A unified approach to xVA with CSA discounting and initial margin
PPRICING OF COUNTERPARTY RISK AND FUNDING WITH CSADISCOUNTING, PORTFOLIO EFFECTS AND INITIAL MARGIN.
FRANCESCA BIAGINI, ALESSANDRO GNOATTO, AND IMMACOLATA OLIVAApril 14, 2020
Abstract.
In this paper we extend the existing literature on xVA along three directions. First, weenhance current BSDE-based xVA frameworks to include initial margin by following the approach ofCr´epey (2015a) and Cr´epey (2015b). Next, we solve the consistency problem that arises when the front-office desk of the bank uses trade-specific discount curves that differ from the discount curve adoptedby the xVA desk. Finally, we address the existence of multiple aggregation levels for contingent claimsin the portfolio between the bank and the counterparty, providing suitable extensions of our proposedsingle-claim xVA framework. Introduction
As a consequence of the 2007-2009 financial crisis, academics and practitioners are revisiting thevaluation of financial products in several aspects. In particular, the value of a product should accountfor the possibility of default of any agent involved in the transaction. Also the trading activity isfunded by resorting on different sources of liquidity. This results in the interest rate multi-curvephenomenon, so that the existence of a unique source of funding, growing at a risk-free interest rate,no longer represents a realistic assumption. Financial regulations, such as Basel III/IV and Emir,are also driving the methodological development. Regulations on collateral imply an increasinglyimportant role of central counterparties.All these issues are represented at the level of valuation equations by introducing value adjustments (xVA), which are further terms to be added or subtracted to an idealized reference price, computedin the absence of the aforementioned frictions, in order to obtain the final value of the transaction.In this work, we propose an xVA framework using BSDEs techniques in a market described by dif-fusions. We revisit concepts such as the self-financing property, absence of arbitrage and replicationof contingent claims in a market with frictions, due to the presence of counterparty risk and multiplefunding curves. Our replication BSDE, introduced under a classical enlarged filtration, is specified upto a random time horizon, given by the minimum between the default time of the counterparty, thedefault time of the bank, and the natural maturity of the contract. We discuss the well posedness ofthe BSDE by considering the associated pre-default BSDE under a reduced filtration along the linesof Cr´epey (2015a), Cr´epey (2015b), Bichuch et al. (2018a), Bielecki and Rutkowski (2015) and Brigoet al. (2018).Given the xVA framework for a single transaction, we then consider the consistency problem betweenxVA pricing equations and the CSA discounting rules. The latter originate from the quoting mecha-nism of market standard instruments. Such instruments are quoted under the assumption that they
Mathematics Subject Classification.
JEL Classification
E43, G12.
Key words and phrases.
CVA, DVA, FVA, CollVA, xVA, EPE, Basel III, Collateral, Initial Margin.
Acknowledgements.
The authors are grateful to Damiano Brigo, Agostino Capponi, Andrea Pallavicini, and the partic-ipants to the SIAM Conference on Financial Mathematics & Engineering in Toronto, Canada, for their comments andsuggestions. Any errors remain the responsibility of the authors. a r X i v : . [ q -f i n . P R ] A p r FRANCESCA BIAGINI, ALESSANDRO GNOATTO, AND IMMACOLATA OLIVA are perfectly collateralized. Since a perfectly collateralized transaction is funded by the collateralprovider, the discounting rate applied to evaluate market instruments is given by a collateral rate,which typically corresponds to an overnight interest rate. The presence of multiple assumptions on thecollateral rate implies the co-existence of quotes with different discounting rates, which are in generalat odds with the unique discounting rate dictated by the xVA pricing BSDE. We solve the consistencyissue by relying on an invariance property of linear BSDEs.A further contribution of our work is the discussion of multiple aggregation levels within the portfolioof claims between the bank and the counterparty. We study the impact of such aggregation levels onthe formulation of pricing equations. In realistic situations, the global portfolio of positions betweenthe bank and the counterparty is in general an aggregation of several subsets of claims, where thestructure of the tree of subsets is dictated by the legal agreements in force between the two agents. Weallow for the presence of multiple agreements for the exchange of margins (margin sets) and multiplenetting sets. We adapt our xVA framework to arbitrary configurations of aggregation levels, whilepreserving the well posedness of the underlying portfolio-wide BSDE.Finally, we present incremental xVA charges for new potential trades under the proposed xVA frame-work: given the presence of portfolio effects in the computation of value adjustments, and given anexisting portfolio of K trades, the xVA charge for a new potential ( K + 1)-th trade is computed as thedifference of the xVAs of two portfolios. In particular, this corresponds to the difference between thexVA of the extended portfolio, consisting of ( K + 1) trades, and the xVA charge of the base portfolioof K trades. Such an approach represents an effective way to describe the non-linearity effect existingin the financial industry framework. Existing discussions concerning the impact of the incrementalcash flow of a trade on the balance sheet of the dealer can be found in Burgard and Kjaer (2013),Castagna (2013), Castagna (2014).Given our focus on discounting and aggregation levels, in this paper we do not discuss capital valuationadjustment (KVA). The issue is treated in recent papers such as Albanese and Cr´epey (2017), Albaneseet al. (2016) and Albanese et al. (2017), see also the ongoing discussion in Andersen et al. (2019).This is beyond the scope of the present paper and leave it for future research.The results of our paper fill a gap in the literature on counterparty credit risk and funding. Herewe present an overview providing insights on the main contributions in this field. Possibly, the firstcontribution on the subject is Duffie and Huang (1996). Before the 2007–2009 financial crisis, wemention the works of Brigo and Masetti (2005) and Cherubini (2005), where the concept of credit val-uation adjustment (CVA) is analyzed. The possibility of default of both counterparties involved in thetransaction, represented by the introduction of the debt valuation adjustment (DVA), is investigated,among others, in Brigo et al. (2011) and Brigo et al. (2014).Apart from the issue of default risk, another important source of concern for practitioners and aca-demics is represented by funding costs. A parallel stream of literature emerged during and after thefinancial crisis, to generalize valuation equations to account for features such as the presence of collat-eralization agreements. In a Black-Scholes economy, Piterbarg (2010) provides valuation formulas inpresence or absence of collateral agreements. Piterbarg (2012) generalizes the issue in a multi-currencyeconomy, see also Fujii et al. (2010) and Fujii et al. (2011). The funding valuation adjustment (FVA)under several alternative assumptions on the Credit Support Annex (CSA) is derived in Pallaviciniet al. (2011), while Brigo and Pallavicini (2014) also discusses the role of central counterparties in thecontext of funding costs. A general approach to funding issues in a semimartingale setting is providedby Bielecki and Rutkowski (2015). SDES OF XVA 3
Both funding and default risk need to be unified in a unique pricing framework. Contributions in thissense can be found in Brigo et al. (2018) by means of the so-called discounting approach . In a series ofpapers, Burgard and Kjaer generalize the classical Black-Scholes replication approach to include manyeffects, see Burgard and Kjaer (2011) and Burgard and Kjaer (2013). A more general BSDE approachis provided by Cr´epey (2015a), Cr´epey (2015b), Bichuch et al. (2018a) and Bichuch et al. (2018b).The equivalence between the discounting and the BSDE-based replication approaches is demonstratedin Brigo et al. (2018).The importance of the topic is reflected by the increasing number of monographs on the subject, seee.g. Brigo et al. (2013). An advanced BSDE-based treatment is provided by Cr´epey et al. (2014).A detailed analysis of how to construct large hybrid models for counterparty risk simulations areprovided in Green (2015), Lichters et al. (2015) and Sokol (2014), while Gregory (2015) provides anaccessible introduction to most aspects of the topic.The paper is organized as follows. In Section 2 we formalize in mathematical terms the main financialconcepts related to the xVA framework. Section 3 describes the results related to the xVA evaluationwhen only one transaction is taken into account. In Section 4 we extend the xVA framework, aswell as the related mathematical background, to the case in which the portfolio consists of multiplecontracts. Section 5 provides an example and some numerical results illustrating most of the previouslyintroduced concepts. In Appendix A we gather some results from the literature used to derive themain results. 2.
The financial setting
We fix a time horizon
T < ∞ for the trading activity. We consider two agents named the bank (B) and the counterparty (C). All processes are modeled over a probability space (Ω , G , G , P ) , where G = ( G t ) t ∈ [0 ,T ] ⊆ G is a filtration satisfying the usual assumptions. Here G is assumed to be trivial.We denote by τ B , resp. by τ C , the time of default of the bank, resp. of the counterparty. Remark . Unless otherwise stated, throughout the paper we assume the bank’s perspective andrefer to the bank as the hedger.
We assume that G = F ∨ H , where F = ( F t ) t ∈ [0 ,T ] is a reference filtration satisfying the usualhypotheses and H = H B ∨ H C , with H j = (cid:16) H jt (cid:17) t ∈ [0 ,T ] for H jt = σ ( H u | u ≤ t ), and H jt := 1 { τ j ≤ t } , j ∈{ B, C } . We set(2.1) τ = τ C ∧ τ B . Remark . We use the following conventions: x + := max { x, } , x − := max {− x, } so that x = x + − x − . Note that this is in contrast to the convention adopted e.g. in Burgard and Kjaer (2011,2013).In the present paper we will extensively make use of the so called
Immersion Hypothesis.
Hypothesis 2.3.
Any local ( F , P ) -martingale is a local ( G , P ) -martingale. We introduce some useful spaces of processes.
Definition 2.4.
Let Q be a probability measure on (Ω , G ) and β ≥ . The subspace of all R d -valued, F -adapted processes X such that E Q (cid:20)(cid:90) T e βt | X t | dt (cid:21) < ∞ (2.2) FRANCESCA BIAGINI, ALESSANDRO GNOATTO, AND IMMACOLATA OLIVA is denoted by H ,dβ,T ( Q ) . We set H ( Q ) := H , ,T ( Q ) . Moreover, (2.3) (cid:107) X (cid:107) H ,dβ,T = (cid:115) E Q (cid:20)(cid:90) T e βt | X t | dt (cid:21) . The subspace of all R d -valued, continuous F -adapted processes X such that E Q (cid:34) sup t ∈ [0 ,T ] e βt | X t | (cid:35) < ∞ (2.4) is denoted by S ,dβ,T ( Q ) . We set S ( Q ) := S , β,T ( Q ) . Moreover, (2.5) (cid:107) X (cid:107) S ,dβ,T = (cid:118)(cid:117)(cid:117)(cid:116) E Q (cid:34) sup t ∈ [0 ,T ] e βt | X t | (cid:35) . Basic traded assets.
Risky assets.
For d ≥ , we denote by S i , i = 1 , . . . , d the ex-dividend price (i.e. the price) ofrisky securities with associated cumulative dividend processes D i . All S i are assumed to be c`adl`ag F -semimartingales, while the cumulative dividend streams D i are F -adapted processes of finite variationwith D i = 0.Let W P = (cid:0) W P t (cid:1) t ∈ [0 ,T ] be a d -dimensional ( F , P )-Brownian motion (hence a ( G , P )-Brownian motion,thanks to Hypothesis 2.3). We introduce the following coefficient functions: µ : (cid:16) R + × R d , B (cid:16) R + × R d (cid:17)(cid:17) (cid:55)→ (cid:16) R d , B (cid:16) R d (cid:17)(cid:17) ,σ : (cid:16) R + × R d , B (cid:16) R + × R d (cid:17)(cid:17) (cid:55)→ (cid:16) R d × d , B (cid:16) R d × d (cid:17)(cid:17) ,κ : (cid:16) R + × R d , B (cid:16) R + × R d (cid:17)(cid:17) (cid:55)→ (cid:16) R d , B (cid:16) R d (cid:17)(cid:17) , (2.6)which are assumed to satisfy standard conditions ensuring existence and uniqueness of strong solutionsof SDEs driven by the Brownian motion W P . The matrix process σ is assumed to be invertible atevery point in time. We assume that dS t = µ ( t, S t ) dt + σ ( t, S t ) dW P t S = s ∈ R d (2.7)on [0 , T ] . Note that we are not postulating that the processes S i are positive. The dividend processes D i are specified via(2.8) ( D t , . . . , D dt ) (cid:62) = (cid:90) t κ ( u, S u ) du, t ∈ [0 , T ] , for κ given in (2.6) such that (cid:82) T | κ ( u, S u ) | du < ∞ P -a.s.2.1.2. Cash accounts.
We assume the existence of an indexed family of cash accounts ( B x ) x ∈ I , wherethe stochastic process r x := ( r xt ) t ≥ is bounded, right-continuous and F -adapted for all x ∈ I . Theset of indices I embodies the type of agreement the counterparties establish in order to mitigate thecounterparty credit risk. We will specify the characteristics of the aforementioned indices later on.All cash accounts, with unitary value at time 0, are assumed to be strictly positive continuous processesof finite variation of the form B xt := exp (cid:26)(cid:90) t r xs ds (cid:27) , t ∈ [0 , T ] . (2.9) SDES OF XVA 5
In particular, B x := ( B xt ) t ∈ [0 ,T ] is also continuous and adapted for all x ∈ I . Defaultable bonds.
Default times are assumed to be exponentially distributed random variableswith time-dependent intensity Γ jt = (cid:90) t λ js ds, j ∈ { B, C } , t ∈ [0 , T ] , where λ j are non-negative measurable bounded deterministic functions such that (cid:90) T λ js ds < ∞ , ∀ t ≥ , j ∈ { B, C } . We introduce two risky bonds with maturity T (cid:63) ≤ T and rate of return ¯ r j + λ j , issued by the bankand the counterparty, with dynamics dP jt = (cid:16) ¯ r jt + λ jt (cid:17) P jt dt − P jt − dH jt , P j = e − (cid:82) T(cid:63) ∧ τj (¯ r ju + λ ju ) du , j ∈ { B, C } . (2.10)2.2. Repo trading.
In line with the existing literature, we assume that the trading activity onthe risky assets is collateralized.
This means that borrowing and lending activities related to riskysecurities are financed via security lending or repo market.
We refer to Bichuch et al. (2018a) foran illustration of cash-driven and security driven repo transactions. Since transactions on the repomarket are collateralized by the risky assets, repo rates are lower than unsecured funding rates. Asargued in Cr´epey (2015a), assuming that all assets are traded via repo markets is not restrictive.We let B , . . . , B d be the cash accounts associated to the risky assets S , . . . , S d .In case that the transactions are fully collateralized, this translates in the following equality ξ it S it + ψ it B it = 0 , i = 1 . . . , d, t ∈ [0 , T ] . (2.11) Remark . It is worth noting that ξ it , i = 1 , . . . , d, may be either positive or negative. Here ξ it > ξ it < i -th asset is shorted, so that the whole amount of collateral is depositedin the riskless asset. Remark . Condition (2.11) plays an important role in precluding trivial arbitrage opportunitiesamong different cash accounts. In fact, by assuming ξ it = 1 for all i = 1 , . . . , d in (2.11), the gainprocess G t = ( G t ) t ∈ [0 ,T ] has dynamics dG it = r it ψ it B it dt + dS it + dD it = dS it − r it S it dt + dD it , (2.12)where D i , i = 1 , . . . , d, is defined in (2.8).Analogously, positions in the bonds satisfy the following condition(2.13) ξ jt P jt + ψ jt B jt = 0 , j ∈ { B, C } , t ∈ [0 , T ] . Alternatively, it is possible to assume that the trading activity on bonds is financed via unsecuredfunding as in Bichuch et al. (2018a). This implies the appearence of further positions in risky bondsin the FVA terms. Our choice results in FVA terms whose form is closer to market standard formulasas provided e.g. in Burgard and Kjaer (2013).It is worth noting that in (2.11) and (2.13) ξ, ψ serve as portion of traded securities constituting theportfolio. The properties of such processes will be pointed out in Section 3.
