Wethereforereferto[2]fora,perhaps,more straightforwardintroductiontothealgorithmsforasinglederivative.

3.1. Phasei:Learningexercisestrategy

As indicatedabove,thecoreofthealgorithm istoapproximatedecisionfunctions,inordertoobtaingood approxima-tions ofthevalueofaportfolioofderivatives.Weapproximatethedecisionfunctions fV, fU and fA,withfullyconnected neuralnetworks.Tobemoreprecise,letN=

|

T

(

0

)

|

,forn∈

{

1,2,...,N

}

andforZ∈

{

V,U

}

,thedecisionfunction fZ

(

Tn,·

)

, isapproximatedbyafullyconnectedneuralnetworkoftheform fθn_:Rd_→

{

_0,1

}

J_,where

θ

n∈Rqn isavectorcontaining all the qn∈Ntrainable parameters innetwork n.The decisionfunction fA

(

Tn,·,·

)

isapproximated by similarneural net-works, withthe onlydifference that theinput also includes informationof which derivativesin theportfolio have been exercisedpriortoTn,i.e., fθn:Rd×

{

0,1

}

J→

{

0,1

}

J.

Sincebinarydecisionfunctionsarediscontinuous,andthereforeunsuitableforgradient-typeoptimizationalgorithms,we useasan intermediate step,theneuralnetwork Fθn_:Rd_→

₍

_0,1

₎

J_{.Instead ofabinarydecision,the outputofthe neural} networkFθn _{canbeviewedastheprobability}4 _{forexercisetobeoptimal.Thisoutputisthenmappedto1forvaluesabove}

(or equalto)0.5, andto 0otherwise,bydefining fθn

₍

_·

₎

_{=a◦ F}θn

(

_·

)

_{,wherea isacomponent-wiseround-off function,i.e.,} for j∈

{

1,2,...,J

}

,andx∈Rd_{,the j:thcomponentofa}

₍

_x

₎

_isgivenby

₍

_a

₍

_x

₎₎

j=I{x_j≥1/2}.ForeachZ∈

{

V,U

}

,ouraimisto adjusttheparameters

θ

1,

θ

2,...,

θ

N suchthat

(

fZ

(

T1,·

)

,fZ

(

T2,·

)

,...,fZ

(

TN,·

))

T≈

(

fθ1,fθ2,...,fθN

)

T=:f, (3.1)

(

fA

(

T1,·,·

)

,fA

(

T2,·,·

)

,...,fA

(

TN,·,·

))

,T≈

(

fθ1,fθ2,...,fθN

)

T=:f, (3.2) where werecall that



₌

{

θ

1,

θ

2,...,

θ

N

}

.Forn∈

{

1,2,...,N

}

,we define thesequence ofneuralnetworks, approximating thedecisionfunctionsatexercisedatesTn,Tn+1,...TN,byfn :=

(

fθn,fθn+1,...,fθN

)

T.Notethattheinputdimensionforthe neural networks is differentwhen we want to approximate fV and fU compared to when we want to approximate fA. To avoidhavingto introduce an extralayer ofnotation, we useforall networks



to denotethe setofparameters, and keepinmindthatthedimensiondependsonthespecificproblemconsidered.Althoughtheaboveprovidesagoodintuition forwhatwe wanttoaccomplish,itisnotclearinwhichsense wewantthefunctionstobesimilar, orhowtoadjustthe parameterstoachievethis.Toapproachamoretractableform,fromacomputationalperspective,fort∈

(

Tn−1,Tn],weinsert

(3.1)in(2.10)and(3.2)in(2.20)to obtain

τ

[fn]

(

Xt,x

)

= N  k=n Tkfθk

(

Xk

)

 N  m=k



1 J− fθm

(

Xm

)



, (3.3)

τ

[f_n]

(

Xt,x_,At,α

₎

₌N k=n Tkfθk

(

Xk,Ak

)

 N  m=k



1 J− fθm

(

Xm,Am

)



. (3.4)

Note that(3.3) isa J_{−dimensional}vector ofX_−stopping timesand(3.4) isa J_{−dimensional}

(

X_,A

)