FRANCESCA BIAGINI, ALESSANDRO GNOATTO, AND IMMACOLATA OLIVA
Unsecured funding account.
Within the bank, the trading desk borrows and lends moneyfrom/to the treasury desk. Borrowing and lending rates are allowed to differ, hence we denote by r f,b , r f,l the rate at which the trading desk borrows from and lends to the treasury desk, respectively.Recalling the notation given in (2.9), we introduce the associated cash accounts B f,b , B f,l . This meansthat if the position of the trading desk is negative, i.e. ψ f = ψ f,b < , the trading desk borrowsfrom the treasury desk at the rate r f,b . Conversely, if the position of the trading desk is positive, i.e. ψ f = ψ f,l > , the trading desk lends money to the treasury desk with remuneration r f,l . Remark . It is worth observing that simultaneously borrowing and lending from the treasury deskis precluded, so we set ψ f,lt ψ f,bt = 0 for all t ∈ [0 , T ] . Collateralization.
In the financial jargon, a margin represents an economic value, either in theform of cash or risky securities, exchanged between the counterparties of a financial transaction, inorder to reduce their outstanding risk exposures. In line with the market practice, we distinguishbetween initial margin and collateral (or variation margin ), that we present in what follows.2.4.1.
Variation margin.
A collateral is posted between the bank and the counterparty to mitigatecounterparty risk. The collateral process C = ( C t ) t ∈ [0 ,T ] is assumed to be G -adapted. We follow theconvention of Bichuch et al. (2018a) and Cr´epey (2015a): • If C t > , we say that the bank is the collateral provider. It means that the counterpartymeasures a positive exposure towards the bank, so it is a potential lender to the bank, hencethe bank provides/lends collateral to reduce its exposure. • If C t < , we say that the bank is the collateral taker. It means that the bank measures apositive exposure towards the counterparty, so it is a potential lender to the counterparty,hence the counterparty provides/lends collateral to reduce its exposure.Let V = ( V t ) t ∈ [0 ,T ] be a generic G -adapted process, representing either the value of the trade includingcounterparty risk and funding adjustments or the clean value process, as it will be clarified later on.We assume that C t := f ( V t ) , t ∈ [0 , T ] , where f : R → R is a Lipschitz function. This assumptionallows to cover realistic collateral specifications, see e.g. Lichters et al. (2015) and Ballotta et al.(2019).If there is a collateral agreement (or a multitude of agreements) between the bank and the counterparty,in evaluating portfolio dynamics we need to make a distinction between the value of the portfolio andthe wealth of the bank, the two concepts being distinguished since the bank is not the legal owner ofthe collateral (prior to default).In this paper collateral is always posted in the form of cash, in line with standard market practice.Moreover, we assume rehypothecation, meaning that the holder of collateral can use the cash to financeher trading activity. This is the opposite of segregation, where the received cash collateral must bekept in a separate account and can not be used to finance the purchase of assets.In line with Section 2.3, we associate the following interest rates to the collateral account: • r c,l with account B c,l , representing the rate on the collateral amount received by the bankwho posted collateral to the counterparty. • r c,b with account B c,b , representing the rate on the collateral amount paid by the bank whoreceived collateral from the counterparty.We simply set r c = r c,l = r c,b in case there is no bid-offer spread in the collateral rate. Possiblechoices for the collateral rate are e.g. EONIA for EUR trades, Fed Fund for USD and SONIA forGBP trades. Such rates are overnight rates with a negligible embedded risk component. The choice SDES OF XVA 7 of such approximately risk-free rates as collateral rates is motivated by market consensus. However,two counterparties might enter a collateral agreement that involves a remuneration of collateral atany other risky rate of their choice. Here we do not assume any requirements on collateral rates.This allows us to cover the quite common situation where the collateral rate agreed between the twocounterparties in the CSA is defined by including a real valued spread over some market publiclyobserved rate, e.g. EONIA − bps, where bps stands for basis points. For the collateral account we have the following equations:(i) if C t >
0, then the bank has lent ψ ct = ψ c,lt < ψ c,lt B c,lt = − C + t , t ∈ [0 , T ];(ii) if C t <
0, then the bank has borrowed ψ ct = ψ c,bt > ψ c,bt B c,bt = C − t , t ∈ [0 , T ] . Initial margin.
The collateralization represented by the variation margin is imperfect, due tothe margin period of risk phenomenon: a defaulted counterparty stops posting collateral. However,bankruptcy procedure requires a certain time interval (typically 10 or 20 days) before the close-outpayments are performed. This results in a period of time where the value of the transaction oscillatesin the absence of an adjustment of the collateral account, hence producing an exposure. This is oneof the reasons for the introduction of initial margins, which constitutes a further form of collateral.According to the EMIR regulation, starting from 2020, most agents participating in an OTC trans-action will be forced to post initial margin, which constitutes an additional form of collateral. Initialmargin, according to Garcia Trillos et al. (2016) is a misnomer, as an initial margin is not only initial,but it is periodically updated during the lifetime of the trade. It is initial in the sense that it is meantto provide a coverage from the initial point in time, where there is a default of the counterparty in acollateralized transaction.It is important to stress that, differently from variation margin, an initial margin can not be rehy-pothecated, but it is instead segregated. From the point of view of the wealth dynamics, this meansthat initial margin received from the counterparty can not be used by the trading desk as a compo-nent of the value of the portfolio. However, the received initial margin represents a loan from thecounterparty that must be remunerated, hence funding costs related to initial margin will appear inthe self-financing condition, see Section 3.1 for further details.We model initial margins with G -adapted processes I T C = ( I T Ct ) t ∈ [0 ,T ] , I F C = ( I F Ct ) t ∈ [0 ,T ] , and wedenote by B I,x , x ∈ { l, b } , the cash accounts associated to I T C , I
F C , respectively.In case the bank is the initial margin provider, the bank posts I T C to the counterparty (TC), i.e.(2.16) ψ I,lt B I,lt = − I T Ct , t ∈ [0 , T ] , or equivalently(2.17) − ψ I,lt dB I,lt = r I,lt I T Ct dt .
In case the bank is the initial margin taker, the bank receives I F C from the counterparty (FC), i.e.(2.18) ψ I,bt B I,bt = I F Ct , t ∈ [0 , T ] , FRANCESCA BIAGINI, ALESSANDRO GNOATTO, AND IMMACOLATA OLIVA or equivalently(2.19) ψ I,bt dB I,bt = r I,bt I F Ct dt .
We highlight that, contrary to the case of variation margin and in line with market practice, I T C and I F C are simultaneously active and do not net each other.More precisely, initial margins are computed via stochastic processes with values in the space of riskmeasures, such as value at risk or expected shortfall, as we will specify in (3.29). Expected shortfallis a popular choice to compute the initial margin for credit derivatives, since it is a coherent riskmeasure. Recently, the International Swaps and Derivatives Association (ISDA) has proposed a novelmethodology, the so called
Standard Initial Margin Model (SIMM), see ISDA (2018). SIMM providessome standardized formulae to evaluate initial margin on non-cleared derivatives, based on usingportfolio sensitivities instead of historical simulations.From a computational point of view, the presence of a risk measure inside portfolio dynamics resultsin an increased complexity both from a theoretical and computational point of view: since xVAequations are generally solved by means of Monte Carlo simulation, a brute force computation offuture initial margin profiles requires nested historical simulation inside the risk neutral forward MonteCarlo simulation. There is a significant stream of research regarding efficient methodologies for theestimation of future initial margin profiles, a popular technique being given by adjoint algorithmicdifferentiation (AAD), see e.g. Fries et al. (2018), Fries (2019b), Fries (2019a), Antonov et al. (2017),Henrard (2017), Capriotti (2011) and references therein.Such issues are beyond the scope of the present study. For our purposes, we assume that the initialmargin is a given real-valued process which is regular enough to guarantee existence and uniquenessof the BSDEs we are going to consider.Another peculiar feature of initial margins is that, in case the counterparty is a clearing house, thenthe bank is always initial margin provider, i.e. I F C = 0 d P ⊗ dt -a.s.In the sequel we use I = ( I t ) t ∈ [0 ,T ] as shorthand for both presented or received initial margins.2.5. Contingent claims.
We introduce the process A = ( A t ) t ∈ [0 ,T ] representing the payment streamof a financial contract. The process A is assumed to be an F -adapted c`adl`ag process of finite variation,as in Cr´epey (2015b). We use the notation ∆ A t := A t − A t − for the jumps of A .The following assumption will be useful later on. Assumption 2.8.
Assume that A ∈ S ( Q ) and A T ∈ L ( F T , Q ) . Remark . Differently from the standard literature, see e.g. Agarwal et al. (2018), we requirestronger integrability conditions for the process A, because of the possible presence of jumps.To include the more general case in which the presence of default events is assumed, we define theprocess ¯ A = (cid:0) ¯ A t (cid:1) t ∈ [0 ,T ] by setting(2.20) ¯ A t := 1 { t<τ } A t + 1 { t ≥ τ } A τ − , where we recall that τ := τ C ∧ τ B .2.6. The close-out condition.
In case of default, cashflows are exchanged between the survivingagent and the liquidators of the defaulted agent. Here we use the term agent as a placeholder forthe bank or for the counterparty. Due to the exchange of cashflows at default time, agents need toperform a valuation of the position at a random time. The object of the analysis can be the value inthe absence of counterparty risk (referred to in the literature as risk-free close-out ) or the value of the
SDES OF XVA 9 trade including the price adjustments due to counterparty risk and funding ( risky close-out ), see e.g.Brigo and Morini (2018). A risky close-out condition guarantees that the surviving counterparty canideally fully substitute the transaction with a new trade entered with another counterparty with thesame credit quality. This comes at the price of a significant increase of the complexity of the valuationequations. Market practice and the existing literature mainly focus on the estimation of the risk-freeclose-out value.We now provide the definition of close-out condition, in line with Bichuch et al. (2018a) Section 3.4.
Definition 2.10.
Let < R j < , j ∈ { B, C } , be the recovery rates of the bank and the counter-party, respectively. The close-out condition θ τ ( V , C, I T C , I
F C ) , expressed from the bank’s perspective,is defined by (2.21) θ τ ( V , C, I T C , I
F C ) := V τ + ∆ A τ + 1 { τ C <τ B } (1 − R C ) (cid:0) V τ + ∆ A τ − C τ − + I F Cτ − (cid:1) − − { τ B <τ C } (1 − R B ) (cid:0) V τ + ∆ A τ − C τ − − I T Cτ − (cid:1) + . The interpretation of θ τ ( V , C, I T C , I
F C ) is in line with Bichuch et al. (2018a) Remark 3.3.
Remark . Eq. (2.21) already encodes two terms giving rise to the credit valuation adjustment (CVA) and debt valuation adjustment (DVA), that we will define in details in Section 3.3.
Single aggregation level xVA framework
Trading strategies and the self-financing property.
In this section we proceed to adaptclassical concepts relating to contingent claim valuation in the present multiple-curve and defaultablesetting. We define the concept of self-financing trading strategies in this context. In the followingsection we then address the issue of viability of the unextended market model featuring only the basictraded assets, thus excluding trading on the contingent claim with dividend process ¯ A . Definition 3.1.
A dynamic portfolio, denoted by ϕ, is given by ϕ = (cid:16) ξ , . . . , ξ d , ξ B , ξ C , ψ , . . . , ψ d , ψ B , ψ C , ψ f,b , ψ f,l , ψ c,b , ψ c,l , ψ I,b , ψ
I,l (cid:17) , where (i) ξ , . . . , ξ d are G -predictable processes, denoting the number of shares of the risky primaryassets S , . . . , S d . (ii) ξ B , ξ C are G -predictable processes, denoting the number of shares of the risky bonds P B and P C . (iii) ψ , . . . , ψ d , ψ B , ψ C are G -adapted processes, denoting the number of shares of the repo accounts B , . . . B d , B B , B C . (iv) ψ f,b is a G -adapted process, denoting the number of shares of the unsecured funding borrowingcash account B f,b . (v) ψ f,l is a G -adapted process, denoting the number of shares of the unsecured funding lendingcash account B f,l . (vi) ψ c,b is a G -adapted process, denoting the number of shares of the collateral borrowing cashaccount B c,b for the received cash collateral. (vii) ψ c,l is a G -adapted process, denoting the number of shares of the collateral lending cash account B c,l for the posted cash collateral. (viii) ψ I,b is a G -adapted process, denoting the number of shares of the initial margin borrowingcash account B I,b for the initial margin received from the counterparty. (ix) ψ I,l is a G -adapted process, denoting the number of shares of the initial margin lending cashaccount B I,l for the initial margin posted to the counterparty.All processes introduced above are such that the stochastic integrals in the sequel are well defined.
Given a dynamic portfolio, we associate it to a financial contract, known in the literature as
CreditSupport Annex (CSA), see e.g. BCBS (2014).
Definition 3.2.
A CSA between the bank and the counterparty is represented through the pair ( C, I ) , where C is the variation margin and I is the initial margin. Definition 3.3. A collateralized hedger’s trading strategy associated to the collateralized contract ¯ A and the CSA ( C, I ) is a quintuplet (cid:0) x, ϕ, ¯ A, C, I (cid:1) , where x ∈ R is the initial endowment and ϕ is adynamic portfolio and I is the initial margin. We can define the wealth process associated to a collateralized hedger’s trading strategy (cid:0) x, ϕ, ¯ A, C, I (cid:1) as follows.
Definition 3.4.
The wealth process (or value process ) V ( ϕ ) = ( V t ( ϕ )) t ∈ [0 ,T ] associated to a collat-eralized hedger’s trading strategy (cid:0) x, ϕ, ¯ A, C, I (cid:1) is given by V t ( ϕ ) := d (cid:88) i =1 (cid:0) ξ it S it + ψ it B it (cid:1) + (cid:88) j ∈{ B,C } (cid:16) ξ jt P jt + ψ jt B jt (cid:17) + ψ f,bt B f,bt + ψ f,lt B f,lt − (cid:16) ψ c,bt B c,bt + ψ c,lt B c,lt + ψ I,lt B I,lt (cid:17) . (3.1) Remark . The sign minus in (3.1) in front of the last term depends on our convention on thecollateral.Note that in (3.1) we are not including the cash account for the received initial margin. This is due tothe fact that the received initial margin is posted in a segregated account and, hence, is not availableas a funding asset to the trading desk. However, the received initial margin will generate funding coststhat will appear in the self-financing condition we are going to introduce.
Definition 3.6.
Given the initial endowment x, a collateralized hedger’s trading strategy (cid:0) x, ϕ, ¯ A, C, I (cid:1) associated to the collateralized contract ¯ A and the CSA ( C, I ) is said to be self-financing if, for any t ∈ [0 , T ] , the wealth process V t ( ϕ ) satisfies V t ( ϕ ) = x + d (cid:88) i =1 (cid:90) (0 ,t ] ξ iu (cid:32) µ i ( u, S u ) du + d (cid:88) k =1 σ i,k ( u, S u ) dW k, P u + κ i ( u, S u ) du (cid:33) + d (cid:88) i =1 (cid:90) t ψ iu dB iu + (cid:88) j ∈{ B,C } (cid:90) t (cid:0) ξ ju dP ju + ψ ju dB ju (cid:1) − ¯ A t + (cid:90) t ψ f,bu dB f,bu + (cid:90) t ψ f,lu dB f,lu − (cid:90) t ψ c,bu dB c,bu − (cid:90) t ψ c,lu dB c,lu − (cid:90) t ψ I,bu dB I,bu − (cid:90) t ψ I,lu dB I,lu . (3.2)The last two terms in (3.2) represent the cash for the received initial margin. In general, we assumezero initial endowment, x = 0, i.e., V t ( ϕ ) = V t (cid:0) , ϕ, ¯ A, C, I (cid:1) for the sake of simplicity.
Definition 3.7.
A collateralized hedger’s trading strategy is admissible if it is self-financing and theassociated value process V ( ϕ ) is bounded. SDES OF XVA 11
Following Definition 5.1 in Bielecki and Rutkowski (2015) and the discussion thereafter, we can givethe definition of replicating strategy.
Definition 3.8.
A self-financing collateralized hedger’s trading strategy (0 , ϕ, ¯ A, C, I ) is said to repli-cate the collateralized contract ¯ A if V ˆ τ ( ϕ ) = 0 , where ˆ τ := τ ∧ T. Absence of arbitrage.
We provide the following definition of arbitrage-free strategy.
Definition 3.9.
Let the assumptions of Section 2 be in force. Then, the market is arbitrage-free if,for (0 , ϕ, , , , we have either P [ V ˆ τ (0 , ϕ, , ,
0) = 0] = 1 or P [ V ˆ τ (0 , ϕ, , , < > , for some stopping time ˆ τ > , where ˆ τ is introduced in Def. 3.8.Remark . In Bielecki and Rutkowski (2015, Definition 3.3) the authors introduce the concept ofa market which is said to be arbitrage-free for the hedger with respect to a class of contingent claims.Their definition is formulated in terms of a netted wealth process , which corresponds to a long-shortstrategy involving the claim ¯ A , where the first position is hedged and the second is unhedged. On theother hand, in Bichuch et al. (2018a) the question concerning absence of arbitrage is first answeredin a setting where only the basic traded assets are considered. This is also referred to as absence ofarbitrage with respect to the null contract in Bielecki et al. (2018). In our setting, the two approachescoincide.We restate, in our notations, an analog of Assumption 4.2 from Bichuch et al. (2018a). Assumption 3.11.
We assume r ft bounded from below and r f,lt ≤ r f,bt , d P ⊗ dt -a.s. Unlike Bichuch et al. (2018a), we do not impose constraints between the unsecured funding rate andthe returns of the risky bonds, since such securities are traded via repo markets. If the positions onthe risky bonds were financed via unsecured funding, then we would need the same sort of restrictionsbetween the rates, i.e., we would need to impose the assumption r f,lt ≤ ( r Bt + λ Bt ) ∧ ( r Ct + λ Ct ) , t ∈ [0 , T ] , P − a.s. Such assumptions would exclude the possibility for the trading desk to create trivial arbitrages betweenthe unsecured funding accounts and the risky bonds.To prove the absence of arbitrage for non-collateralized, non-defaultable contracts we define the cu-mulative dividend price process, see e.g. Bielecki and Rutkowski (2015).
Definition 3.12.
The cumulative dividend price associated to the i -th asset is given by (3.3) S i,cldt := S it + B it (cid:90) (0 ,t ] dD iu B iu , i = 1 , . . . d , t ∈ [0 , T ] . Proposition 3.13.
Let Assumption 3.11 hold. Moreover, assume that r f,lt ≥ r it , i = 1 , . . . , d , r f,lt ≥ r jt , j ∈ { B, C } , P -a.s., for all t ∈ [0 , T ] , and that there exists a probability measure Q ∼ P such that thediscounted asset price processes ˜ S i,cldt := S i,cldt B it , i = 1 , . . . d, ˜ P jt := P jt B jt , j ∈ { B, C } , (3.4) are local martingales. Then, the market consisting of the basic traded assets (0 , ϕ, , , is free ofarbitrage opportunities. Proof.
See Section A.1 in Appendix A. (cid:3)
From now on, we assume the following.
Assumption 3.14.
There exists an equivalent martingale probability measure Q ∼ P under which theprocesses ˜ S i,cldt , ˜ P jt in (3.4) are local martingales with dynamics d ˜ S i,cldt = 1 B it (cid:0) dS it − r it S it dt + dD it (cid:1) = d (cid:88) k =1 σ i,k ( t, S t ) B it dW k, Q t , i = 1 , . . . , d,d ˜ P jt = 1 B jt (cid:16) dP jt − r jt P jt − dt (cid:17) = − ˜ P jt − dM j, Q t , j ∈ { B, C } . More precisely, we assume the existence of an equivalent probability measure Q ∼ P with Radon-Nikodym density(3.5) ∂ Q ∂ P (cid:12)(cid:12)(cid:12) G t = E (cid:18)(cid:90) · β s dW P s (cid:19) (cid:89) j ∈ { B,C } exp (cid:40)(cid:90) t ln (cid:32) r js − r js λ j, P s (cid:33) dH js − (cid:90) t (¯ r js − r js ) ds (cid:41) , t ∈ [0 , T ] , where the process β = ( β t ) t ∈ [0 ,T ] , with β t := ( σ ( t, S t )) − ( µ ( t, S t ) − r t ) , is such that the stochasticexponential E (cid:0)(cid:82) · β s dW P s (cid:1) is a martingale and the dynamics of defaultable bonds under Q are givenby dP j, Q t = r jt P jt dt − P jt − dM j, Q t , j ∈ { B, C } , where the process(3.6) M j, Q t = M j, P t + (cid:90) t (1 − H ju )( λ j, P u − λ j, Q u ) du, λ j, Q t = r ft − λ j, P u − r jt , t ∈ [0 , T ] , is a ( G , Q )-martingale, for j ∈ { B, C } . Then Q is an ELMM for the discounted asset price process in(3.4).3.3. Contingent claim valuation.
In this section we consider the problem of pricing and hedging afinancial contract with payment stream ¯ A . To this purpose, we first write a BSDE for the candidatevalue process V as a consequence of our assumptions so far. After that, we proceed to address theissue of existence and uniqueness for the solutions of such BSDEs. Finally, we discuss if the process V , emerging as solution to such BSDEs, provides us with an arbitrage free price for the contingentclaim with dividend process ¯ A .Under Assumption 3.14 the dynamics of a self-financing collateralized trading strategy (cid:0) x, ϕ, ¯ A, C, I (cid:1) is dV t ( ϕ ) = d (cid:88) i =1 ξ it B it d ˜ S i,cldt + (cid:88) j ∈{ B,C } ξ jt B jt d ˜ P jt − d ¯ A t + ψ f,lt dB f,lt + ψ f,bt dB f,bt − ψ c,lt dB c,lt − ψ c,bt dB c,bt − ψ I,lt dB I,lt − ψ I,bt dB I,bt . (3.7)By using the repo constraints (2.11), (2.14), (2.15) and (2.17), the portfolio value satisfies V t ( ϕ ) = ψ f,lt B f,lt + ψ f,bt B f,bt + C t + I T Ct , t ∈ [0 , T ] , since I F C is segregated. Thanks to Remark 2.7 we obtain the identities ψ f,lt = (cid:0) V t ( ϕ ) − C t − I T Ct (cid:1) + (cid:16) B f,lt (cid:17) − , (3.8) SDES OF XVA 13 ψ f,bt = − (cid:0) V t ( ϕ ) − C t − I T Ct (cid:1) − (cid:16) B f,bt (cid:17) − (3.9)for t ∈ [0 , T ] . Observe that by (2.14) and (2.15) − ψ c,lt dB c,lt = − ψ c,lt r c,lt B c,lt dt = + r c,lt C + t dt, (3.10) − ψ c,bt dB c,bt = − ψ c,bt r c,bt B c,bt dt = − r c,bt C − t dt, (3.11)respectively. By (3.8), (3.9), (3.10), (3.11), (2.17) and (2.19), we can rewrite the wealth dynamics asfollows dV t ( ϕ ) = d (cid:88) i =1 ξ it B it d ˜ S i,cldt + (cid:88) j ∈{ B,C } ξ jt B jt d ˜ P jt − d ¯ A t + (cid:104) r f,lt (cid:0) V t ( ϕ ) − C t − I T Ct (cid:1) + − r f,bt (cid:0) V t ( ϕ ) − C t − I T Ct (cid:1) − + r c,lt C + t − r c,bt C − t + r I,lt I T Ct − r I,bt I F Ct (cid:105) dt. (3.12)We now introduce for convenience an auxiliary artificial interest rate process r = ( r t ) t ∈ [0 ,T ] , assumedto be right-continuous, bounded and F -adapted. This rate is not necessarily linked to a traded asset,but it can be interpreted as an interest rate level, used to express all other rates as spreads over thisartificial rate. When needed, we will explicitly state when the rate r becomes a market rate. Usingthe artificial rate r, we can conveniently rewrite the portfolio dynamics as follows dV t ( ϕ ) = d (cid:88) i =1 ξ it B it d ˜ S i,cldt + (cid:88) j ∈{ B,C } ξ jt B jt d ˜ P jt − d ¯ A t + (cid:104) ( r f,lt − r t ) (cid:0) V t ( ϕ ) − C t − I T Ct (cid:1) + − ( r f,bt − r t ) (cid:0) V t ( ϕ ) − C t − I T Ct (cid:1) − +( r c,lt − r t ) C + t − ( r c,bt − r t ) C − t + ( r I,lt − r t ) I T Ct − r I,bt I F Ct + r t V t ( ϕ ) (cid:105) dt , (3.13)where we added and subtracted the term r t V t ( ϕ ) dt. Remark . The term ( r I,lt − r t ) I T Ct measures a funding benefit from the posted initial margin overthe reference rate level r . We would like to stress that, in general, spreads over r can be negative,representing that we may have funding costs, even when the bank is collateral provider. Such asituation is faced by banks, which clear swaps with the London Clearing House (LCH). If r is chosento represent the EONIA overnight rate, then the rate applied by LCH is r I,l = r − bps, where bpsstands for basis points . On top of such a negative benefit, the bank needs to take into account thecost of raising the amount I T C , hence initial margin can generate funding costs in both directions,from the point of view of fund-raising and from the point of view of collateral remuneration, hencerepresenting a significant source of costs for the bank.We restate the portfolio dynamics in the form of a BSDE under the enlarged filtration G . We set Z kt := d (cid:88) i =1 ξ it σ i,k ( t, S t ) , (3.14a) U jt := − ξ jt P jt − , (3.14b) f ( t, V, C, I T C , I
F C ) := − (cid:104) ( r f,lt − r t ) (cid:0) V t ( ϕ ) − C t − I T Ct (cid:1) + − ( r f,bt − r t ) (cid:0) V t ( ϕ ) − C t − I T Ct (cid:1) − (3.14c) +( r c,lt − r t ) C + t − ( r c,bt − r t ) C − t + ( r I,lt − r t ) I T Ct − r I,bt I F Ct (cid:105) . The full contract G -BSDE for the portfolio’s dynamics has then the form on { τ > t } − dV t ( ϕ ) = d ¯ A t + (cid:0) f ( t, V, C, I T C , I
F C ) − r t V t ( ϕ ) (cid:1) dt − (cid:80) dk =1 Z kt dW k, Q t − (cid:80) j ∈{ B,C } U jt dM j, Q t V τ ( ϕ ) = θ τ ( V , C, I T C , I
F C ) . (3.15)We prove in Theorem 3.33 that there exists a unique solution ( V, Z, U ) for the G -BSDE (3.15), andthe process V assumes the following form on { τ > t } V t ( ϕ ) = B rt E Q (cid:34) (cid:90) ( t,τ ∧ T ] d ¯ A u B ru + (cid:90) τ ∧ Tt f ( u, V, C, I T C , I
F C ) B ru du + 1 { τ ≤ T } θ τ ( V , C, I T C , I
F C ) B rτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t (cid:35) , (3.16)where B rt := exp (cid:16)(cid:82) t r u du (cid:17) , t ∈ [0 , T ] . Remark . Our BSDE formulation (3.15) is in line with Definition 1.2 in Cr´epey (2015b) withhedging error term identically zero, but with a specific choice of the driving martingales, given bythe Brownian motions (cid:0) W , Q , . . . , W d, Q (cid:1) (cid:62) and the compensated jump processes M j, Q , j ∈ { C, B } .As in Cr´epey (2015b), the full BSDE is defined up to a random time horizon. The formulation canbe simplified by obtaining the equivalent F -BSDE by means of the Hypothesis 2.3 between F and G .The close-out condition is still expressed in terms of the general value process V . We characterize theprocess V in Subsection 3.3.1.3.3.1. Clean Value under F . A financial product can be traded between any two counterparties. Sinceevery agent has a different credit quality and different funding costs, this means in general that asingle product (e.g. a 10 year EUR swap) has as many potential values as the number of possiblecombinations of agents in the market. It would be highly impractical for a broker to publish allpossible market quotes for all possible counterparties. In fact, when we look at market quotes, wetypically see a single value (more precisely a bid and offer price). Such quotes are clean prices, i.e.,they do not represent real market prices.A clean price is an ideal value process that would be acceptable between two agents entering a perfectlycollateralized transaction. Perfect collateralization however is not enough to produce a clean price:we also need to explicitly assume that the two agents entering the transaction are default-free. Thisis necessary because, even in the presence of a perfect ideal collateral agreement, counterparty risk isnot perfectly annihilated: when a counterparty defaults, she stops posting collateral.However, default is not automatically legally recognized: typically, bankruptcy procedures requiresome days (e.g. 10 or 20 days) before the close-out payments are exchanged. This creates a period oftime where the counterparty is not officially defaulted but without any collateral adjustment. Suchperiod of time is known as margin period of risk.