_−stoppingtimesvector, whichdependsonf_n onastructurallevelbutalsoontherandomnessofthestochasticprocessXt,x _(andAt,α_{for(3.4)).For} notational convenience, we usetheshort handnotation

τ

¯_n =

τ

[f_n]

(

Xt,x

₎

_(or

_τ

_ˆ

n =

τ

[fn]

(

Xt,x,At,α

)

,when approximating fA), andforelement j∈

{

1,2,...,J

}

,

τ

¯_{n, j}=

(

τ

¯_n

)

j (or

τ

ˆn, j=

(

τ

ˆn

)

j). Weare now readyto define ourobjective,which,for

Tn∈T

(

0

)

,istofind

θ

nsuchthattheexpectedfuturecash-flowsaremaximized.Thecash-flowscanbedividedintothree categories:

1. Thecash-flowsobtainedbythederivativesexercisedatthepresenttimeTn; 2. Thecash-flowsobtainedatlaterexercisedatespriortodefaultofthecounterparty;

3. Thecash-flowsobtainedatdefaultofthecounterparty,accordingtotheclose-outagreement. Forn∈

{

1,2,...,N

}

and j∈

{

1,2,...,J

}

,wedenotedimension j ofdecisionfunction fθn _by



fθn



j=fθnj , and



Fθn



j=Fjθn

Given that no default has occurred prior to Tn, (and the exercise state An=

α

for the risky portfolio with netting) the expectedcash-flows,thatwewanttomaximize,aregivenbelow.

Risk-freeportfolio: ETn



_J j=1 fθn_j

(

Xn

)

gj

(

Tn,Xn

)

+



1− f_jθn

(

Xn

)



D_Tn,τ¯V n+1, jgj



¯

τ

V n+1, j,Xτ¯V n+1, j



, (3.5)

Riskyportfoliowithoutnetting: ETn



J j=1fθnj

(

Xn

)

gj

(

Tn,Xn

)

+



1− fθn

(

_X n

)



1D_τ¯U n+1, jDTn,τ¯ U n+1, jgj



¯

τ

U n+1, j,Xτ¯U n+1, j



+



1−1D_τ¯U n+1, j



D_t,τD



RVj

(

τ

D,X_τD

)

++V_j

(

τ

D,X_τD

)

−



, (3.6)

4 However the interpretation as a probability may be helpful, one should be careful since it is not a rigorous mathematical statement. It should be clear that there is nothing random about the stopping decisions, since the stopping time is H t −measurable. It can also be interpreted as a measure on how certain we can be that exercise is optimal.

Riskyportfoliowithnetting: ETn



J j=1

α

jfθn_j

(

Xn,

α

)

gj

(

Tn,Xn

)

+

α

j



1− f_jθn

(

Xn,

α

)



1D_τˆU n+1, jDTn,τˆ U n+1, jgj



ˆ

τ

U n+1, j,XτˆU n+1, j



+R



_ J k=1

α

k



1− fθn k

(

Xn,

α

)



1−1D_τˆU n+1,k



D_t,τDV_k

(

τ

D,X_τD

)



+ +



J =1

α





1− fθn 

(

Xn,

α

)



1−1D_τˆU n+1,



Dt,τDV_

(

τ

D,XτD

)



−



. (3.7)

Wewanttooptimize

θ

n,suchthattheaboveareascloseaspossible(inmeansquaredsense)to



V

(

Tn,Xn

)

,



U

(

Tn,Xn,1

)

and



A

₍

_T

n,Xn,1,

α

)

,respectively.

Remark3.1. Theobjectivesfortheriskyportfolios,in(3.6)and(3.7),bothdependontherisk-freevaluation ofthe deriva-tives. Therefore,inorder toapproximate therisky decisionfunctions, we firstneed to approximatethe risk-freeexercise strategy,andtherisk-freederivativevalues.Inthenextsubsection,weexplainhowVjcanbeapproximated.