During such interval of time the value of the claimdeviates from the value of the collateral account thus creating a credit exposure.Hence, to preclude margin period of risk and obtain the ideal clean price process, we consider a parallelmarket model with perfect collateralization but no default risk.
Assumption 3.17 (Clean market) . A clean market under F without bid-offer spreads is defined by (i) no bid-offer spread in the funding accounts, i.e., r f,lt = r f,bt = r f ; (ii) no bid-offer spread in the collateral accounts, i.e., r c,lt = r c,bt = r c ; (iii) the collateral rate is equal to the fictious rate, i.e., r c = r ; SDES OF XVA 15 (iv) there is no default, i.e. ˆ τ = T, and risky bonds are excluded from the market; (v) there is no exchange of initial margin; (vi) perfect collateralization, i.e., ˆ V t ≡ C t , for all t ∈ [0 , T ] , where we use ˆ V to denote the valueprocess of a collateralized hedging strategy in the fictious market without default-risk. Note that (vi) in Assumption 3.17 implies that the portfolio weights in the cash accounts are of theform ψ ct = − ˆ V t B ct , ψ ft ≡ , for all t ∈ [0 , T ] , meaning that the position is totally funded by the collateralization scheme, and ˆ V = ( ˆ V t ) [0 ,T ] is an F -adapted process.The portfolio dynamics under Q resulting from (3.13) under Assumption 3.17 are given by d ˆ V t ( ϕ ) = d (cid:88) k =1 ˆ Z kt dW k, Q t − dA t + r t ˆ V t ( ϕ ) dt, where ˆ Z kt := d (cid:88) i =1 ˆ ξ it σ i,k ( t, S t ) . (3.17) Remark . In (3.17) we introduced the F -predictable processes ˆ Z k , k = 1 , . . . , d, that represent thehedging position only for the clean price process, opposed to the processes Z k , k = 1 , . . . , d, from thefull portfolio dynamics that represent hedging positions for the clean price and the value adjustments.The aforementioned processes can be explicitly computed, see e.g. .Inserting the terminal condition ˆ V T = 0 , we can rewrite (3.17) in the classical F -BSDE form − d ˆ V t ( ϕ ) = dA t − r t ˆ V t ( ϕ ) dt − (cid:80) dk =1 ˆ Z kt dW k, Q t ˆ V T ( ϕ ) = 0 . (3.18)We now perform two different tasks. First, we show that, given the processes A and r, it is possible tofind a family of control processes ˆ Z k , k = 1 , . . . , d and a process ˆ V satisfying the clean BSDE (3.18),i.e., we prove an existence and uniqueness result for the solution of (3.18). Then, we establish thatthe process ˆ V provides the arbitrage free clean price. Theorem 3.19.
Under Assumption 2.8 on A, there exists a unique solution (cid:16) ˆ V , ˆ Z (cid:17) ∈ S ( Q ) ×H ,d ( Q ) to the clean BSDE (3.18) .Proof. We note that the clean BSDE (3.18) is similar to the linear BSDE studied e.g. in El Karouiet al. (1997), where the driver is the multidimensional Brownian motion (cid:0) W , Q , . . . , W d, Q (cid:1) (cid:62) .We can apply Theorem A.7 by observing that M = W Q , Q t = t , U = A , ˆ V = Y and h ( t, Y t , Z t ) = − r t ˆ V t , which clearly fulfills the uniform Lipschitz condition. Also the condition h ( · , , ∈ S ( Q ) istrivially satisfied. We also observe that X = S = diag ( S , . . . , S d ), hence we have m t = σ ( t, S t ) , so that γ t = S − σ ( t, S t ) , for γ satisfying the ellipticity condition (A.2). According to Theorem A.7we have ˆ V ∈ H ( Q ) and ˆ V − A ∈ S ( Q ). Now, Assumption 2.8 allows us to conclude that alsoˆ V ∈ S . (cid:3) Next we show that the process ˆ V in Theorem 3.19 provides the arbitrage-free price for the contractwith cashflow stream A . Theorem 3.20.
Let Q ∼ P be an equivalent probability measure such that all processes ˜ S i,cld , i = 1 , . . . , d, are local Q -martingales. Let (cid:16) ˆ V , ˆ Z (cid:17) be the unique solution of (3.18) . Then, under Assumption 2.8 on A, we have ˆ V t ( ϕ ) := E Q (cid:34) B rt (cid:90) ( t,T ] dA u B ru (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) , for all t ∈ [0 , T ] . (3.19) Proof.
The proof follows because ˆ Z ∈ H ,d ( Q ) by Theorem 3.19, B is bounded and thanks to As-sumption 2.8. (cid:3) Remark . Our concept of clean value is in line with the concept of third-party valuation of Bichuchet al. (2018a). Here we introduce the concept of clean value by means of a replicating strategy in afictious idealized market. Our constructive approach is in line with the market standard. Formula(3.19) encodes the idea of
CSA discounting.
Since the rate r is the remuneration of collateral in astylized perfect collateral agreement, we do not need to postulate the existence of a risk-free rate.Bichuch et al. (2018a) define the clean value by introducing an additional valuation measure differentfrom Q . Working with the pricing measure Q also avoids the issue of estimating parameters underdifferent measures.So far, our discussion of the clean market focused on a dividend process specified under the referencefiltration F . As stressed e.g. in Cr´epey (2015b), this assumption is too restrictive to e.g. covercredit derivatives or wrong-way risk. Though, our objective is to focus on multiple aggregation levelsand different discounting regimes, hence we choose to avoid the technicalities that are involved ingeneralizations of the immersion hypothesis.
Lemma 3.22.
Let ˜ X be an F -adapted process. Under the hypothesis 2.3 between F and G , we have ∆ ˜ X τ = 0 -a.s.Proof. This follows by Lemma 2.2 in Cr´epey (2015b). (cid:3)
The following assumption is crucial for next results.
Assumption 3.23.
We assume a risk-free close-out valuation under F , namely we set V t = ˆ V t ( ϕ ) in (2.21) . Full value G -BSDE. Definition 3.24.
We define the following valuation adjustments:
CV A t := B rt E Q (cid:20) { τ ≤ T } { τ C <τ B } (1 − R C ) 1 B rτ (cid:16) ˆ V τ ( ϕ ) − C τ − + I F Cτ − (cid:17) − (cid:12)(cid:12)(cid:12)(cid:12) G t (cid:21) ,DV A t := B rt E Q (cid:20) { τ ≤ T } { τ B <τ C } (1 − R B ) 1 B rτ (cid:16) ˆ V τ ( ϕ ) − C τ − − I T Cτ − (cid:17) + (cid:12)(cid:12)(cid:12)(cid:12) G t (cid:21) ,F V A t := B rt E Q (cid:34) (cid:90) τ ∧ Tt ( r f,lu − r u ) (cid:0) V u ( ϕ ) − C u − I T Cu (cid:1) + − ( r f,bu − r u ) (cid:0) V u ( ϕ ) − C u − I T Cu (cid:1) − B ru du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t (cid:35) ,ColV A t := B rt E Q (cid:34) (cid:90) τ ∧ Tt ( r c,lu − r u ) C + u − ( r c,bu − r u ) C − u B ru du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t (cid:35) ,M V A t := B rt E Q (cid:34) (cid:90) τ ∧ Tt ( r I,lu − r u ) I T Cu − r I,bu I F Cu B ru du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t (cid:35) . On { τ > t } , we define XV A t := − CV A t + DV A t + F V A t + ColV A t + M V A t , (3.20) SDES OF XVA 17 and set
XV A τ = − θ τ + ˆ V τ on { τ ≤ t } , where θ τ is defined in (2.21) .Remark . Upon inspection of the FVA term in Definition 3.24, we observe that, in general, thexVA-BSDE has a recursive nature. The exposure is proportional to the full value of the transaction V and not only to the clean value ˆ V . This implies a high complexity of the numerical scheme. Somepractitioner’s papers, such as Burgard and Kjaer (2013), avoid the recursivity issue by means of ad-hocchoices of the funding strategies, such as the funding strategy called semi-replication with no shortfallon default. However, the bank usually needs to fund the clean value and the value adjustments. Hence,this feature cannot be ignored in a comprehensive mathematical model.Our recursive FVA representation in Definition 3.24 is in line with the one presented in Piterbarg(2010). To clarify the latter point, let us consider the following
Example . Set I T Ct = I F Ct = 0, r f,b = r f,l = r f , r c,b = r c,l = r c and τ C = τ B = ∞ . Then thedriver of the full BSDE is given by(3.21) f ( t, V, C,
0) := − (cid:16) ( r ft − r t ) ( V t ( ϕ ) − C t ) + ( r ct − r t ) C t (cid:17) , t ∈ [0 , T ] . In this case, the integral representation (3.16) of V is of the form V t ( ϕ ) = B rt E Q (cid:34) (cid:90) ( t,T ] dA u B ru + (cid:90) Tt f ( u, V, C, B ru du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) , t ∈ [0 , T ] . (3.22)If we set r t = r ft d P ⊗ dt -a.s. then we obtain by (3.21) that V t ( ϕ ) = B r f t E Q (cid:34) (cid:90) ( t,T ] dA u B r f u + (cid:90) Tt ( r fu − r cu ) C u B r f u du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) , t ∈ [0 , T ] . (3.23)This corresponds to equation (3) in Piterbarg (2010). If we set r t = r ct d P ⊗ dt -a.s. in (3.21), we obtain V t ( ϕ ) = B r c t E Q (cid:34) (cid:90) ( t,T ] dA u B r c u − (cid:90) Tt ( r fu − r cu ) ( V t ( ϕ ) − C u ) B r c u du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) , t ∈ [0 , T ] , (3.24)which corresponds to equation (5) in Piterbarg (2010).3.4. Well Posedness of the pricing BSDE.
In this section we address the issue of existence anduniqueness for the solution of the G -BSDE (3.15). We follow the approach of Cr´epey (2015b).From Assumption 3.23 we have V = ˆ V . Since ˆ V is an F -adapted process, we know from Lemma 3.22that ∆ ˆ V τ = 0. Following Cr´epey (2015b), an application of Theorem 67b in Dellacherie and Meyer(1982) implies that there exists an F -predictable process with the same value as ˆ V in τ , hence ˆ V canbe chosen to be F -predictable. The same argument holds true for the collateral process C, which weassumed to be a Lipschitz function of the clean value, and for the initial margin I , be it posted orreceived. In summary, both exposuresˆ V t − C t + I F Ct , ˆ V t − C t − I T Ct , are assumed to be F -predictable from now on. We set θ Ct := (1 − R C ) (cid:16) ˆ V t − C t + I F Ct (cid:17) − ,θ Bt := (1 − R B ) (cid:16) ˆ V t − C t − I T Ct (cid:17) + , (3.25) and rewrite the close-out condition as XV A τ = 1 { τ We call pre-default XVA-BSDE the following F -BSDE on [0 , T ] with null terminalcondition in T : − dXV A t = ¯ f ( ˆ V t − XV A t ) dt − (cid:80) dk =1 ¯ Z kt dW k, Q t XV A T = 0 , (3.27) where ¯ f ( ˆ V t − XV A t ) := − f ( t, ˆ V − XV A, C, I T C , I F C ) − ( r t + λ C, Q t + λ B, Q t ) XV A t − λ C, Q t θ Ct + λ B, Q t θ Bt , (3.28) for θ B , θ C defined as in (3.25) and λ B, Q , λ C, Q introduced in (3.6) . We now discuss existence and uniqueness for the solution of (3.27). First, we observe that the driver(3.28) also depends on the initial margins. We set(3.29) I it := ρ t ( ˆ V t : T − XV A t : T ) t ∈ [ t,T ] , i ∈ { T C, F C } where XV A t : T := ( XV A s ) s ∈ [ t,T ] is the process defining the pre-default value adjustment and ˆ V t : T :=( ˆ V s ) s ∈ [ t,T ] is the clean value assumed to be a given exogenous process in S ( Q ) , both evaluatedup to the contract’s maturity, since they are used to measure the potential future exposure, and ρ t = ρ ( ω, t ; x ) , t ∈ [0 , T ] is a process with values in the space of risk measures. Remark . It is worth noting that, for the sake of simplicity, we assume the same ρ for both i ∈ T C, F C . This hypothesis can be easily generalized.We also assume the following Assumption 3.29. (i) For any X, Y ∈ S ( Q ) , the process ( ρ s ( X t : T − Y t : T )) s ∈ [0 ,T ] is in H ( Q ) . There exists a constant C ¯ f > and a family of measures ( ν s ) s ∈ [0 ,T ] on R such that ν t ([ t ; T ]) =1 , for every t ∈ [0 , T ] , and, for any x, y , y ∈ S ( Q ) , we have (3.30) (cid:12)(cid:12) ρ t ( x t : T − y t : T ) − ρ t ( x t : Y − y t : T ) (cid:12)(cid:12) ≤ C ρ E (cid:20)(cid:90) Tt | y s − y s | ν t ( ds ) (cid:12)(cid:12) F t (cid:21) , dt ⊗ d P a.e.Moreover, there exists a constant k > such that for every continuous path x : [0 , T ] → R , we have (cid:90) T (cid:90) Ts | x s | ν u ( ds ) du < k sup t ∈ [0 ,T ] | x t | . (ii) we assume that f satisfies Assumption A.9 with y = ˆ V − XV A i , z = C and λ = ρ i , for i = 1 , . We are able to prove the following result. Lemma 3.30. Let ( XV A i , Z i ) ∈ S ( Q ) × H ,d ( Q ) , i = 1 , , be solutions to the F -BSDE (3.27) , with f i ( t, ˆ V − XV A i , C, ρ i ) satisfying Assumption 3.29, for i = 1 , . Moreover, define δXV A := XV A − XV A , SDES OF XVA 19 δZ := Z − Z ,δf := f ( ˆ V − XV A ) − f ( ˆ V − XV A ) ,δρ := ρ s ( ˆ V s : T − XV A s : T ) − ρ s ( ˆ V s : T − XV A s : T ) , for any s ∈ [ t, T ] . Then, there exists a constant C > such that for µ > , we have for β large enough (cid:107) δXV A (cid:107) S β,T ≤ C (cid:20) e βT E (cid:2) | δξ | (cid:3) + µ (cid:18) (cid:107) δ f (cid:107) H ,dβ,T + C f (cid:107) δ ρ (cid:107) H ,dβ,T (cid:19)(cid:21) , (cid:107) δZ (cid:107) H ,dβ,T ≤ C (cid:20) e βT E (cid:2) | δξ | (cid:3) + µ (cid:18) (cid:107) δ f (cid:107) H ,dβ,T + C f || δρ (cid:107) H ,dβ,T (cid:19)(cid:21) , where (cid:107) δXV A (cid:107) S β,T and (cid:107) δZ (cid:107) H ,dβ,T are defined in (2.5) and (2.3) , respectively.Proof. First, we observe that f i ( ˆ V − XV A i ) , i = 1 , , given in (3.28), consists of three terms. The firstone is the full G -BSDE driver f, given in (3.14c) and expressed in terms of the collateral C, which is aLipschitz function of the clean value by definition, and the (posted/received) initial margin I, which isLipschitz by (3.29) and (3.30). The second term depends on the short rate r and the jump intensities λ B, Q , λ C, Q , which are bounded by definition. The last term relies upon the close-out conditions θ B , θ C given in (3.25), which are Lipschitz functions, by following the same arguments as before. Therefore,the driver satisfies Assumption A.9. Moreover, we observe that, for any ( XV A , Z ) , ( XV A , Z ) ∈S ( Q ) × H ,d ( Q ) , we have | f ( ˆ V − XV A ) − f ( ˆ V − XV A ) |≤ | f ( ˆ V − XV A ) − f ( ˆ V − XV A ) | + | f ( ˆ V − XV A ) − f ( ˆ V − XV A ) |≤ | f ( s, ˆ V − XV A , C, ρ ( ˆ V s : T − XV A s : T )) − f ( s, ˆ V − XV A , C, ρ ( ˆ V s : T − XV A s : T )) | + | ( r s + λ C, Q s + λ B, Q s )( XV A − XV A ) | + | δf |≤ ( ¯ C f + ¯ C s ) (cid:0) | δXV A | (cid:1) + ¯ C f (cid:16) | ρ ( ˆ V s : T − XV A s : T ) − ρ ( ˆ V s : T − XV A s : T ) | + | δρ | (cid:17) + | δf | where the last inequality holds true thanks to Assumption A.9, adding and subtracting ρ ( ˆ V s : T − XV A s : T ) and choosing a suitable constant ¯ C s ≥ r s + λ C, Q s + λ B, Q s . The result follows by applying the argument provided in the proof of Lemma A.10 with terminalcondition ξ = 0 . (cid:3) The a-priori estimates specified in Lemma 3.30 imply the following result. Proposition 3.31. Under Assumptions 2.8, A.9 and 3.29, the F -BSDE (3.27) is well posed and hasa unique solution ( XV A, Z ) ∈ S ( Q ) × H ,d ( Q ) . Proof. The proof is analogous to the one in Theorem 2.1 of Agarwal et al. (2018). (cid:3) Now, given the uniqueness of the solution to (3.27) we have the following result. Proposition 3.32. Let (cid:0) XV A, ¯ Z (cid:1) be the unique solution of the pre-default XVA-BSDE (3.27) . Define X t := XV A t J t + 1 { τ ≤ t } ϑ τ , t ∈ [0 , τ ∧ T ] , (3.31) where J t := 1 { t<τ } = 1 − H t . Then, under Assumptions 2.8 and 3.29, the process (cid:16) X, ˜ Z, ˜ U (cid:17) solves the G -BSDE on { τ > t } − dX t = − (cid:104) f ( t, ˆ V − XV A, C, I T C , I F C ) + r t XV A t (cid:105) dt − (cid:80) dk =1 ˜ Z kt dW k, Q t − (cid:80) j ∈{ B,C } ˜ U jt dM j, Q t X τ = 1 { τ ≤ T } (cid:16) ˆ V τ ( ϕ ) − θ τ ( ˆ V , C, I T C , I F C ) (cid:17) (3.32) with respect to the filtration G .Moreover, the ( G , Q ) -martingale components of the XVA-BSDE satisfy on { t < τ } d (cid:88) k =1 (cid:90) t ˜ Z ku dW k, Q u = d (cid:88) k =1 (cid:90) t ¯ Z ku dW k, Q u , (3.33) (cid:88) j ∈{ B,C } ˜ U jt dM j, Q t = − (cid:16) ( ϑ t − XV A t ) dJ t + λ C, Q t ( − θ Ct − XV A t ) dt + λ B, Q t ( θ Bt − XV A t ) dt (cid:17) , (3.34) where ˜ Z ∈ H ,d ( Q ) and ˜ U ∈ H , ( Q ) . In particular, X t = XV A t , t ∈ [0 , T ] , where XV A isintroduced in Definition 3.24.Proof. We start from (3.31) and apply the product rule, hence dX t = d (cid:0) XV A t J t (cid:1) + d (cid:0) { τ T C , I F C ) + ( r t + λ C, Q t + λ B, Q t ) XV A t + λ C, Q t θ Ct − λ B, Q t θ Bt (cid:105) dt + d (cid:88) i =1 ¯ Z kt { t<τ } dW k, Q t − (cid:0) ϑ t − XV A t (cid:1) dJ t We note that the process (cid:80) di =1 (cid:82) · ¯ Z ku { u<τ } dW k, Q u is a ( G , Q )-martingale, since ¯ Z is in H ,d ( Q ) dueto the immersion hypothesis. From Lemma 5.2.9 in Cr´epey et al. (2014) we deduce that the process,expressed in differential form − (cid:16) ( ϑ t − XV A t ) dJ t + λ C, Q t ( − θ Ct − XV A t ) dt + λ B, Q t ( θ Bt − XV A t ) dt (cid:17) (3.35)is also a ( G , Q )-local martingale. Moreover, we observe that, since ˆ V ∈ S ( Q ) , also C ∈ S ( Q ) ,C being a Lipschitz function of ˆ V . Additionally, the initial margin, be it posted or received, lies in H ( Q ) by assumption. Summing up, both θ B and θ C , and hence ϑ belong to the space H ( Q ). Onthe other hand, XV A ∈ S ( Q ). Recalling that both λ C, Q and λ B, Q are bounded, it follows that thecompensated jump term (3.35) is a square integrable martingale. Then, we have that (3.34) must holdfor some ˜ U j and we conclude that the process XV A solves the XVA-BSDE (3.32) under the filtration G . (cid:3) We can finally combine the solution of the BSDE (3.27) for the clean value with the result above tosolve the G -BSDE (3.15). Theorem 3.33. Let V t := ˆ V t − XV A t , t ∈ [0 , T ] , on { τ > t } , where ˆ V and XV A are definedin (3.19) and (3.20) , respectively. Then, under Assumptions 2.8 and 3.29, the triplet ( V, Z, U ) ∈S ( Q ) × H ,d ( Q ) × H , ( Q ) solves the G -BSDE (3.15) , where Z and U are given by Z kt = ˆ Z kt − ˜ Z kt , k = 1 , . . . , d, (3.36) SDES OF XVA 21 U jt = − ˜ U jt , j ∈ { B, C } . (3.37) Moreover, the process V satisfies (3.16) .Proof. By Hypothesis 2.3, on { t < τ } we haveˆ V t ( ϕ ) = B rt E Q (cid:34) (cid:90) ( t,T ] dA u B ru (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) = B rt E Q (cid:34) (cid:90) ( t,T ] dA u B ru (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t (cid:35) . So we consider ˆ V under G . We also observe that, on { t < τ } , we have ¯ A t = A t and recall the pricedecomposition V t = ˆ V t − XV A t . Using (3.18) and (3.32), we write the dynamics of V on { t < τ }− dV t = d ¯ A t + (cid:104) f ( t, ˆ V − XV A, C, I T C , I F C ) dt − r t (cid:16) ˆ V t − XV A t (cid:17)(cid:105) dt − d (cid:88) k =1 (cid:16) ˆ Z kt − ˜ Z kt (cid:17) dW k, Q t − (cid:88) j ∈{ B,C } (cid:16) − ˜ U jt (cid:17) dM j, Q t with terminal condition at τ V τ = ˆ V τ − XV A τ = ˆ V τ − (cid:16) θ τ − ˆ V τ (cid:17) = θ τ . Since Z = ˆ Z − ˜ Z ∈ H ,d ( Q ) and U = − ˜ U ∈ H , ( Q ) by Theorem 3.19 and Proposition 3.32, weobtain that ( V, Z, U ) solves the G -BSDE (3.15) and satisfies the required integrability conditions.Finally, we are now able to prove that (3.16) is equivalent to (3.15).Here we assume to work only on { τ > t } . Since V t = ˆ V − XV A t and thanks to Definition 3.24 we have V t ( ϕ ) = ˆ V t ( ϕ ) + B rt E Q { τ ≤ F C ) B ru du (cid:12)(cid:12)(cid:12) G t (cid:21) . Assumption 3.23 and (2.20) ensure that V t ( ϕ ) = ˆ V t ( ϕ ) + B rt E Q (cid:34)(cid:90) τ ∧ Tt f ( u, V, C, I T C , I F C ) B ru du + 1 { τ ≤ T } θ τ ( ˆ V ( ϕ ) , C, I T C , I F C ) − ˆ V τ ( ϕ ) B rτ (cid:12)(cid:12)(cid:12) G t (cid:35) . Now, we apply (3.19), the tower property and Hypothesis 2.3, so that V t ( ϕ ) = B rt E Q (cid:34)(cid:90) τ ∧ Tt f ( u, V, C, I T C , I F C ) B ru du + 1 { τ ≤ T } θ τ ( ˆ V ( ϕ ) , C, I T C , I F C ) B rτ (cid:12)(cid:12)(cid:12) G t (cid:35) + B rt E Q (cid:34)(cid:90) ( t,T ] dA u B ru − { τ ≤ T } (cid:90) ( t,T ] dA u B rτ (cid:12)(cid:12)(cid:12) G t (cid:35) . Finally, again by (2.20), we have V t ( ϕ ) = B rt E Q (cid:34)(cid:90) ( t,T ] d ¯ A u B ru + (cid:90) τ ∧ Tt f ( u, V, C, I T C , I F C ) B ru du + 1 { τ ≤ T } θ τ ( ˆ V ( ϕ ) , C, I T C , I F C ) B rτ (cid:12)(cid:12)(cid:12) G t (cid:35) . (cid:3) We now provide an explicit formula for the value adjustments under the filtration F . This represen-tation is particularly useful from a computational point of view: risk factors can be simulated underthe smaller filtration F and the computation of value adjustment does not require the simulation ofdefault times. It is an immediate consequence of Proposition 3.31. Corollary 3.34. Let (cid:0) XV A, ¯ Z (cid:1) be the unique solution to the pre-default XVA-BSDE under F (3.27) .Define the process ˜ r = (˜ r t ) t ∈ [0 ,T ] by setting ˜ r := r + λ C, Q + λ B, Q . Under Assumptions 2.8 and 3.29 thestochastic process XV A admits the following representation. XV A t = − CV A t + DV A t + F V A t + ColV A t + M V A t , (3.38) where CV A t := B ˜ rt E Q (cid:20) (1 − R C ) (cid:90) Tt B ˜ ru (cid:16) ˆ V u ( ϕ ) − C u + I F Cu (cid:17) − λ C, Q u du (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) ,DV A t := B ˜ rt E Q (cid:20) (1 − R B ) (cid:90) Tt B ˜ ru (cid:16) ˆ V u ( ϕ ) − C u − I T Cu (cid:17) + λ B, Q u du (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) ,F V A t := B ˜ rt E Q (cid:34) (cid:90) Tt ( r f,lu − r u ) (cid:0) V u ( ϕ ) − C u − I T Cu (cid:1) + − ( r f,bu − r u ) (cid:0) V u ( ϕ ) − C u − I T Cu (cid:1) − B ˜ ru du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) ,ColV A t := B ˜ rt E Q (cid:34) (cid:90) Tt ( r c,lu − r u ) C + u − ( r c,bu − r u ) C − u B ˜ ru du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) ,M V A t := B ˜ rt E Q (cid:34) (cid:90) Tt ( r I,lu − r u ) I T Cu − r I,bu I F Cu B ˜ ru du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) . In the literature there has been an intense debate regarding the problem given by the possible overlapbetween FVA and DVA, see e.g. Hull and White (2012), Andersen et al. (2019), Brigo et al. (2019)and references therein. This problem, however, is due to accounting inconsistencies that do not affectour pricing equations. We limit ourselves to mention that a sound treatment of the issue is providedby Brigo et al. (2019) and that their solution can be embedded in our framework at the cost of furthernotations. 4. Multiple aggregation level xVA framework Multiple discounting regimes. In this section we analyze the market practice of CSA discount-ing in the context of our general G -BSDE. CSA discounting means that a transaction is consideredas a clean transaction, in line with our previous Assumption 3.17 in Section 3. SDES OF XVA 23 In Section 3.3.1 we assumed that the clean value refers to an idealized fully collateralized transactionwhere the collateral rate is simply r . The situation in practice is more complicated. The marketpractice adopted for the computation of clean prices involves a multitude of discount curves. Possibleexamples from the market practice are • The (clean) value of a perfectly uncollateralized derivative might be discounted by a bank bymeans of a bank-specific funding curve with associated short rate r f (this could correspondto the Libor rate for a bank belonging to the Libor panel), see e.g. Piterbarg (2010). • The (clean) value of a derivative collateralized in a foreign currency is discounted on themarket at a rate depending on cross currency bases, see the formulas and derivations in Table1 in Moreni and Pallavicini (2017).It is quite natural to ask why banks employ multiple discount regimes for clean values and, on topof that, xVA corrections. The main reason is purely pragmatic and non-mathematical: from theperspective of a trading desk it is convenient to treat multiple CSAs by means of different discountregimes, because this allows to deal with portfolio market risk via traditional trading-desk techniques,such as curve trades (i.e. e.g. buying/selling interest rate swaps on different buckets/maturities alongthe curve). Hedging the expectation of an integral such as the FVA term in practice is much morecomplicated. A possible approximate treatment involves discretizing the time integral and treatingthe resulting Riemann sum over time as a portfolio of claims. In view of the aforementioned difficulty,market operators prefer to obtain an additive price representation, where discount curves are used toreduce the magnitude of the (funding related) xVA terms, which are more difficult to hedge.From now on, we shall assume that the bank has two internal desks, dubbed the front-office desk and the xVA desk, respectively. The front-office desk is responsible for the calculation of clean valuesand for the trading activity required to hedge market risk of the clean values. The xVA desk insteadcomputes and hedges all the value adjustments and is forced, according to internal rules of the bank,to adopt for each transaction the clean value dictated by the front-office desk. The fact that the xVAdesk is a clean-value-taker implies that care is needed when computing xVAs, in order to avoid doublecounting effects.The xVA desk has to deal with two different clean values for the same transaction. On the one side,the clean value performs an arbitrage-free pricing under the F -BSDE. On the other side, we have theclean value prescribed by the front office function, which constitutes the official clean value acceptedwithin the bank. The xVA desk is then faced with the following challenge: Problem 4.1 (xVA-CSA consistency problem) . Produce a price decomposition of V in terms of cleanvalue and xVA such that(i) the representation of V is coherent with the G -BSDE (3.15) , and(ii) the clean price in the representation corresponds to the one prescribed by the front-officefunction. We assume a portfolio consisting of K > A m = ( A mt ) t ∈ [0 ,T ] andvalue processes ˆ V m = (cid:16) ˆ V mt (cid:17) t ∈ [0 ,T ] , for m = 1 , . . . , K, and provide a price representation in terms ofmultiple discounting rules. Based on Assumption 3.17, we treat each discounting rule as based ona different clean market: every (possibly) trade-specific clean valuation results from an underlying(possibly) trade-specific clean market. In line with Assumption 3.17, in every trade-specific cleanmarket the collateralization scheme is perfect, but now the remuneration of collateral is performed ata different interest rate. Assumption 4.2. A clean market under F without bid-offer spreads with multiple CSAs is defined by(i) No bid-offer spreads in the funding accounts, i.e., r f,lt = r f,bt = r f . (ii) No bid-offer spreads in the collateral accounts, i.e., r c,lt = r c,bt = r c . (iii) There is no default, i.e. ˆ τ = T, and risky bonds are excluded from the market.(iv) There is no exchange of initial margin.(v) ˆ V m and A m are F -adapted processes.(vi) Perfect collateralization, i.e., ˆ V mt = C mt d P ⊗ dt -a.s.