Although(3.5)-(3.7)areaccuraterepresentationsoftheoptimizationproblems,theygiveussomepracticalproblems.In general, we havenoaccessto fV, fU and fA whichcontrol

τ

¯V_n+1,

τ

¯U_n+1 and

τ

ˆ_nA₊₁.Anotherproblemisthat,in general,we havenoaccesstothetruedistributionsoftheportfoliovaluesforcomparison.However,ifTN˜ isthematurity ofderivative

j∈

{

1,2,...,J

}

,itisoptimaltoexercise aslongasthepay-off valueispositive,bythedefinitionofthedecisionfunctions. Wecanthereforeset

fθ˜N

j

(

·

)

:=I{gj(TN˜,·)>0}, and f θk

j

(

·

)

:=0, fork>N˜.

Furthermore,atTN,thematurityoftheportfolio,thepositivepartofthepay-off valueequalsthederivativevalue(ifno defaultin

(

TN−1,TN],intheriskycases).AtTN−1,Equation(3.5)thenbecomes

ETN−1



_J j=1 fθN−1_j

(

XN

)

gj

(

TN−1,XN−1

)

+DN−1,N



1− fθN−1j

(

XN−1

)



gj

(

TN,XN

)

. (3.8)

Recall thatifTN isgreaterthanthe maturityofcontract j,wehavegj

(

TN,·

)

≡ 0.Sinceall componentsin(3.8) areknown exceptforthedecisionfunction fθN−1_{,wewanttofind}

θ

_N

−1,suchthataMonte-Carloapproximationsof(3.8)ismaximized. GivenM∈Nsamples,distributedasX,whichform∈

{

1,2,...,M

}

isdenotedbyx=

(

xt

(

m

))

t∈[0,T],weapproximate(3.8)by

1 M M  m=1 J  j=1 fθN−1 j

(

xN−1

(

m

))

gj

(

TN−1,xN−1

(

m

))

+DN−1,N



1− fjθN−1

(

xN−1

(

m

))



gj

(

TN,xN

(

m

)

)

. (3.9)

Theonlyunknownentityin(3.9)istheparameter

θ

N−1inthedecisionfunction fθN−1=

(

f1θN−1,...,f

θN−1

j

)

T.Furthermore,we wishtofind

θ

N−1suchthat(3.9)ismaximized,sinceitrepresentstheaveragecash-flowin[tN−1,tN].Once

θ

N−1isoptimized, weusethisparametertosetupasimilarexpressionfortheexpectedcash-flowon[tN−2,tN],whichismaximizedbyfinding an optimal

θ

N−2.Thisprocedureis theniteratively continueduntilalso

θ

N−3,

θ

N−4,...,

θ

1 are optimized.Theprocedure is similarfortheriskyportfolios,butbasedon(3.6)or(3.7) instead.Thisimpliesthatwealsoneedtosampledefaultevents ofthecounterparty.Wedenoteby

θ

n∗theoptimizedversionofparameter

θ

nandthesequenceofoptimizedparametersfor thenetworksatexercisedatesTn,Tn+1,...,TNaredefinedas



∗

n:=

{

θ

n∗,

θ

n∗+1,...,

θ

N∗

}

,

andfornotationalconvenience,wedefinethecompletesequenceofparametersas



∗:=



∗ 1.

Remark3.2. Sinceweareconsideringaportfolioinwhichallthederivativesmayhaveadifferentsetofexercisedates,we havethatforTn∈T

(

0

)

,thereareJnEx∈

{

1,2,...,J

}

derivativesthatmaybeexercised.Therefore,weonlyneedtocompute

JEx

n ofthe J dimensions of fθn andcan by defaultset theremaining J− JnEx dimensionsof fθn to0. Thiscan be done by appropriatelyadjustingsomeweightsandbiases.

Tokeeptheflowofthepaper,thedetailsofthealgorithmsandtheparameters

θ

naregivenintheAppendix.

3.2. PhaseII:Learningpathwisederivativevaluesandportfolioexposures

Asmentioned inRemark3.1,therisk-freederivativevaluesneedtobeapproximatedpathwiseinordertoapproximate theriskydecisionfunctions.Moreover,thepathwisederivativevaluesarerequiredtoapproximatetheexposureprofilesfor the risk-free,as well astherisky portfolios. Inthissubsection, we focuson therisk-free portfolios, buttheextension to riskyportfoliosisstraightforward.