(vii) There exists a specific collateral rate ˆ r m with cash account B ˆ r m t for each claim A m , m =1 , . . . , K. Observe that (vii) in Assumption 4.2 ensures that the repo-like relation(4.1) ˆ V mt + ψ mt B ˆ r m t = 0holds for each claim A m , m = 1 , . . . , K. Note also that ψ f = 0 . Remark . To provide a concrete example, Assumption 4.2 covers the situation where the tradingdesk of the bank enters into two perfectly collateralized transactions with two different counterparties,the first one being e.g. a clearing house such as LCH, the other one being another clearing housesuch as Eurex. Although the dividend process of the claim is the same for both transactions, thecollateral remuneration provided by the trade with Eurex and the trade with LCH is different. Thespread in the collateral remuneration between EUREX and LCH is called Eurex-LCH basis, see e.g.Mackenzie Smith (2017) for a more detailed discussion. This will result in the two clean values beingcomputed by means of different discounting rates.In summary, the market practice of discounting cashflows according to trade-specific collateral ratesimplies that, within the bank, a single transaction will be discounted at least according to two differentregimes. Initially, the front-office determines the clean value by discounting cash flows through an idealmarket collateral rate ˆ r m . Hence the front-office clean value ˆ P mt , m = 1 , . . . , K, is obtained from the F -BSDE − d ˆ P mt = − (cid:80) dk =1 ˆ Z k,mt dW k, Q t + dA mt − ˆ r m ˆ P mt dt, ˆ P mT = 0 . (4.2)On the other side, the xVA desk first computes the clean value ˆ V mt , m = 1 , . . . , K, as the solution tothe F -BSDE (3.18), i.e. by solving(4.3) − d ˆ V mt = − (cid:80) dk =1 ˆ Z k,mt dW k, Q t + dA mt − r t ˆ V mt dt, ˆ V mT = 0 , for each claim A m . From a valuation perspective, if clean values represented the prices of real trans-actions, the presence of multiple discounting rules would immediately imply the presence of triv-ial arbitrage opportunities in the market. Only the endogenous price (4.3) is compatible with thearbitrage-free setting of Section 3. On the other hand, the xVA desk is forced to provide results interms of the multiple discounting regimes imposed by the front-office. The two approaches can becombined in an arbitrage-free setting by means of the following invariance property of linear BSDEs. Lemma 4.4. Let ( ˆ V m , ˆ Z ,m , . . . , ˆ Z d,m ) be the unique solution of the F -BSDE (4.3) . Under Assump-tion 4.2 for A m , m = 1 , . . . , K, the value process ˆ V m admits the two equivalent representations SDES OF XVA 25 i) xVA-discounting representation ˆ V mt = B rt E Q (cid:34)(cid:90) ( t,T ] dA mt B ru (cid:12)(cid:12)(cid:12) F t (cid:35) , (4.4) ii) CSA-discounting representation (4.5) ˆ V mt = ˆ P mt − DiscV A mt , where DiscV A mt represents the discounting valuation adjustment, defined as (4.6) DiscV A mt := B ˆ r m t E Q (cid:34)(cid:90) Tt ( r u − ˆ r m ) ˆ V mu B ˆ r m u du (cid:12)(cid:12)(cid:12) F t (cid:35) , and ˆ P m is the value process in the solution ( ˆ V m , ˆ Z ,m , . . . , ˆ Z d,m ) of the F -BSDE (4.2)(4.7) ˆ P mt = B ˆ r m t E Q (cid:34)(cid:90) ( t,T ] dA mt B ˆ r m u (cid:12)(cid:12)(cid:12) F t (cid:35) . Proof. The integral representation (4.4) is immediate. To obtain (4.5) we rewrite the F -BSDE (4.3)adding and subtracting the term ˆ r m ˆ V mt , i.e., − d ˆ V mt = − (cid:80) dk =1 ˆ Z k,mt dW k, Q t + dA mt − ( r t − ˆ r mt ) ˆ V mt dt − ˆ r mt ˆ V mt dt ˆ V mT = 0 , m = 1 , . . . , K. The value process of the solution is given byˆ V mt = B ˆ r m t E Q (cid:34)(cid:90) ( t,T ] dA mt B ˆ r m u (cid:12)(cid:12)(cid:12) F t (cid:35) − B ˆ r m t E Q (cid:34)(cid:90) Tt ( r u − ˆ r mu ) ˆ V mu B ˆ r m u du (cid:12)(cid:12)(cid:12) F t (cid:35) , where we recognize the first expectation as ˆ P m , whereas the second one provides DiscV A m . (cid:3) This lemma gives a price decomposition which is compatible with the presence of multiple discountingrules for different claims. The full contract G -BSDE for the portfolio of claims ( A m ) m ∈ { ,...,K } can bewritten as − dV t ( ϕ ) = (cid:80) Km =1 d ¯ A mt + (cid:0) f ( t, V, C, I T C , I F C ) − r t V t ( ϕ ) (cid:1) dt − (cid:80) dk =1 Z kt dW k, Q t − (cid:80) j ∈ { B,C } U jt dM j, Q t ,V τ ( ϕ ) = θ τ (cid:16)(cid:80) Nm =1 ˆ V m , C, I (cid:17) , (4.8)where ¯ A mt is defined in (2.20), Z t = (cid:0) Z t , . . . , Z dt (cid:1) , and U t = (cid:0) U Bt , U Ct (cid:1) , represent the control processesgiven by G -predictable processes, and f ( t, V, C, I T C , I F C ) is the G -BSDE driver given by (3.14c). Theclose-out condition is V τ ( ϕ ) = K (cid:88) m =1 ˆ P mτ + 1 { τ C <τ B } (1 − R C ) (cid:32) K (cid:88) m =1 ˆ P mτ − C τ − + I F Cτ − − K (cid:88) m =1 DiscV A mτ (cid:33) − − { τ B <τ C } (1 − R B ) (cid:32) K (cid:88) m =1 ˆ P mτ − C τ − − I T Cτ − − K (cid:88) m =1 DiscV A mτ (cid:33) + . (4.9)By using the same arguments given for Theorem 3.33 and taking into account Definition 3.24, weobtain the following result, with the help of Lemma 4.4. Proposition 4.5. Under Assumption 4.2 and 3.29, the G -BSDE (4.8) admits the following integralrepresentation V t ( ϕ ) = K (cid:88) m =1 B ˆ r m t E Q (cid:34)(cid:90) ( t,T ] dA mu B ˆ r m u (cid:12)(cid:12)(cid:12) F t (cid:35) − XV A t = K (cid:88) m =1 ˆ P mt − (cid:92) XV A t , (4.10) on the event { τ > t } , t ∈ [0 , T ] , where (cid:92) XV A := XV A + DiscV A, (4.11) with XV A t := F V A t + ColV A t + M V A t − CV A t + DV A t = F V A t + ColV A t + M V A t − B rt E Q { τ Multiple aggregation levels. We can use our setting to analyze multiple aggregation levels.We start from the example illustrated in Figure 1. The set of trades between the bank and thecounterparty can be typically split into several subsets reflecting multiple aggregation levels.One can distinguish between funding/margin sets and netting sets. Funding/margin sets are tradedbetween the bank and the counterparty that share the same funding policy. This corresponds todifferent CSAs: for example, one CSA (Margin Set 2) could group all trades for which collateral isexchanged in USD (e.g. for foreign exchange derivatives), whereas another CSA (Margin Set 3) couldbe relevant for all instruments collateralized in EUR. Finally, trades that are not collateralized, butwhose exposures are netted among each other, can be also grouped in a separate margin/funding set,corresponding to Margin Set 1 in Figure 1. SDES OF XVA 27 The protection provided by collateralization agreements might however be imperfect, hence a legalagreement between the bank and the counterparty might allow for the netting of residual post collateralexposures arising from different margin sets. This corresponds to Netting Set 1 in Figure 1. CounterpartyNettingSet 1 e.g. a firstsubsidiary NettingSet 2 e.g. a secondsubsidiary MarginSet 2 USD collater-alized Trades MarginSet 1 UnsecuredTrades MarginSet 3 EUR collater-alized Trades Legacy EURMargin Set Monthlymargin calls. New EURMargin Set Daily mar-gin calls. Figure 1. A possible hierarchical structure of aggregation levels.Another typical source of multiple aggregation levels is the historical stratification of legal agreements:in Figure 1 we have a second netting set, corresponding to a second subsidiary of the parent coun-terparty, where legacy trades are covered by an old CSA agreement involving monthly margin calls,whereas all trades entered after a certain date are covered by a newer CSA agreement involving dailymargin calls.A further level of complexity could arise when the parent and the subsidiaries have different defaulttimes: this introduces further complications when modeling the close-out condition because one mighthave e.g. a situation where the default of a subsidiary is covered by the parent. Such issues are leftfor future research. From a practical point of view it is also difficult to find calibration instrumentsfor default probabilities, since subsidiaries typically do not enjoy a liquid CDS market.The example we discussed highlights the fact that the portfolio-wide G -BSDE depends on the structureof all legal agreements between the bank and the counterparty.In line with the previous sections, we assume that the portfolio P of trades between the bank andthe counterparty consists of K trades, that we identify by means of the respective payment processes,i.e., P = (cid:8) A , . . . A K (cid:9) . We use again ˆ V m to denote the clean reference value of the claims A m , m =1 , . . . K , representing also their credit exposure before collateral is applied. We construct a bottom-upaggregation hierarchy of claims by means of the following definitions. Definition 4.7. A margin (or funding) set M is a set of claims whose aggregated clean values (expo-sures) are fully or partially covered by a CSA (collateral agreement). We let N M denote the numberof margin sets in the portfolio P . Assumption 4.8. For every claim A m ∈ P , m = 1 , . . . K, we assume that the margin set for initialand variation margin coincide.Remark . It is worth noting that uncollateralized trades that can be netted among each other canbe treated as margin sets with zero initial and variation margin. Moreover, trades that can not be aggregated with other trades can be treated as separate margin sets consisting of the single tradethemselves. Finally, we observe that all trades within a margin set share the same funding source. Definition 4.10. A netting set N is a set of margin sets whose post-margin exposures can be aggre-gated. We let N N denote the number of netting sets in the portfolio P . The structure of the portfolio is illustrated in Figure 1, where the first row illustrates the compositionof all margin sets as groups of claims and the second line describes the netting sets as groups of marginsets, P = { A , . . . , A N } (cid:124) (cid:123)(cid:122) (cid:125) M ∪ { A N +1 , . . . , A N } (cid:124) (cid:123)(cid:122) (cid:125) M ∪ . . . ∪ { A N N M− +1 , . . . , A N N M } (cid:124) (cid:123)(cid:122) (cid:125) M N M = {M , . . . , M M } (cid:124) (cid:123)(cid:122) (cid:125) N ∪ {M M +1 , . . . , M M } (cid:124) (cid:123)(cid:122) (cid:125) N ∪ . . . ∪ {M M N M− +1 , . . . , M M N M } (cid:124) (cid:123)(cid:122) (cid:125) N N N (4.12)where we have N N M = K .Given the structure we introduced, we can generalize the CVA and DVA formulas at the portfolio-widelevel as follows. The portfolio exposure within a margin set is given by |M m | (cid:88) m =1 (cid:16) ˆ P mτ − DiscV A mτ (cid:17) − C τ − − I T Cτ − , m = 1 , . . . , N M . We aggregate the margin-set-level exposure at the netting set level to obtain the netting-set-levelpositive exposure |N m | (cid:88) m =1 |M m | (cid:88) m =1 (cid:16) ˆ P m,m τ − DiscV A m,m τ (cid:17) − C M m τ − − I T C, M m τ − − , m = 1 , . . . , N N , and similarly for the netting-set-level negative exposure.Finally, we sum exposures over netting sets to obtain the portfolio-wide CVA over all K claims as CV A Kt := N N (cid:88) m =1 B rt E Q (cid:20) { τ F V A Kt := N N (cid:88) m =1 B rt × E Q (cid:90) τ ∧ Tt B ru ( r f,l, N m u − r u ) V m u ( ϕ ) − |N m | (cid:88) m =1 (cid:16) C M m , N m u + I T C, M m , N m u (cid:17) + − ( r f,b, N m u − r u ) V m u ( ϕ ) − |N m | (cid:88) m =1 (cid:16) C M m , N m u + I T C, M m , N m u (cid:17) − du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t . (4.15)Similar expressions are obtained for ColVA and MVA, again over all K claims in the portfolio: ColV A Kt := N N (cid:88) m =1 |N m | (cid:88) m =1 B rt E Q (cid:20) (cid:90) τ ∧ Tt B ru (cid:20) ( r c,l, M m , N m u − r u ) (cid:16) C M m , N m u (cid:17) + − ( r c,b, M m , N m u − r u ) (cid:16) C M m , N m u (cid:17) − (cid:21) du (cid:12)(cid:12)(cid:12)(cid:12) G t (cid:21) , (4.16) M V A Kt := N N (cid:88) m =1 |N m | (cid:88) m =1 B rt E Q (cid:20) (cid:90) τ ∧ Tt B ru (cid:104) ( r I,l, M m , N m u − r u ) I T C, M m , N m u − r I,b, M m , N m u I F C, M m , N m u (cid:105) du (cid:12)(cid:12)(cid:12) G t (cid:105) . (4.17)By regrouping all portfolio-wide value adjustments (4.13), (4.14), (4.15), (4.16) and (4.17) we obtainan xVA correction for the entire portfolio that accounts for multiple discounting regimes and multipleaggregation levels. We set XV A Kt := F V A Kt + ColV A Kt + M V A Kt − CV A Kt + DV A Kt , (4.18) (cid:92) XV A Kt := XV A Kt + N N (cid:88) m =1 |N m | (cid:88) m =1 |M m | (cid:88) m =1 DiscV A m,m ,m t , (4.19)and finally write the whole portfolio value as V Kt ( ϕ ) := K (cid:88) m =1 ˆ P mt − (cid:92) XV A Kt . (4.20)To avoid further heavy notations, we omit the statement of such BSDE.A natural question involves the well-posedness of the portfolio-wide valuation BSDE for V . Upondirect inspection of (4.13)- (4.17) we observe the following:(i) The presence of multiple margin sets is represented by the introduction of multiple collateralaccounts. If we assume that each margin account (be it of variation margin or initial margintype) satisfies the same assumptions from Section 2.4.1 and Section 2.4.2, then existence anduniqueness for a portfolio wide-BSDE in the context of one netting set and multiple marginsets immediately follow from our discussion so far as an application of our arguments fromSection 3.4.(ii) Multiple netting sets are simply accounted for by summing value adjustments over all nettingsets, each netting set possibly featuring multiple margin sets. Being a sum of well-posed netting set specific BSDEs, the well posedness of the full portfolio BSDE is then again animmediate consequence of our arguments from Section 3.4.In other words, netting sets correspond to different BSDEs, whereas margin sets appear as additionalterms in the driver.In summary, our assumptions underlying the single-claim xVA framework from Section 3 are sufficientto guarantee the well posedness of the BSDE for V also in the presence of multiple aggregation levels.4.3. Incremental xVA charge. Consider the situation where the portfolio of contingent claimsbetween the bank and the counterparty consists of K claims, so that the full portfolio value is givenby (4.20). From the discussion so far it is evident that the portfolio-wide value adjustment (cid:92) XV A K does not coincide with the sum of K distinct xVA processes for the K distinct claims. This is dueboth to the presence of different aggregation levels (margin sets and netting sets) and the non-lineareffects induced by different rates for borrowing and lending.Let us assume now that the counterparty wishes to enter into a further ( K + 1)-th trade with thebank. If entered, the newly introduced ( K + 1)-th claim would contribute to the global riskiness of theportfolio between the bank and the counterparty. It is natural to ask then what is the price the bankshould charge to the newly introduced ( K + 1)-th claim given the presence of the already existing K claims. One could consider two different approaches.(i) Stand-alone scenario: the ( K +1)-th contingent claim and the corresponding xVA are eval-uated in isolation. This corresponds to computing the integral representation with discountingadjustment (4.10) from Proposition 4.5 for the case of the single ( K + 1)-th contingent claim,i.e. only for m = K + 1. This scenario underestimates diversification benefits, due to existingdeposited margins and netting agreements.(ii) Incremental xVA charge: to account for portfolio effects involving margin and nettingsets, two different scenarios are compared.(a) Base scenario: The value of the portfolio is given by V Kt ( ϕ ) as in formula (4.20). Thiscorresponds to the value of the portfolio before the inclusion of the candidate new trade.(b) Full scenario: The value of the portfolio is given by V K +1 t ( ϕ ), computed in line withformula (4.20). This corresponds to the value of the portfolio after the inclusion of thecandidate ( K + 1)-th contingent claim.The bank determines the price to be charged to the counterparty as the difference betweenthe value of the portfolio under the full and the base scenario, i.e. the bank charges the incremental value ∆ V K +1 t , defined as∆ V K +1 t ( ϕ ) := V K +1 t ( ϕ ) − V Kt ( ϕ ) . (4.21)The incremental value (4.21) represents the prevailing market practice. From the perspective of thecounterparty it has the interesting implication that the counterparty, who wishes to invest in the( K + 1)-th claim, when setting up an auction on the ( K + 1)-th claim, will be offered different pricingproposals by the different banks participating in the auction, due to the different structures of theexisting portfolios.By analyzing (4.21) we can isolate the impact of the ( K + 1)-th trade as follows.∆ V K +1 t ( ϕ ) := V K +1 t ( ϕ ) − V Kt ( ϕ )(4.22) = K +1 (cid:88) m =1 ˆ P mt − (cid:92) XV A K +1 t − K (cid:88) m =1 ˆ P mt + (cid:92) XV A Kt SDES OF XVA 31 = ˆ P K +1 t − (cid:16) XV A K +1 t − XV A Kt (cid:17) − DiscV A K +1 t = ˆ P K +1 t − ∆ XV A t − DiscV A K +1 t , where, in the last step, we implicitly defined the incremental xVA charge (4.23) ∆ XV A t := XV A K +1 t − XV A Kt as the adjustment to be charged on the ( K + 1)-th claim, given the presence of the already existing K claims in the portfolio.Our discussion motivates the introduction of the concept of non-linearity effect. Definition 4.11. The non-linearity effect on the ( K + 1) -th contingent claim is defined as N L t (cid:0) V K +1 (cid:1) := V K +1 t ( ϕ ) − ∆ V K +1 t ( ϕ ) , (4.24) where V K +1 t ( ϕ ) is determined by solving the stand-alone G -BSDE and ∆ V K +1 t ( ϕ ) is the incrementalcharge as defined in (4.22) . The non-linearity effect coincides with the difference of the incremental xVA charge and the stand-alone xVA, in fact: N L t (cid:0) V K +1 (cid:1) := V K +1 t ( ϕ ) − ∆ V K +1 t ( ϕ )= (cid:16) ˆ P K +1 t − XV A t − DiscV A K +1 t (cid:17) − (cid:16) ˆ P K +1 t − ∆ XV A t − DiscV A K +1 t (cid:17) = ∆ XV A t − XV A t . (4.25) Remark . Let us observe the following. • In the present setting the clean valuation of the contingent claim is still linear, hence the cleanvalue of the portfolio still corresponds to the sum of the clean values of the single claims. • We typically have ∆ XV A t − XV A t (cid:54) = 0. The stand-alone xVA of the ( K + 1)-th claim ishigher than ∆ XV A . • N L t (cid:0) V K +1 (cid:1) = 0 only when there are no portfolio/netting effects.5. Examples and numerical illustrations We conclude the paper by presenting a simple example using a lognormal model for a single riskyasset. Under the setting and assumptions of the previous sections we consider a single risky asset S = ( S t ) t ∈ [0 ,T ] that pays dividends at a rate κ = ( κ t ) t ∈ [0 ,T ] , with dividend process D t = (cid:82) t κ s S s ds, t ∈ [0 , T ] . The asset price is assumed to have the P -dynamics dS t = S t (cid:16) µ t dt + σ t dW P t (cid:17) , (5.1)where µ t , σ t are deterministic functions of time such that the SDE (5.1) has a unique strong solution.Under the martingale measure Q defined by (3.5) the risky asset evolves according to dS t = S t (cid:16) ( r rt − κ t ) dt + σ t dW P t (cid:17) , (5.2)where r r = ( r rt ) t ∈ [0 ,T ] is the repo rate associated to the asset S . We now consider a simple contingentclaim, namely a forward written on the asset S . The dividend process of the claim A = ( A t ) t ∈ [0 ,T ] , is given by A t = 1 { t = T } ( S T − K ) , (5.3) for K a positive constant. We recall that the clean value ˆ V satisfying (3.18) represents a fictiousvalue process for the claim under the assumption of a perfect collateralization scheme that annihilatescounterparty risk, see Assumption 3.17.According to Theorem 3.20 the arbitrage free price of the forward isˆ V t ( ϕ ) = E Q (cid:34) B rt (cid:90) ( t,T ] dA u B ru (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) = B rt E Q (cid:20) S T − K B rT (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) . (5.4)Assume now that the bank enters a forward with a counterparty without any collateral agreementand without any previous existing trade: there is no exchange of variation or initial margin, meaningthat C = I T C = I F C = 0, d Q ⊗ dt -a.s. Exposures on such transactions are to be funded by theinternal treasury desk of the bank, hence, due to internal rules of the bank, the front office deskdecides to discount cashflows via a synthetic unsecured discount curve with associated short rateprocess r f = ( r ft ) t ∈ [0 ,T ] defined via r f = r f,l + r f,b , where r f,l and r f,b are defined in Section 2.3.Hence the official clean price from the bank perspective isˆ P t = E Q (cid:34) B ft (cid:90) ( t,T ] dA u B fu (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) = B ft E Q (cid:34) S T − K B fT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) . (5.5)The xVA desk is forced by the internal policy of the bank to employ (5.5) as the official clean pricefor the transaction. However, using Proposition 4.5 it is possible to compute a consistent price whichis then given by V t ( ϕ ) = V t ( ϕ ) = ˆ P t − XV A t − DiscV A t , where XV A t = − CV A t + DV A t + F V A t = − B rt E Q (cid:20) { τ 0) + r t XV A t (cid:105) dt − (cid:80) dk =1 ˜ Z kt dW k, Q t − (cid:80) j ∈{ B,C } ˜ U jt dM j, Q t ,XV A τ = 1 { τ ≤ T } (cid:16) ˆ V τ ( ϕ ) − θ τ ( ˆ V , , (cid:17) . (5.8)We observe that the non-linearity effect N L t ( V ) = 0 is of course zero, since the portfolio betweenthe bank and the counterparty consists of a single contingent claim.Assume now that the counterparty is interested in a second product, e.g. a second forward contracton the risky asset S with maturity T and opposite direction, so that A t = 1 { t = T } ( K − S T ) . (5.9) SDES OF XVA 33 In line with the previous reasoning, the clean values from the perspective of the xVA desk and thefront-office desk are respectivelyˆ V t ( ϕ ) = B rt E Q (cid:20) K − S T B rT (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , ˆ P t = B rt E Q (cid:34) K − S T B fT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:35) . (5.10)Given the presence of the first forward contract in the portfolio, the full value of the portfolio, nowincluding the second claim, is V t ( ϕ ) = ˆ P t + ˆ P t − XV A t − DiscV A t − DiscV A t , where XV A t = − CV A t + DV A t + F V A t = − B rt E Q (cid:20) { τ 0) + r t XV A t (cid:105) dt − (cid:80) dk =1 ˜ Z kt dW k, Q t − (cid:80) j ∈{ B,C } ˜ U jt dM j, Q t XV A τ = 1 { τ ≤ T } (cid:16) ˆ V τ ( ϕ ) + ˆ V τ ( ϕ ) − θ τ ( ˆ V + ˆ V , , (cid:17) (5.12)is given by (5.6).Given the presence of the first claim in the portfolio, the XVA charge on the second claim is ∆ XV A = XV A − XV A , whereas the non-linearity is N L t ( V ) = XV A t + B rt E Q (cid:20) { τ Computation of the discounting valuation adjustment. We conclude the paper by presentingtwo numerical illustrations. The source code for our examples is available at https://github.com/AlessandroGnoatto/BiaginiEtAlExamples . The first one aims at providing evidence regarding thelast claim of the previous section, namely that the estimation of DiscVA is a feasible task. Wepostulate that the risky asset evolves according to (5.1). We assume, for the sake of simplicity, thatall parameters are constant and we set r r = r = 0 . κ = 0, σ = 0 . S = 100. In line with theprevious section we consider the forward contract (5.3) written on 1000 units of S with strike K = 80and T = 1.For such a simple claim we can compute prices without resorting to simulations. In particular, thexVA desk computes the price ˆ V according to (5.4), i.e. ˆ V = 20795 . 22 EUR, obtained by assuminga perfect collateralization with collateral rate r . However, the claim is perfectly uncollateralized,hence the front office function employs for valuation the unsecured discount rate r f = 0 . 05 and hencecomputes the price P according to (5.5). The front office price is P = 19980 . 62 EUR. As we havealready seen, the front office price is not consistent with the portfolio-wide valuation of the xVA desk,however the xVA desk can solve the consistency problem by computing DiscV A as in (5.7).To estimate (5.7) we performed a Monte Carlo simulation with 10 paths over a time discretizationcovering the interval [0 , T ] with 10 time steps and a uniform mesh grid. At each point in time on theresulting discretization grid we compute pathwise the exposure: this means that, at each realizationof the underlying, t i and path ω j we are able to do an analytic, exact computation of the conditionalexpectation (5.4). Since the claim is very simple it admits a closed-form valuation formula and sowe do not need to resort on regression estimates of the conditional expectation. We remark howeverthat this is a technical detail, since the use of regression estimates does not alter the validity of thenumerical procedure. By repeating the calculation above over all paths and over all points in time,we generate a surface of values, where each row represents a single path and each column representsa single point in time. To construct a Monte Carlo estimate of (5.7) we simply approximate theintegral with respect to time by means of a standard quadrature (e.g. via a simple trapezoidal rule)and then we compute the sample average over all paths. The resulting Monte Carlo estimator is then DiscV A = − . . If we sum the (analytic) price used by the front office P and the DiscV A estimator, we obtain 20785 . 33 EUR, that we compare with the aforementioned value ˆ V = 20795 . Of course, the equality sign is a slight abuse of notation, since we computed a Monte Carlo approximation. SDES OF XVA 35 The small difference we observe between the two values is explained by the two numerical errors of ourprocedure: we have the numerical error induced by the quadrature used to approximate the integralwith respect to time and the Monte Carlo error.5.1.2. Portfolio effects, non-linearity effect. The second experiment we propose aims at showing therelevance of portfolio effects in xVA computations. Again a simple example will suffice to provideenough intuition. To illustrate the issue we simplify the treatment by assuming that there is a uniquerisk-free rate r involved in all valuations. We consider again the bank trading two forwards on thestock S as before. We assume again T = 1 and set K = S = 100. We suppose that the forwardis written on 1000 units of the stock, and that only the counterparty can default. In summary, thewhole xVA adjustment is solely given by the CVA. We suppose that the counterparty has a constanthazard rate λ C, Q = 0 . 04 and a recovery rate R C = 0 . S . After that, we perform apathwise simulation of the exposure of the forward, which we then numerically integrate with respectto time and average over all paths. This procedure produces a Monte Carlo estimation of the CVAunder the filtration F according to Corollary 3.34. We obtain an estimate for CV A = 148 . 17 EUR.Let us now introduce the second forward mentioned above, where we assume again that T = T = 1and set K = 90. We also suppose that the second forward is written on the same quantity of sharesof S , namely 1000. We observe that, due to the different strikes, the second forward does not perfectlyoffset the first one. We first assumed that the second forward is the only claim in the portfolio and thusobtained an estimate of the stand alone CVA of 309 . 22 EUR, so that the sum of the CVAs of the twoforwards, ignoring portfolio effects, is 457 . 49 EUR. Such value clearly overestimates the outstandingcredit exposure between the bank and the counterparty.By relying again on a Monte Carlo simulation under the same assumptions as above, we compute theportfolio-wide CVA, i.e. CV A and we obtain the estimate CV A = 232 . 69 EUR. We observe thenthat the incremental CVA, ∆ CV A = 84 . 52 EUR. Finally, the non linearity from Definition 4.11 isestimated by N L ( V ) = 232 . − . 52 = 148 . 17 EUR.The example we propose shows quite clearly the relevance of portfolio effects: if the xVA desk ignoredportfolio effects, the xVA charge would be 457 . 49 EUR. By applying the incremental approach to xVAcharge instead, when the second forward is included in the portfolio, there is only an additional chargeof 84 . 52 EUR. This is due to the fact that the two credit exposure partially compensate each other.In Figure 2 we provide a further visualization of the portfolio effects. We compute the Monte Carlosample average of the negative and positive part of the credit exposure of the forwards under con-sideration: such quantities are usually termed expective negative (resp. positive) exposure . Also, wecompute the 95%-quantile of the exposure. Red lines correspond to the first forward with strike K whereas green lines refer to the second forward with strike K . Finally, the portfolio resulting from thecombination of the two forwards is represented by a blue line. We can clearly observe that combiningthe two claims has a beneficial effect in terms of reduction of the exposure: in particular we observethat the 95%-quantile is constant. References Agarwal, A., De Marco, S., Gobet, E., Lpez-Salas, J., Noubiagain, F., and Zhou, A. (2018). Backwardstochastic differential equations arising in initial margin requirements. Preprint hal-01686952v2. 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Fx modelling in collateralized markets: foreign measures, basiscurves, and pricing formulae. Preprint (available at https://arxiv.org/abs/1508.04321).Nie, T. and Rutkowski, M. (2016). Bsdes driven by multidimensional martingales and their applicationsto markets with funding costs. Theory Probab. Appl. , 60(4):604–630.Pallavicini, A., Perini, D., and Brigo, D. (2011). Funding Valuation Adjustment: a consistent frame-work including CVA, DVA, collateral,netting rules and re-hypothecation. ArXiv e-prints , pagearXiv:1112.1521.Piterbarg, V. (2010). Funding beyond discounting: collateral agreements and derivatives pricing. RiskMagazine , 2:97–102.Piterbarg, V. (2012). Cooking with collateral. Risk Magazine , 2:58–63.Sokol, A. (2014). Long-Term Portfolio Simulation: For XVA, Limits, Liquidity and Regulatory Capital .Risk Books, London. SDES OF XVA 39 Appendix A. Existence and Uniqueness of BSDEs In this section we review some results on existence and uniqueness for some BSDEs. Our mainreferences are Nie and Rutkowski (2016), which in turn extends results from Carbone et al. (2008),and Agarwal et al. (2018).Let M = (cid:0) M , . . . M d (cid:1) (cid:62) be a d -dimensional, real-valued, continuous and square integrable martin-gale on a filtered probability space (Ω , F , F , Q ) , where the filtration is assumed to satisfy the usualhypotheses and we assume that the predictable representation property holds with respect to M for( F , Q )-martingales. We use (cid:104) M (cid:105) to denote the quadratic variation of M . Assumption A.1 (Nie and Rutkowski (2016) Assumption 3.1) . There exists an R d × d -valued process m and an F -adapted, continuous, bounded, increasing process Q with Q = 0 such that, for all t ∈ [0 , T ] , (cid:104) M (cid:105) t = (cid:90) t m u m (cid:62) u dQ u . (A.1)If M = W is a one-dimensional standard Brownian motion, then Q t = t , whereas m corresponds tothe identity matrix. Next we introduce the driver of the BSDE via the following Assumption A.2 (Nie and Rutkowski (2016) Assumption 3.2) . Let h : Ω × [0 , T ] × R × R d (cid:55)→ R bean F ⊗ B ([0 , T ]) ⊗ B ( R ) ⊗ B ( R d ) -measurable function such that h ( · , · , y, z ) is an F -adapted process forany fixed ( y, z ) ∈ R × R d . The BSDEs of interest in view of financial applications are forward-backward SDEs (FBSDEs). Fol-lowing Nie and Rutkowski (2016), we introduce a generic (forward) factor matrix-valued process givenby X t := X t . . . X t . . . . . . X dt , t ∈ [0 , T ] , where the auxiliary processes X i , i = 1 , . . . , d, are assumed to be F -adapted. The processes X i represent market risk factors or traded assets. We assume that the function h of Assumption A.2 canbe written as h ( ω, t, y, z ) = g ( ω, t, y, X t z ) , for g satisfying Assumption A.2. Definition A.3 (Nie and Rutkowski (2016) Definition 4.1) . We say that an R d × d -valued process γ satisfies the ellipticity condition if there exists a constant Λ > such that d (cid:88) i =1 (cid:16) γ t γ (cid:62) t (cid:17) ij a i a j ≥ Λ (cid:107) a (cid:107) (A.2) for all a ∈ R d and t ∈ [0 , T ] . Assumption A.4 (Nie and Rutkowski (2016) Assumption 4.2) . The R d × d -valued F -adapted process m in (A.1) is given by m t m (cid:62) t = X t γ t γ (cid:62) t X (cid:62) t , where γ = [ γ ] ij is a d -dimensional square matrix of F -adapted processes satisfying the ellipticitycondition (A.2) . In the following we recall some definitions from Nie and Rutkowski (2016). Definition A.5. We say that the function h : Ω × [0 , T ] × R × R d (cid:55)→ R satisfies • the uniform Lipschitz condition if there exists a constant L such that for any t ∈ [0 , T ] andall y , y ∈ R , z , z ∈ R d | h ( t, y , z ) − h ( t, y , z ) | ≤ L ( | y − y | + (cid:107) z − z (cid:107) ) ; • the uniform m -Lipschitz condition if there exists a constant ˆ L such that for any t ∈ [0 , T ] andall y , y ∈ R , z , z ∈ R d | h ( t, y , z ) − h ( t, y , z ) | ≤ ˆ L (cid:16) | y − y | + (cid:13)(cid:13)(cid:13) m (cid:62) t ( z − z ) (cid:13)(cid:13)(cid:13)(cid:17) ; • the uniform X -Lipschitz condition if there exists a constant ˜ L such that for any t ∈ [0 , T ] andall y , y ∈ R , z , z ∈ R d | h ( t, y , z ) − h ( t, y , z ) | ≤ ˜ L ( | y − y | + (cid:107) X t ( z − z ) (cid:107) ) . Lemma A.6 (Nie and Rutkowski (2016) Lemma 4.2) . If Assumption A.4 holds and the generator h is uniform X -Lipschitz, then h is uniform m -Lipschitz with ˆ L = ˜ L max (cid:110) , Λ − (cid:111) , where Λ is theconstant defined in (A.2) . Theorem A.7 provides the existence and uniqueness result, which is relevant for our purposes. Theorem A.7 (Nie and Rutkowski (2016) Theorem 4.1) . Assume that the function h can be repre-sented as h ( t, y, z ) = g ( t, y, X t z ) , where the function g : Ω × [0 , T ] × R × R d (cid:55)→ R satisfies the uniformLipschitz condition. Let the process h ( · , , belong to the space H ( Q ) , the random variable η belongto L ( F T , Q ) and U be a real-valued F -adapted process such that U ∈ H ( Q ) and U T ∈ L ( F T , Q ) .Assume that the process m satisfies Assumption A.4 for some constant Λ > . Then the BSDE dY t = Z (cid:62) t dM t − h ( t, Y t , Z t ) dQ t + dU t ,Y T = η, (A.3) has a unique solution ( Y, Z ) such that ( Y, m (cid:62) Z ) ∈ H ( Q ) × H ,d ( Q ) . Moreover the processes Y and U satisfy E Q (cid:34) sup t ∈ [0 ,T ] | Y t − U t | (cid:35) < ∞ . We now recall the results of Agarwal et al. (2018). Assumption A.8 (Agarwal et al. (2018) Assumption (A)) . For any X ∈ S ( Q ) , (Λ t ( X t : T )) t ∈ [0 ,T ] defines a stochastic process that belongs to H ( Q ) . There exists a constant C Λ > and a family ofmeasures ( ν t ) t ∈ [0 ,T ] on R such that for every t ∈ [0 , T ] ν t has support included in [ t, T ] , ν ([ t, T ]) = 1 , and for any y , y ∈ S ( Q ) , we have | Λ t ( y t : T ) − Λ t ( y t : T ) | ≤ C Λ E (cid:20)(cid:90) Tt | y s − y s | ν t ( ds ) (cid:12)(cid:12)(cid:12) F t (cid:21) , dt ⊗ d P a.e.Moreover, there exists a constant k > such that for every β ≥ and every continuous path x :[0 , T ] → R , (cid:90) T e βs (cid:90) Ts | x u | ν s ( du ) ds ≤ k sup t ∈ [0 ,T ] e βs | x t | . Assumption A.9 (Agarwal et al. (2018) Assumption (S)) . For any y, z, λ ∈ R × R d × R , f ( · , y, z, λ ) is an F -adapted stochastic process with values in R and there exists a constant C f > such that P -a.s., SDES OF XVA 41 for all ( s, y , z , λ ) , ( s, y , z , λ ) ∈ [0 , T ] × R × R d × R , | f ( s, y , z , λ ) − f ( s, y , z , λ ) | ≤ C f ( | y − y | + | z − z | + | λ − λ | ) . Moreover, E (cid:104)(cid:82) T | f ( s, , , | ds (cid:105) < ∞ . Lemma A.10 (Agarwal et al. (2018) Lemma 2.2) . Let ( Y , Z ) , ( Y , Z ) ∈ S ( Q ) × H ,d ( Q ) , besolutions to the McKean anticipative BSDEs (hereafter, MKABSDE) Y t = ξ + (cid:90) Tt f ( s, Y s , Z s , Λ s ( Y s : T )) ds − (cid:90) Tt Z s dW s , t ∈ [0 , T ] , associated to the parameters ( f , Λ , ξ ) and ( f , Λ , ξ ) . We assume f satisfies Assumption A.9 andthat Λ satisfies Assumption A.8. Let us define δY := Y − Y , δZ := Z − Z , δξ := ξ − ξ . Finally,let us define for s ∈ [ t, T ] ,δ f := f ( s, Y s , Z s , Λ ( Y s : T )) − f ( s, Y s , Z s , Λ ( Y s : T )) and δ Λ s := Λ s ( Y s : T ) − Λ s ( Y s : T ) . Then, there exists a constant C > such that for µ > , we have for β large enough (cid:107) δY (cid:107) S β,T ≤ C (cid:20) e βT E (cid:2) | δξ | (cid:3) + µ (cid:18) (cid:107) δ f (cid:107) H ,dβ,T + C f (cid:107) δ Λ (cid:107) H ,dβ,T (cid:19)(cid:21) , (cid:107) δZ (cid:107) H ,dβ,T ≤ C (cid:20) e βT E (cid:2) | δξ | (cid:3) + µ (cid:18)(cid:13)(cid:13) δ ¯ f (cid:13)(cid:13) H ,dβ,T + C f || δ Λ (cid:107) H ,dβ,T (cid:19)(cid:21) . Theorem A.11 (Agarwal et al. (2018) Theorem 2.1) . Under Assumptions A.8 and A.9, for anyterminal condition ξ ∈ L T ( F T , Q ) , the BSDE Y t = ξ + (cid:90) Tt f ( s, Y s , Z s , Λ( Y s : T )) ds − (cid:90) Tt Z s dW s , t ∈ [0 , T ] has a unique solution ( Y, Z ) ∈ S ( Q ) × H ,d ( Q ) . A.1. Proof of Proposition 3.13. Since we are only trading in the basic risky assets, the position inthe initial margin is zero hence, by (2.11) and (3.1), the value process is of the form V t ( ϕ ) = ψ f,bt B f,bt + ψ f,lt B f,lt , t ∈ [0 , T ] . (A.4)Recalling that simultaneous borrowing and lending at the same time is not allowed, we have by (A.4)that ψ f,l = ( V t ( ϕ )) + (cid:16) B f,lt (cid:17) − , ψ f,b = − ( V t ( ϕ )) − (cid:16) B f,bt (cid:17) − , t ∈ [0 , T ] . Moreover, we can rewrite the funding term of the generic i -th risky assets as follows (cid:90) t ψ iu dB iu = − (cid:90) t ξ iu S iu B iu dB iu = − (cid:90) t r iu ξ iu S iu du, t ∈ [0 , T ] . Upon substitution in the self-financing condition (3.2), we obtain dV t ( ϕ ) = d (cid:88) i =1 ξ it (cid:0) dS it + dD it − r it S it dt (cid:1) + (cid:88) j ∈{ B,C } ξ jt (cid:16) dP jt − r jt P jt − dt (cid:17) − r f,bt ( V t ( ϕ )) − dt + r f,lt ( V t ( ϕ )) + dt. We now use the inequality r f,lt ≤ r f,bt from Assumption 3.11, hence dV t ( ϕ ) = d (cid:88) i =1 ξ it (cid:0) dS it + dD it − r it S it dt (cid:1) + (cid:88) j ∈{ B,C } ξ jt (cid:16) dP jt − r jt P jt − dt (cid:17) + r f,lt ( V t ( ϕ )) + dt − r f,bt ( V t ( ϕ )) − dt ≤ d (cid:88) i =1 ξ it (cid:0) dS it + dD it − r it S it dt (cid:1) + (cid:88) j ∈{ B,C } ξ jt (cid:16) dP jt − r jt P jt − dt (cid:17) + r f,lt V t ( ϕ ) dt. Introducing ˜ V lt ( ϕ ) := (cid:16) B f,lt (cid:17) − V t ( ϕ ) , we have then the inequality d ˜ V lt ( ϕ ) ≤ d (cid:88) i =1 ξ it (cid:16) B f,lt (cid:17) − (cid:0) dS it + dD it − r it S it dt (cid:1) + (cid:88) j ∈{ B,C } ξ jt (cid:16) B f,lt (cid:17) − (cid:16) dP jt − r jt P jt − dt (cid:17) or equivalently, d ˜ V lt ( ϕ ) ≤ d (cid:88) i =1 ξ it B it B f,lt (cid:0) dS it + dD it − r it S it dt (cid:1) B it + (cid:88) j ∈{ B,C } ξ jt B jt B f,lt (cid:16) dP jt − r jt P jt − dt (cid:17) B jt , and so, by (3.4), we arrive at the inequality(A.5) d ˜ V lt ( ϕ ) ≤ d (cid:88) i =1 ξ it B it B f,lt d ˜ S i,cldt + (cid:88) j ∈{ B,C } ξ jt B jt B f,lt d ˜ P jt . We observe that the right-hand side in (A.5) is a local martingale, which is bounded from below, byDefinition 3.7 and Assumption 3.11 on r f . This implies that the aforementioned right-hand side is asupermartingale. Absence of arbitrage follows along the usual lines. (Francesca Biagini) LMU M¨unchen, Mathematics Institute,Theresienstr. 39, D-80333 Munich, Germany E-mail address , Francesca Biagini: [email protected] (Alessandro Gnoatto) University of Verona, Department of Economics,via Cantarane 24, 37129 Verona, Italy E-mail address , Alessandro Gnoatto: [email protected] (Immacolata Oliva) University of Rome, Department of Methods and Models for Economics, Territoryand FinanceVia del Castro Laurenziano 9, 00161 Rome, Italy E-mail address , Immacolata Oliva:, Immacolata Oliva: