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Frontiers of Mathematical Finance
Vol. 2, No. 3, September 2023, pp. 283-312
doi:10.3934/fmf.2023014
DERIVATIVES RISKS AS COSTS
IN A ONE-PERIOD NETWORK MODEL
Dorinel Bastide
1,2, St´
ephane Cr´
epey
∗3, Samuel Drapeau
4and
Mekonnen Tadese
5
1BNP Paribas Stress Testing Methodologies & Models, France
2LaMME, Universit´e d’Evry/Universit´e Paris-Saclay CNRS UMR 8071, France
3LPSM, Sorbonne Universit´e and Universit´e Paris Cit´e, CNRS UMR 8001, France
4Shanghai Jiao Tong University, Shanghai, China
5Woldia University, Mathematics Department, Ethiopia
(Communicated by Robert Jarrow and Dilip Madan)
Abstract. In counterparty credit risk complete markets, collateral and capi-
tal requirements would be indifferent to banks. The quantification by banks of
market incompleteness based on various XVA metrics ([11]) has emerged as the
unintended consequence of the FRTB banking reform ([26]) and of the more
demanding regulatory capital requirements ([38]). The related risks are in fact
reckoned today as the major risks for banks, well ahead market risk ([35, Figure
65 page 67]). The XVA metrics have been introduced and traditionally used
by investment banks for pricing and collateral/capital optimization purposes.
We demonstrate in this paper that they can be fruitfully used for risk man-
agement, suggesting a sound approach to regulatory requirements. We present
a one-period cost-of-capital XVA setup encompassing bilateral and centrally
cleared trading in a unified framework, with explicit formulas for most quan-
tities at hand. We illustrate possible uses of this framework for running stress
test exercises on financial networks with one and two clearinghouses from a
clearing member’s perspective or for optimizing the porting of the portfolio
of a defaulted clearing member using Monte-Carlo technique with correspond-
ing confidence errors in elliptical models. A continuous-time extension of this
approach is provided in the companion paper [7].
2020 Mathematics Subject Classification. Primary: 91-10, 91B05, 91G15, 91G45; Secondary:
62P05, 91G70.
Key words and phrases. Financial network, counterparty credit risk, central clearing counter-
parties, XVA, risk management, reverse stress test, economic capital, uncertainty quantification.
This work benefited from the support of the grant When Credit Meets Liquidity: The Clearing
Member Default Resolution Issue, under the aegis of the Europlace Institute of Finance, France.
The research of St´ephane Cr´epey benefited from the support of the Chair Stress Test, RISK
Management and Financial Steering, led by the French ´
Ecole polytechnique and its Foundation
and sponsored by BNP Paribas.
This article is not meant to represent the position or opinions of BNP Paribas or its members.
∗Corresponding author: St´ephane Cr´epey.
283
284 D. BASTIDE, S. CR´
EPEY, S. DRAPEAU AND M. TADESE
1. Introduction. In the wake of the 2008–09 global financial crisis, clearing through
central counterparties (CCPs) has become mandatory for standardized derivatives,
other ones remaining under bilateral setup with higher capital requirements.
One role of the CCPs1is to provide to their clearing members fully collateralized
hedges of their market risk with their clients. But this comes at a cost to the
clearing members, which pass it to their corporate clients in the form of XVA
(cross-valuation adjustment) add-ons. Bearing in mind that the risks of a hedge
are, by definition, of the same magnitudes as the ones of the originating position and
that standardized derivatives usable as hedging assets have to be traded through
CCPs, the XVA footprint of not only bilateral but also centrally cleared trading
is significant and should be analyzed in detail, which is the topic of this paper.
[11, Section 6] provides a continuous-time XVA analysis in the realistic situation
of a bank dealing with an arbitrary number of clients and CCPs. For the sake
of tractability, this is mimicked here in a stylized one-period setup, fine-tuned to
applications including risk assessment in the context of stress test exercises2or
optimizing the porting of the portfolio of defaulted clearing members.
The first type of application is motivated by the default in 2020 of Ronin Capital,
a broker/dealer firm that had clearing exposures on both CCP services Fixed Income
Clearing Corporation (FICC) GSD3segment (123 members) and CME Futures (56
members of which 24 common with FICC GSD). If all members are assumed to
be only exposed to these CCPs and their cleared clients, we can illustrate these
relationships by the network depicted in Figure 1. Any common member on those
two CCPs needs to ensure conservative risk assessment that can be achieved in the
proposed framework by accounting for common memberships on the two CCPs. If
such common memberships are ignored, they can lead to lower loss estimates giving
wrong risk view on potential losses.
The second type of application is an illustration of the results of defaulted port-
folio porting as it has been the case for the trader Einer Aas on NASDAQ OMX4
that has defaulted on 2018 with loss spill-over effect on surviving members.
The paper is outlined as follows. Section 2sets the stage. Section 3develops the
corresponding XVA analysis. Section 4sets up an elliptical market and credit model
amenable to efficient XVA computations. Section 5introduces the case studies.
Section 6provides numerical results of stress test exercises. Section 7shows how to
optimize the porting of defaulted members portfolios. Section 8concludes.
2. General setup. We consider a finite set of market participants, also suscep-
tible to serve as clearing members of CCPs. Derivative transactions can then be
concluded between two individual participants, or between a set of participants5,
pooled in the form of a CCP, and a clearing member of this CCP.
The trades of a clearing member bank with a CCP are partitioned between
proprietary trades, which are in effect hedges of the bilateral trading exposure of
the bank, and back-to-back hedges of so-called cleared client trades, through which
non-member clients gain access to the clearing services of a CCP: see Figure 2.
The contractual cash flows from cleared and bilateral clients to a reference clearing
member, dubbed the bank hereafter, are promised in successive turns from the
1See [15] and [16] for general CCP and XVA references, as well as [21] for a CCP survey.
2as required by [38, Article 302].
3Government Securities Division.
4Optionsm¨aklarna/Helsinki Stock Exchange.
5two or more, in practice from a few units to a few hundreds.
DERIVATIVES RISKS AS COSTS 285
Figure 1. Network consisting of two CCPs (in red), 123 members
for CCP1 seen on the left hand side, and 56 members for CCP2
on the right hand side, with 24 common members displayed as the
group of members in the middle of the two CCPs (155 members in
total, in blue), and with 179 cleared clients (in green).
bank to the CCP (cash flows denoted by Pand Pon Figure 2), from the CCP to
other clearing members, and from the latter to their own clients. As a consequence,
the CCP is flat in terms of market risk, as is also each of the clearing members.
CCPs are typically siloed into different services, each devoted to a specific class of
derivatives. We first consider a setup with a single CCP service, the extension to
several CCPs being done in Section 3.3.
2.1. Defaults settlement rule. As reasserted in the wake of the 2008–09 global
financial crisis by the Volcker rule, a dealer bank should be hedged as much as
possible, at least in terms of market risk6. Jump-to-default risk, on the other
hand, is hardly hedgeable in practice. Instead it is mitigated through netting and
collateralization. Namely, designated netting sets of transactions between two given
counterparties (two individual participants or a participant and the CCP) are jointly
collateralized, i.e. guaranteed against the default of one or/and the other party. The
collateral (or guarantee) comprises a variation margin, which tracks the mark-to-
market (counterparty-risk-free value) of the netting set between the two parties,
and nonnegative amounts of initial margin posted by each party to the other, which
provide a defense against the risk of slippage of the value of the netting set away
from its (frozen) variation margin during its liquidation period. In the case of
transactions with a CCP, there is an additional layer of collateral in the form of the
(funded) default fund contributions of the clearing members, which is meant as a
defense against extreme and systemic risk. For each participant, variation margin is
rehypothecable and fungible across all its netting sets. Initial margin is segregated
at the netting set level. Default fund contributions are segregated at the clearing
member level.
6cf. paragraph number 1851 in section 619 from [32].
286 D. BASTIDE, S. CR´
EPEY, S. DRAPEAU AND M. TADESE
CCP
CCP
CCP
cleared
contracts
bilateral
contracts
mirroring trades of
cleared contracts
proprietary trades
CM
CM CM
CM
CM
P
P
Figure 2. Promised cash flows between market participants. The
reference clearing member bank is on the left.
The general rule regarding the settlement of contracts of a defaulted netting set,
to be instantiated in practical setups on a case by case basis7, is that:
Principle 2.1. If a counterparty in default is indebted toward the other beyond its
posted margin, then this debt is only reimbursed at the level of this posted margin
(assuming zero recovery rate of the defaulted party for simplicity in this paper);
otherwise the debt between the two parties is fully settled.
Here debt is understood on a counterparty-risk-free basis.
Remark 2.1. One intuitively expects client default cash flows of the form C=
(1 −R)(D−M)+, where the“debt” Drepresents the pre-default value of the client
derivative portfolio to the bank, Mthe margin posted by the client to the bank, and
Rthe recovery rate of the client. Technically, such an (1 −R)(D−M)+effectively
arises as
D−M+R(D−M)+−(D−M)−= (1 −R)(D−M)+,
where M+R(D−M)+−(D−M)−is what the bank obtains from the client and
Dwhat the bank pays on the hedge of the portfolio. In the special case where case
R= 0, what the bank obtains from the client simplifies to M−(D−M)−=D∧M,
in line with Principle 2.1, and the above expression to
D−D∧M= (D−M)+.
We emphasize that a counterparty credit default loss C= (1 −R)(D−M)+(or
simply (D−M)+if R= 0) should not be taken as an assumption, but only arises
7cf. e.g. Assumption 3.2 below.
DERIVATIVES RISKS AS COSTS 287
as the result of a computation accounting for the cash flows of the portfolio and its
hedge, derived in a specific market setup under the umbrella of the guiding principle
2.1 (or the corresponding extension to nonzero recovery, skipped for simplicity in
this work). The exact outcome in fact depends on the refined specification of the
setup at hand: see e.g. Assumption 3.2 below and the ensuing formulas (7) (in a
single CCP setup) and (15) (under the multiple CCP extension) for the counterparty
credit default loss Cin the market setup of this work. Such formulas cannot be safely
guessed, they should only be derived from first principles.
Remark 2.2. The above is of course a very crude description of default cash flows.
Nonzero, possibly random, recoveries could be introduced at no harm from a the-
oretical viewpoint, as already pointed out above and done in the continuous-time
setup of [11, Section 3.3]. Nonzero recoveries are of course more realistic. But, from
a qualitative viewpoint that is our main objective in this work, they only soften the
impacts of the defaults. Random recoveries are in line with the uncertainty about
the actual level of recovery rates that are only observed a posteriori and can re-
flect the possibility of liquidating various forms of collateral, account for the output
of liquidation procedures, legal resolutions, and other complex and unobservable
features. For our purposes in this work, random recoveries could be used for em-
phasizing some extra dependencies via correlations with other random modeling
features. This is all ignored hereafter for avoiding to blur the main features.
Principle 2.1 also applies to a netting set of transactions between a clearing mem-
ber and a CCP. However, in our stylized setup, a CCP is nothing but the collection
of its clearing members. Our CCP has no resources of its own (in particular, it
cannot post any default fund contribution, or “skin-in-the-game”8). As long as it
is non-default, i.e. as long as at least one of its clearing members is non-default,
our CCP can only handle the losses triggered by the defaults of some of its clearing
members by redirecting these losses on the surviving ones. This participation of
the surviving members to the losses triggered by the defaults of the other mem-
bers corresponds in our framework to the usage by the CCP of their default fund
contributions, both funded (as already introduced above) and unfunded. As will
be detailed in equations below, the funded default fund contributions are used for
covering losses triggered by the defaults of clearing members over their margins.
The unfunded default fund contributions correspond to additional refills that can
be required by the CCP, often up to some cap in principle, without bounds in our
model, in case the funded default fund contributions of the surviving members are
not enough.
2.2. XVA framework. In a nutshell, the main XVAs are the CVA, the FVA/MVA,
and the KVA, where:
i. the CVA is the expected cost for the bank of the default risk of its clients;
ii. the FVA/MVA is the expected cost for the bank of its own default risk or, more
precisely, of the implications of this risk in terms of rehypothecable/segregated
collateral funding spreads for the bank;
iii. the KVA is the cost for the bank of having to remunerate its shareholders
at some hurdle rate for their capital at risk, capital which is required by the
8such additional protection layer, though quite common in practice, is of marginal magnitude
compared to the other protection layers. By omitting skin-in-the-game component, the obtained
results are conservative in terms of risk management and the various formulations are simplified.
288 D. BASTIDE, S. CR´
EPEY, S. DRAPEAU AND M. TADESE
regulator as a provision against the residual risk left uncovered by i. and ii.
(as default risk cannot be hedged by the bank).
Going into details, assume that at time 0 all the banking participants, including
the reference clearing member bank9, with no prior endowments, enter transactions
with their clients and hedge their positions, both bilaterally between them and
through the CCP. As seen above, the CCP and each bank are flat in terms of
market risk. However, as market participants are assumed to be defaultable with
zero recovery, in order to account for counterparty credit risk and its funding and
capital consequences, the reference bank (and each clearing member bank alike)
requires from its corporate clients a pricing rebate (considering conventionally the
bank as the “buyer”) with respect to the mark-to-market (counterparty-risk-free)
valuation of the deals. The corporate clients of the bank are assumed to absorb the
ensuing prices via their corporate business, which is their primary motivation for
these deals.
A reference probability measure R?, relevant for grounding both stress test ex-
ercises and risk management analysis such as economic capital calculation, with
corresponding expectation operator denoted by E?, is used for the linear valuation
of cash flows, using the risk-free asset as our num´eraire everywhere. This choice
of a num´eraire simplifies equations by removing all terms related to the (assumed
risk-free) remuneration of all cash and collateral accounts. The funding issue is then
refocused on the risky funding side of the problem, i.e. funding costs in what follows
really means excess funding costs with respect to a theoretical situation where the
bank could equally borrow and lend at the risk-free rate.
More precisely, as suitable for XVA calculations [6, Remark 2.3]: given a physical
probability measure defined on the full model σalgebra Aand equivalent to a
given risk-neutral measure on the financial sub σalgebra Bof A, we take R?equal
to the risk-neutral measure on Band equal to the physical probability measure
conditionally on B10.
Following the general XVA guidelines of [11, Section 1], the XVA pricing rebate
required by the reference clearing member bank from its corporate clients, dubbed
funds transfer price (FTP), comes in two parts: first, the expected counterparty
default losses and funding expenditures of the bank, an amount that flows into
the bank liabilities and which we refer to as contra-asset valuation (CA = CVA +
FVA + MVA as we will see); second, a cost of capital risk premium (KVA), which
instead is loss-absorbing11 and is also used by the management of the bank as
retained earnings for remunerating the shareholders of the bank for their capital at
risk within the bank. All in one, the bank buys the deals from its clients at the
(aggregated) price (MtM −FTP), where MtM is their counterparty-risk-free value
and
FTP = CA
|{z}
Expected costs
+ KVA
|{z}
Risk premium
.(1)
Let EC denote an economic capital of the bank corresponding to the minimum
level of capital at risk that the bank should hold from a regulatory (i.e. solvency)
perspective. If KVA <EC, then the bank shareholders need to provide the missing
9cf. Figure 2.
10these two conditions uniquely characterize R?[10, Proposition 2.1].
11hence, not a liability.
DERIVATIVES RISKS AS COSTS 289
amount (EC −KVA) of capital at risk, so that the actual level of capital at risk of
the bank is
max(EC,KVA),
while shareholder capital at risk reduces to
max(EC,KVA) −KVA = (EC −KVA)+.(2)
3. Theoretical XVA analysis. In this section we detail each term in the equa-
tions above, in the realistic setup of a bank involved into an arbitrary combination
of bilateral and centrally cleared portfolios, in a tractable one-period setup with
period length T. In the one-period XVA model of [6, Section 3], there were no
CCPs and the bank was assumed to have access to a “fully collateralized back-to-
back hedge of its market risk”, ensuring by definition and for free to the bank a
cash-flow (P − MtM) at time 1, irrespective of the default status of the bank and
its client. There, Pdenoted the contractual cash flows from the (assumed unique)
client to the bank and MtM was the corresponding counterparty-risk-free value. In
the present paper we reveal the mechanism of such a “fully collateralized hedge of
the market risk” of the bank, which can be achieved through central clearing, but
at a certain cost that we analyze.
All proofs are deferred to Section A.
3.1. Cash flows. Given disjoint sets of indices I30, C, and Bfor the clearing
members (including the reference bank labeled by 0) and for the respective cleared
and bilateral netting sets of the bank with its (individual) counterparties, we denote
by:
•J0, shortened as J, and Ji, i ∈I\{0}, the survival indicator random variables
of the bank and of the other clearing members at time 1; γ=R?(J= 0),the
default probability of the bank;
• J = maxiJi, the survival indicator random variable of the CCP (i.e. of at
least one clearing member),
• Pi, MtMi=E?Pi, and IMi,i∈I, the contractual cash flows, variation margin,
and initial margin from the clearing member ito the CCP corresponding to
the cleared clients account of the member i;
• Pi, MtMi=E?Pi, and IMi,i∈I, the contractual cash flows, variation mar-
gin, and initial margin from the clearing member ito the CCP corresponding
to the proprietary (also dubbed house) account of the clearing member i;
•DFi,i∈I, the funded default fund contribution posted by the clearing mem-
ber ito the CCP;
•Jb,b∈B, the survival indicator random variable of the counterparty of the
bilateral netting set bof the reference bank; Pb, VMb, and IMb, the associated
contractual cash flows, variation margin, and initial margin from the corre-
sponding counterparty to the bank; and IMb, the initial margin from the bank
to this counterparty;
•Jc,c∈C, the survival indicator random variable of the client of the cleared
trading netting set cof the bank, and Pc, MtMc=E?Pc12, and IMc, the
associated contractual cash flows, variation margin, and initial margin from
the corresponding client to the bank13;
12reflecting the fact that members of CCPs are fully collateralized.
13note that a bank does not post any initial margin on its cleared client netting sets.
290 D. BASTIDE, S. CR´
EPEY, S. DRAPEAU AND M. TADESE
• L, the loss of the CCP, i.e. the loss triggered by the defaults of its clearing
members beyond their posted collateral14, which is borne by the surviving
members (if any)15;
•µ=Jµ, the proportion of these losses allocated to the reference clearing
member bank (based on remaining survivors).
Moreover, in case i= 0 (so regarding the reference clearing member bank), we
typically skip the index i(as in J0=J).
Assumption 3.1. Pi(Pi+Pi) = 0 (the CCP is flat in terms of market risk),
PcPc=P0(by definition of cleared trades and of their mirroring trades), and
PbPb=P0(the reference bank is flat in terms of market risk).
Assumption 3.1 yields the clearing conditions regarding the contractually
promised cash flows, which applies to each banking participant (written there for
the reference bank) and to the CCP.
Moreover, in line with Principle 2.1 that monitors the default cash flows:
Assumption 3.2. On the CCP survival event {J = 1}, the CCP receives from
each clearing member i
Ji(Pi+Pi) + (1 −Ji)Pi∧(MtMi+ IMi) + Pi∧(MtMi+ IMi)
+(Pi−(MtMi+ IMi))++ (Pi−(MtMi+ IMi))+∧DFi.
(3)
On the bank survival event {J= 1}(⊆ {J = 1}), the bank receives on each cleared
netting set cand bilateral netting set b
JcPc+ (1 −Jc)Pc∧(MtMc+ IMc)and JbPb+ (1 −Jb)Pb∧(VMb+ IMb),(4)
whereas it pays to the CCP
X
cPc+X
bPb=X
cJcPc+ (1 −Jc)Pc+X
bJbPb+ (1 −Jb)Pb.(5)
We need one more condition, regarding the funding side of the problem:
Assumption 3.3. At time 0 the amounts CA and KVA sourced from the corporate
clients of the bank are deposited on reserve capital and capital at risk accounts of
the bank. The bank can use the amounts CA and max(EC,KVA)16 on its reserve
capital and capital at risk accounts for its variation margin borrowing purposes.
Funds needed beyond CA+max(EC,KVA) for variation margin posting purposes are
borrowed by the bank at its credit spread γabove OIS. Instead, the bank must borrow
entirely the amounts to post as initial margin and funded default fund contributions,
but this can be achieved at some blended funding spread eγ≤γ.
The rationale for funding variation margin but not initial margin from CA +
max(EC,KVA) is set out before Equation (15) in [4]. The motivation for the as-
sumption eγ≤γis provided in [3, Section 5], along with numerical experiments
suggesting that eγcan be several times lower than γ.
14variation margin, initial margin, and (funded) default fund contributions.
15see the last paragraph of Section 2.1.
16where max(EC,KVA) −KVA = (EC −KVA)+is provided by the bank shareholders, cf. (2).
DERIVATIVES RISKS AS COSTS 291
Lemma 3.4. The borrowing needs of the bank for reusable and segregated collateral
amount to, respectively,
X
b
(MtMb−VMb)−CA −max(EC,KVA)+and
IM + IM + DF + X
b
IMb.(6)
Lemma 3.5. On the bank survival event {J= 1}, the counterparty default losses
Cand the funding expenses Fof the bank are given by
C=X
b
(1 −Jb)(Pb−VMb−IMb)++X
c
(1 −Jc)(Pc−MtMc−IMc)++µL,(7)
where
L=X
i
(1 −Ji)(Pi−MtMi−IMi)++ (Pi−MtMi−IMi)+−DFi+,(8)
and
F=eγIM + IM + DF
+eγX
b
IMb+γX
b
(MtMb−VMb)−CA −max(EC,KVA)+.(9)
3.2. Valuation. Let Edenote the expectation with respect to the bank survival
measure Rassociated with R?, i.e., for any random variable Y,
EY= (1 −γ)−1E?(JY).(10)
(expectation of Yconditional on the survival of the bank). As (readily) seen in [6,
Section 3]:
Lemma 3.6. For any random variable Yand constant Y, we have
Y=E?(JY+ (1 −J)Y)⇐⇒ Y=EY.(11)
Under a cost-of-capital XVA approach, the bank charges its future losses to its
corporate clients at a CA level making `=J(C+F − CA), the trading loss of the
shareholders of the bank, R?centered. In addition, given a target hurdle rate h
assumed in [0,1] (and typically of the order of 10%), the management of the bank
ensures to the bank shareholders dividends at the height of htimes their capital at
risk (EC −KVA)+(cf. (2)), where we model EC as ES(`), the expected shortfall
of the trading loss `17 computed under the bank survival measure Rat a quantile
level18 α(e.g. α= 99% and α= 99.75% in our experiments), i.e., under the dual
representation of the expected shortfall19:
EC = sup E[`χ] ; χmeasurable, 0 ≤χ≤(1 −α)−1,and E[χ] = 1,(12)
which for atomless `coincides20 with E[`|`≥VaR(`)],where VaRis the Rvalue-at-
risk (lower quantile) at the level α. Note that, in view of (12), an expected shortfall
of a centered random variable is nonnegative.
17assumed Rintegrable.
18under normal distribution assumptions, such ES at percentile level 99.75% allows reaching
similar loss level as with a VaR (quantile) risk metric at the level 99.9%. In practice, regulatory
and economic capital indeed aims at capturing extreme losses that can occur once every 1000
years, cf. paragraph 5.1 from [25] for the detailed instructions.
19see e.g. [17, Theorem 4.1].
20see Corollary 5.3 and representation thanks to expression (3.7) from [2].
292 D. BASTIDE, S. CR´
EPEY, S. DRAPEAU AND M. TADESE
The definitions of the XVA metrics corresponding to the above specifications are
given in Table 1. Hence in view of (7) and (9):
XVA Expression Full name and description
KVA E?J h(EC −KVA)++ (1 −J)KVA,
where EC = ESJ(C+F − CA)capital valuation adjustment
CA CVA + MVA + FVA contra-asset valuation
CVA BCVA + CCVA credit valuation adjustment
BCVA
E? JX
b
(1 −Jb)(Pb−VMb−IMb)+
+(1 −J)BCVA!credit valuation adjustment
for bilateral exposures
CCVA
E? JX
c
(1 −Jc)(Pc−MtMc−IMc)+
+µL+ (1 −J)CCVA!credit valuation adjustment
for clearing activity exposures
MVA BMVA + CMVA margin valuation adjustment
BMVA E? JeγX
b
IMb+ (1 −J)BMVA!margin valuation adjustment
for bilateral exposures
CMVA E?JeγIM + IM + DF+ (1 −J)CMVAmargin valuation adjustment
for clearing activity exposures
FVA
E? JγX
b
(MtMb−VMb)−CA
−max(EC,KVA)++ (1 −J)FVA!funding valuation adjustment
Table 1. XVA definitions, cf. Section 2.2 (with C,Fand Lgiven
by Lemma 3.5).
CA = E?JC+F+ (1 −J)CA,(13)
i.e. E?JC+F − CA)= 0, as desired21 . The terminal cash flows of the form
(1 −J)×· ·· in Table 1expressions and in (13) are thus consistent with the desired
shareholder centric perspective. They can also be interpreted as the amounts of
reserve capital and risk margin lost by the bank shareholders, hence valued as such
by CA, as their property is transferred to the liquidator of the bank if the bank
defaults.
Due to these terminal cash flows, the above definition is in fact a fix-point system
of equations. The split of the underlying CA equation (13) into the collection of
equations in Table 1is motivated by both interpretation and numerical considera-
tions. From an interpretation viewpoint, it is useful to provide the more granular
view on the costs of the bank provided by the split of the global CA amount between,
on the one hand, bilateral and centrally cleared trading default risk components
BCVA and CCVA and, on the other hand, bilateral and centrally cleared trading
21see after Lemma 3.6.
DERIVATIVES RISKS AS COSTS 293
funding risk components BMVA and CMVA for segregated initial margin, whereas
the FVA cost of funding variation margin is holistic in nature (can only be appre-
hended at the level of the bank balance-sheet as a whole), via the feedback impact
of CA + max(EC,KVA) into the FVA. From a numerical viewpoint, the collection
of smaller problems in Table 1may be easier to address than the global equation
(13). Each of the smaller problems can also be handled by a dedicated desk of the
bank, namely the CVA desk, for the BCVA and CCVA, and the Treasury of the
bank, for the BMVA, CMVA and the FVA.
Passing in the above equations to the bank survival measure Rbased on Lemma
3.6 shows that the corresponding fixed point problem is in fact well-posed and yields
explicit formulas for all the quantities at hand.
Theorem 3.7. The explicit XVA formulas of Table 2hold and we have
J(C − CVA) =JX
c
(1 −Jc)(Pc−MtMc−IMc)++µL − CCVA
+X
b
(1 −Jb)(Pb−VMb−IMb)+−BCVA.
(14)
In particular, all the XVA (and also EC) numbers are nonnegative22.
XVA Explicit formula
CCVA E X
c
(1 −Jc)(Pc−MtMc−IMc)++µL!
CMVA eγIM + IM + DF
BCVA E X
b
(1 −Jb)(Pb−VMb−IMb)+!
BMVA eγX
b
IMb
EC ESJ(C − CVA)
FVA γ
1 + γ X
b
(MtMb−VMb)−(CCVA + CMVA + BCVA + BMVA) −EC!+
KVA h
1 + hEC
Table 2. XVA explicit formulas (with C,Fand Lgiven by Lemma 3.5).
Remark 3.8. The reason why funding disappears from the bank trading loss,
i.e. J(C+F −CA) = J(C − CVA), is because, in a one-period setup, the collateral
borrowing requirements (6) of the bank are simply constants. Hence funding triggers
no risk to the bank, but only a deterministic cost. In the dynamic setup of [3],
funding generates both costs and risk.
3.3. Extension to several CCPs or CCP services. In the realistic case where
the reference bank is a clearing member of several services of one or several CCPs,
we index all the CCP related quantities in the above by an additional index ccp in
a finite set disjoint from I∪C∪B. Then, with CA = CCVA + CMVA + BCVA +
BMVA + FVA as before:
22cf. [11, Sections 1 and 7.1].
294 D. BASTIDE, S. CR´
EPEY, S. DRAPEAU AND M. TADESE
Proposition 3.9. The counterparty default loss Cacross several counterparties and
several CCPs is given by
C=X
ccp,c
(1 −Jc)(Pccp
c−MtMccp
c−IMccp
c)++X
ccp
µccpLccp
+X
b
(1 −Jb)(Pb−VMb−IMb)+,
(15)
where
Lccp =X
i
(1 −Ji)(Pccp
i−MtMccp
i−IMccp
i)+
+ (Pccp
i−MtMccp
i−IMccp
i)+−DFccp
i+.
(16)
The funding expenses Facross several CCPs and several counterparties are given
by
F=eγX
ccp IMccp + IMccp + DFccp
+eγX
b
IMb+γX
b
(MtMb−VMb)−CA −max(EC,KVA)+.
(17)
The only XVA definitions and explicit formulas that change with respect to Tables 1
and 2(on top of Cand Fgeneralized as above) are the ones for CCVA and CMVA,
the way detailed in Tables 3and 4. Moreover,
J(C − CVA) =JX
ccp,c
(1 −Jc)(Pccp
c−MtMccp
c−IMccp
c)++X
ccp
µccpLccp −CCVA
+X
b
(1 −Jb)(Pb−VMb−IMb)+−BCVA.
(18)
XVA Expression Full name and description
CCVA
E? JX
ccp,c
(1 −Jc)(Pccp
c−MtMccp
c−IMccp
c)+
+X
ccp
µccpLccp + (1 −J)CCVA!credit valuation adjustment
for clearing activity exposures
CMVA E? JX
ccp eγIMccp + IMccp + DFccp + (1 −J)CMVA!margin valuation adjustment
for clearing activity exposures
Table 3. CCVA and CMVA definitions with several CCPs (also,
Cand Fare now given by Proposition 3.9, as also Lccp ).
XVA Explicit formula
CCVA E X
ccp,c
(1 −Jc)(Pccp
c−MtMccp
c−IMccp
c)++X
ccp
µccpLccp !
CMVA X
ccp eγIMccp + IMccp + DFccp
Table 4. CCVA and CMVA explicit formulas with several CCPs
(also, Cand Fare now given by Proposition 3.9, as also Lccp ).
DERIVATIVES RISKS AS COSTS 295
Before passing to the case studies, we specify the calculation of economic capital
under the member survival measure.
Lemma 3.10. If R(`=VaR(`)) = 0, where `=J(C − CVA), then
EC = E?C − CVAC − CVA ≥VaR(`),J=1.(19)
4. Market and credit model. We introduce a market and credit model, written
under R?, with parameters that can capture dependence between portfolio changes,
joint defaults and possible averse exacerbated changes of the portfolio due to their
owner default known as wrong-way risk.
For any j∈I∪B∪C, denoting by Fjthe marginal cdf of a financial participant
j’s default time τj, ∆Pj:= Pj−MtMj,Sthe Student-t cdf with 3 degrees of
freedom, nomja (signed) nominal of the portfolio of the market participant j,σj
its annualized relative volatility, and ∆la positive liquidation period accounting for
the time taken by the CCP to novate or liquidate23 defaulted portfolios, we define
τj=F−1
j(S(Xj)) ,
∆Pj
nomjσj√∆l
=Yj,(20)
where
Xj=√ρcrT − qρwwr
jXj+q1−ρcr −ρwwr
jTj,
Yj=pρmktE+qρwwr
jXj+q1−ρmkt −ρwwr
jEj.
(21)
Here ρcr, ρmk t and the ρwwr
jare positive credit/credit, market/market and credit/
market correlation coefficients, while T,Tj,E,Ejand Xjare i.i.d. random variables
following Student-t distributions with degree of freedom 3 such that:
• T represents the common systemic factor for default times across members,
• E represents the common systemic factor for portfolio variations across mem-
bers,
• Xjis the common factor co-driving portfolio variations and default time of
market participant j,
• Tjis the idiosyncratic factor for market participant j’s default time,
• Ejis the idiosyncratic factor for market participant j’s portfolio variations.
Remark 4.1. In practice, margin computations rely on historical estimates based
on several market stressed periods. Our approach, instead, aims at reflecting ex-
treme market shocks with fat tailed Student-t distributions of degree of freedom
ν= 3, and volatility level within a reasonable range of [20%,40%]. Our static for-
mulation depicts stationary increments of the defaulted portfolios’ value over the
liquidation period.
In view of the above, the setup is well defined if and only if24
ρwwr
j<min 1−ρcr,1−ρmk t.(22)
23cf. Section 7.
24otherwise, the model for both default time and portfolio variation factors is undefined due
to their idiosyncratic coefficient term q1−ρcr −ρwwr
jand q1−ρmkt −ρwwr
j. Also we discard
the limit cases where ρwwr
j= 1 −ρcr or ρwwr
j= 1 −ρmk t as they lead to a zero contribution of
the idiosyncratic factors, which would be unrealistic.
296 D. BASTIDE, S. CR´
EPEY, S. DRAPEAU AND M. TADESE
The “minus” sign in front of the common credit-market factor −pρwwr
jfor the
default time component in (21) ensures that the corresponding common factor ac-
celerates defaults, whilst increasing the market exposure due to the +pρwwr
jfactor
in the second part of (21).
Remark 4.2. Our model of latent variables (X, Y ) has a (multivariate) elliptical
distribution, i.e. (X, Y ) = AZ, where, with N=|I∪B∪C|the number of market
participants in the financial network, Z=T,E,T1,X1,E1,...,TN,XN,EN>and A
is the matrix implicit in (21) [19, Chapter 6]. [9] introduce a dynamic model locally
elliptical in the sense of elliptical on each next time step given the information
at the beginning of the time step. Under simplifying assumptions including their
equation (32) and Assumption 2, they obtain (in our notation) explicit CMVA and
approximate CCVA formulas. In their case, defaults are triggered by ∆Pj(in our
notation) falling below a Merton-like threshold. In our static setup with extra latent
variables for defaults, we do not have such explicit formulas. However, Monte Carlo
simulation is quite efficient and required anyway for stress test exercises that aim
at identifying scenarios leading to extreme losses with adequate description such as
the identification of defaulted members and their corresponding losses.
Hereafter, we describe two possible applications of our XVA framework which
will be illustrated by numerical case studies in the above model. To these ends,
two networks will be defined to serve the numerical illustrations, one rather educa-
tional on the use of the XVA metrics and the other one reflecting the more realistic
situation depicted by Figure 1.
In the numerical applications that follow, all members play in turn the role of the
reference bank in the theoretical XVA framework of Sections 2-3. The CVA and
KVA computations require a Monte-Carlo routine run under R?in combination
with a rejection technique in order to yield simulations under the survival measures
associated with different clearing members. For obtaining confidence intervals re-
garding the expected shortfalls that are embedded in the KVA computations, the
simulations are split into several batches, from which the mean of the (partial) EC
estimates yields the final EC estimate, while their standard deviation is used to
define a confidence interval.
5. Case studies setup. In the examples that follow, market participants are iden-
tified by a number and can then be included in one of several of the considered CCPs.
We restrict ourselves to cleared client trades, so that the nonvanishing XVA metrics
reduce to the CCVA, the CMVA, and the KVA.
5.1. Single CCP setup and initial XVA costs. We consider a single CCP
service with 20 members labeled by i∈0·· ·n= 19, only trading for cleared clients
(i.e. without bilateral or centrally cleared proprietary trading). Each member faces
one client. The ensuing financial network is depicted by Figure 3.
All clients are assumed to be risk-free. For any member i, its posted IM to the
CCP is calculated based on the idea of a VM call not fulfilled over a time period
∆s<∆lat a confidence level α∈(1/2,1), using a VaR metric25 applied to the
non-coverage of VM call taken also to follow a scaled Student-t distribution Sνwith
νdegrees of freedom, with cdf Sν:
IMi=VaRnomiσip∆sSν=|nomi|σip∆sSν−1(α).(23)
25under the member survival measure.
DERIVATIVES RISKS AS COSTS 297
CCP
B0
B1
B2
B3 B4 B5 B6 B7
B8
B9
B10
B11
B12
B13
B14
B15
B16
B17
B18
B19
C15
C5
C4
C3
C2
C1
C0
C6
C7
C8
C9
C10
C14
C13
C12
C11
C16
C17
C18
C19
Figure 3. Financial network composed of 1 CCP, its 20 members
(labeled by B) and one client per member
The default fund is calculated at the CCP level as
Cover2 = SLOIM(0) + SLOIM(1),(24)
for the two largest stressed losses over IM (SLOIMi) among members, identified
with subscripts (0) and (1), where SLOIM is calculated as the value-at-risk VaR0
at a confidence level α0> α of the loss over IM, i.e.
SLOIMi=VaR0nomiσip∆sSν−IMi=|nomi|σip∆sSν−1(α0)−Sν−1(α).
(25)
The total amount (24) is then allocated between the clearing members to define
their (funded) default fund contributions as DFi=SLOIMi
PjSLOIMj
Cover2.
The nomj’s of other clearing members are not observable by a given one. How-
ever, following [22] and [18], |nom|(i)denoting the i-th largest absolute nominal
amount for i∈0···n= 19, a parameterization of the form
|nom|(i)=βe−β0(i+1) , β, β 0>0 (26)
can be fit to the total default fund held by the CCP26 and the sum of its five largest
default fund contributions27, made public each quarter for most of the CCPs. The
parameters βand β0inferred from the default fund data are used to depict a similar
pattern on the absolute nominal sizes28. The participants and portfolios parameter
inputs are detailed in Table 5, where id is the identifier of the CM, DP stands for
26item referenced as 4.3.15 in [24], Value of pre-funded default resources (excluding initial and
retained variation margin) held for each clearing service in total, post-haircut. in the quantitative
disclosure documents.
27item referenced as 18.4.2 in [24]:For each segregated default fund with 25 or more members;
Percentage of participant contributions to the default fund contributed by largest five clearing
members in aggregate.; or item referenced 18.4.1 for CCP services with less than 25 members
28as if the default fund amounts are proportional to the portfolio sizes.
298 D. BASTIDE, S. CR´
EPEY, S. DRAPEAU AND M. TADESE
cm id 0 1 2 3 4 5 6 7 8 9
DP (bps) 50 60 70 80 90 200 190 180 170 160
size -242 184 139 105 -80 -61 -46 35 26 -20
vol (%) 20 21 22 23 24 25 26 27 28 29
cm id 10 11 12 13 14 15 16 17 18 19
DP (bps) 150 140 130 120 110 100 90 80 70 60
size -15 -11 -9 -6 5 -4 -3 2 2 -1
vol (%) 30 31 32 33 34 35 36 37 38 39
Table 5. Member characteristics and portfolio parameters, or-
dered by decreasing member size.
the one year probability of default of the member expressed in percentage points,
size represents the overall portfolio size of the member detained within the CCP,
and vol is the annual volatility used for the portfolio variations.
The portfolios listed in the Table 5relate to the members towards the CCP
(which are mirroring the ones between the members and their clients). The signs
of the nominals are distributed so that Pjnomj= 0, consistent with the clearing
condition (first identity in Assumption 3.1, here without proprietary trades).
The parameters of the XVA costs calculations are summarized in Table 6. Note
that the chosen period length of T= 5 years covers the bulk (if not the final
maturity) of most realistic CCP portfolios.
One-period length T5 years
Liquidation period at default ∆l5 days
Portfolio variations correlation ρcr 30%
Credit factors correlation ρmkt 20%
Correlation between credit factors and portfolio variations ρwwr 20%
IM covering period (MPoR) ∆s2 days
IM quantile level 95%
Funding blending ratio eγ/γ 25%
SLOIM calculation29 for DF Cover-2 VaR 97%
Funded DF allocation rule ∝SLOIM
Lallocation rule (3µ)∝DFi
Quantile level used for clearing members EC calculation 99.75%
Hurdle rate hused for KVA computations 10.0%
Number of Monte-Carlo simulation (for CCVA and KVA computations) 10M
Number of batches (for KVA computations) 100
Table 6. XVAs calculation configuration
For each member, the CCVA, CMVA and KVA costs are calculated and reported
in Table 7. For KVA, two calculations have been performed, one based on ES
at 99th percentile level and another one based on 99.75th percentile level. The
amount in square bracket is the corresponding quantile level from which average
is calculated and numbers in parenthesis represent the 95% confidence interval in
relative difference from calculated metric for both CCVA and KVA. All the XVA
numbers decrease with the member size.
To assess the average behavior w.r.t. ρcr ,ρmkt and ρwwr of the CCVA and
KVA, we vary these correlations between 5% and 95%, with 5% step and display
DERIVATIVES RISKS AS COSTS 299
cm id CMVA CCVA KVA (99%) KVA (99.75%)
0 0.0687 0.0778 (0.3%) 0.2734 [0.1396] (0.6%) 0.5138 [0.2923] (1%)
1 0.0656 0.0805 (0.4%) 0.3407 [0.1422] (0.7%) 0.7022 [0.3806] (1.1%)
2 0.0604 0.0635 (0.4%) 0.2725 [0.1132] (0.7%) 0.5624 [0.3039] (1.1%)
3 0.0544 0.0503 (0.5%) 0.2191 [0.0903] (0.9%) 0.4549 [0.2439] (1.5%)
4 0.0485 0.0356 (0.5%) 0.1654 [0.0625] (0.9%) 0.3565 [0.185] (1.2%)
5 0.0834 0.0252 (0.5%) 0.1339 [0.0507] (0.8%) 0.2856 [0.1513] (1.2%)
6 0.0623 0.021 (0.5%) 0.1085 [0.0429] (0.8%) 0.2284 [0.1211] (1.3%)
7 0.0467 0.0187 (0.5%) 0.0883 [0.0365] (0.8%) 0.1836 [0.098] (1.1%)
8 0.0341 0.0146 (0.5%) 0.0685 [0.0282] (0.9%) 0.1432 [0.0755] (1.5%)
9 0.0256 0.0113 (0.5%) 0.0549 [0.0223] (1%) 0.1157 [0.06] (1.6%)
10 0.0187 0.009 (0.5%) 0.0429 [0.0173] (1%) 0.0908 [0.0467] (1.7%)
11 0.0132 0.007 (0.7%) 0.0328 [0.0132] (1.3%) 0.0701 [0.0354] (2.2%)
12 0.0104 0.006 (0.6%) 0.0279 [0.0111] (1.3%) 0.0598 [0.0299] (2.3%)
13 0.0066 0.0042 (0.9%) 0.0198 [0.0077] (1.9%) 0.0438 [0.0206] (3.3%)
14 0.0052 0.0037 (0.9%) 0.0174 [0.0066] (1.8%) 0.0389 [0.0177] (3.1%)
15 0.0039 0.0032 (1.3%) 0.0151 [0.0054] (2.5%) 0.0351 [0.0146] (4.3%)
16 0.0027 0.0025 (1.4%) 0.012 [0.0042] (2.7%) 0.0285 [0.0113] (4.4%)
17 0.0017 0.0018 (2%) 0.0088 [0.0029] (4%) 0.0218 [0.0078] (6.4%)
18 0.0015 0.0019 (2%) 0.0093 [0.0029] (4.1%) 0.0233 [0.008] (6.5%)
19 0.0007 0.0011 (3.7%) 0.006 [0.0015] (6.9%) 0.017 [0.0041] (9.7%)
Table 7. Initial XVA costs: estimates, [value-at-risk underlying
the KVA estimate] and (95% confidence level errors).
in Figures 4and 5the corresponding metrics, aggregated over all clearing members
successively considered as the reference bank. For such tests, the default correlation
ρcr and ρmkt are both set to 4% when they are not changed between 5% and 95%.
This is to allow for runs with ρwwr
i= 95% satisfying the condition (22).
The KVA depicts an increase w.r.t. ρcr but also w.r.t. ρwwr and very limited
change w.r.t. ρmkt . The correlation ρwwr has more impact than ρcr and ρmk t
(right panels in Figures 4and 5). As seen on the left panels of Figures 4and 5,
there are very marginal changes for the aggregated CCVA w.r.t. ρcr and ρmkt , but
a significant positive impact of ρwwr. This is understandable for the sensitivity
to ρcr and ρmkt as, apart for modulations of the measure with respect to which
each individual CCVA is assessed, the CCVA aggregated over clearing members
is essentially an expectation of the CCP loss L(cf. the first line of Table 2). The
individual CCVAs (as per the first line of Table 2) of each clearing member, however,
may depend on ρcr and ρmkt (on top of ρwwr) in a strong and nontrivial manner,
via the allocation coefficient µ.
5.2. Two CCPs network setup. We now consider the case of Figure 1where
there are two CCPs with some common members and stress test is considered from
the perspective of one of these common members. The motivation for this case is
to provide a realistic example mimicking in a simplified way the trading firm Ronin
Capital, which had memberships on both FICC GSD30 segment, hereafter denom-
inated by CCP1, and CME Futures segment, hereafter denominated by CCP2, in
March 2020. It is well known that a VaR type risk measure is not sub-additive, in
particular for credit portfolios as illustrated in Example 5.4 in [2] and Example 2.25
in [19] for a portfolio of defaultable bonds, so that for a common member adding
VaR estimates of trading losses on two CCPs separately can lead to underestimated
30Government Securities Division
300 D. BASTIDE, S. CR´
EPEY, S. DRAPEAU AND M. TADESE
wwr
0.2 0.4 0.6 0.8
cr
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Total CCVA overall members
wwr
0.2
0.4 0.6 0.8
cr
0.2
0.4
0.6
0.8
3
4
5
6
7
Total KVA overall members
Figure 4. CCVA and KVA w.r.t. credit factors correlation and
credit and portfolio variation factors correlation.
wwr
0.2 0.4 0.6 0.8
mkt
0.2
0.4
0.6
0.8
0.4
0.6
0.8
1.0
1.2
1.4
Total CCVA overall members
wwr
0.2 0.4 0.6 0.8
mkt
0.2
0.4
0.6
0.8
3
4
5
6
7
Total KVA overall members
Figure 5. CCVA and KVA w.r.t. market factors correlation and
credit and portfolio variation factors correlation.
levels with respect to the actual VaR of the global exposition of the member. As
such, stress test exercises accounting for common memberships could reveal a larger
value-at-risk compared to the exercise where stress tests are conducted separately
on each CCP.
To perform the analysis, the following setup is considered:
•all members have only clearing client positions31, with 123 members on CCP1
and 56 members on CCP2, out of which 24 are common to both CCPs,
•all clients are assumed default free,
•both CCPs use configuration as per Table 6,
31Ronin Capital had in fact only a house account and was thus not clearing any client position.
DERIVATIVES RISKS AS COSTS 301
•the sizes of the positions are assumed exponentially distributed in the sense
that from the most exposed member to the least one, absolute value of posi-
tions decrease exponentially with the form in (26) as depicted by Figures 6
and 7respectively,
•the proportion of the default fund detained by the 5 biggest members is 25%
for CCP1 and 61% for CCP232,
•the size of the default fund of CCP1 is assumed to be twice the one of the
default fund of CCP2.
Figure 6. Decreasing
absolute nomiper
member for CCP1
Figure 7. Decreasing
absolute nomiper
member for CCP2
All data used are either public sources or have been anonymized, with default
intensities ranging from 10 bps to 400 bps and portfolio volatilities ranging from 20
to 30. Similar configuration as given in Table 6is used, apart from the number of
Monte-Carlo simulations reduced to 2M for memory capacity reasons.
The clearing conditions are ensured by setting the sum of the portfolio sizes nomi
to zero on each CCP. The situation of member 3, exposed to both CCPs, as the
defaulting member, corresponds roughly to the situation of Ronin Capital in 2018.
In particular, an annual probability of default of 0.1% corresponds roughly to a
BBB rating, that was assigned to Ronin Capital in 2018 for its issuances33.
6. Stress test exercises. As outlined in the capital requirements regulation de-
tailed in [38] article 290, financial institutions must conduct regular stress test
exercises of their credit and counterparty exposures. Paragraph 8 of this article
also stipulates the reverse stress test34 requirement to
[...] identify extreme, but plausible, scenarios that could result in signif-
icant adverse outcomes.
This is complemented by article 302 on the exposure financial institutions may have
towards CCPs:
32taken from the quantitative disclosure of both CCPs as of third quarter of 2020.
33https://www.spglobal.com/marketintelligence/en/news- insights/blog/
banking-essentials- newsletter-july- edition-2.
34see dedicated definition p.12 in [29] and articles 97, 98 p. 37 in [36] for official regulatory
definitions.
302 D. BASTIDE, S. CR´
EPEY, S. DRAPEAU AND M. TADESE
Institutions shall assess, through appropriate scenario analysis and stress
testing, whether the level of own funds held against exposures to a CCP,
including potential future credit exposures, exposures from default fund
contributions and, where the institution is acting as a clearing mem-
ber, exposures resulting from contractual arrangements as laid down in
Article 304, adequately relates to the inherent risks of those exposures.
In practice, stress test exercises aim at assessing the capacity of financial institu-
tions to absorb financial and economic shocks. In regular exercises, such as the ones
conducted by the European Banking Authority, the shocks are usually considered
under so called central and baseline macro-economic scenarios corresponding to a
median quantile and adverse scenario usually taken as a 90th percentile reflecting
severe yet plausible scenario that can occur once every 10 years35. Additionally,
extreme scenarios can be considered for measuring the capital adequacy36 for ab-
sorbing extremely severe losses around confidence level at 99.9%. From a clearing
member perspective, this requires to have the capacity of scanning certain points
of its trading loss distribution. In our framework, this boils down to identifying
particular levels of the distribution of the trading loss `=J(C − CCVA −BCVA)
of the reference clearing member bank, where the different terms are detailed in
Proposition 3.9.
The other type of stress test exercises, referenced as reverse stress test37 [13],
consists in identifying the probability of reaching a given loss level as well as de-
scribing the scenario configuration such as projected defaults and loss magnitude
leading to such loss levels. The distribution must span a sufficient large spectrum of
losses, including the ones targeted by the exercise, but it also has to be sufficiently
rich numerically to allow identifying combinations of events leading to such losses.
Confidence intervals of corresponding extreme scenario probabilities should com-
plement the analysis to ensure the reliability of the used model and numerical
methods.
Regulators have the ability to challenge financial institutions on these elements
and demand for improvements38.
6.1. Scenarios identification for reverse stress test. We now briefly explain
how to identify and exploit the scenarios leading to contribute the most to economic
capital, in the spirit of [7]. We denote by Mthe number of Monte-Carlo scenario
for which J= 1, i.e. survival of the reference bank. Its trading loss C − CVA for a
simulation mis given by Cm−CVA, where m∈1·· ·Menumerates the simulated
scenarios for which the reference member bank ends up in survival state.
To get an estimate of the economic capital based on expected shortfall, relying
on [2, Definition 2.6 and Proposition 4.1], we calculate, for a high confidence level
35such confidence levels are suggested by the Federal Reserve outlining p.10 in [30] the various
recession periods of the United States listed in their Table 1 p. 14. The 2021 instructions in [34]
also indicate p.72 that stressed market risk factors are based on shocks specified in [39], citing [12,
p. 29], with the US recessions periods as stressful economic episodes.
36cf. paragraph 5.1 p.11 from [25].
37see also dedicated definition on p.12 in [29] and articles 97, 98 p. 37 in [36] for official
regulatory definitions.
38this may entail re-assessment of the Pillar 2 guidance additional capital requirement set in
the annual Supervisory Review and Evaluation Process reported by Banks, cf. [37] for a brief
definition and use and [28] for more extensive details as well as [31] for similar requirements.
DERIVATIVES RISKS AS COSTS 303
α∈(1
2,1) and [x] denoting the integer part of any real x,
c
ES (C − CVA) := 1
M−[αM]
M
X
m=[αM]+1 nC(m)−CVAo,(27)
where the C(m)−CVA’s are the simulated trading losses of the reference bank ranked
in increasing order.
To obtain the contribution of any simulated scenario m(with Cm≥ C([αM]) ) to
the economic capital estimated by (27), we compute
c
ES−m(C − CVA)
:= 1
M−1−[α(M−1)] nM−[αM]c
ES (C − CVA) −(Cm−CVA)o.(28)
The contribution δmc
ES (C − CVA) of scenario mto c
ES (C − CVA) is then given by:
δmc
ES (C − CVA) = c
ES (C − CVA) −c
ES−m(C − CVA) .(29)
To illustrate the various flavors of stress test exercises that can be conducted by a
CCP member, we report numerical results for the two network examples introduced
in Section 5. We start with a reverse stress test exercise on example covered by Table
5. For this first illustration, a specific extreme loss is targeted and the corresponding
probability of loss reaching at least such target level is estimated. We then consider
the example illustrated by Figure 1where projected loss levels for specific confidence
levels are indicated for the members with common memberships on the two CCPs.
6.2. Numerical results. In Table 8, we report, for the example summarized in
Table 5, the 99.9th percentile trading loss levels, referenced as extreme quantile, with
corresponding (asymmetric) confidence intervals based on the approach proposed
in [20, Section G.2]. This is done for every clearing member successively playing
the role of the reference bank in the setup of Sections 2-3. We also compute the
probabilities of reaching a loss equal to 1.5 times the obtained extreme quantile
level, referenced as RST scenario, with corresponding confidence levels39.
Our description of the scenarios leading to such losses includes the identified
defaulted members, the generated losses and the allocated loss coefficient of the
reference clearing member (CM1 in this example). Table 9provides the description
of the 20 worst scenarios, contributing the most to the EC estimation for the second
biggest member, that is CM140. Most of these scenarios are driven by significant
losses stemming from CM0’s default, reflecting the highly concentrated position
of CM0. We observe that several scenarios illustrate the cases where more than
one clearing member default such as 2nd to 5th scenarios for which not only CM0
generates most of the loss but other defaulting members generate significant losses
yet of less magnitude compared to CM0.
39the calculation of the latter confidence intervals of the probability of being above a quantile
relies on the same numerical approach based on batches used for KVA calculations. Also, the
batch approach leads to reasonably tight confidence intervals for the RST scenario probabilities.
40its theoretical number of scenarios above the RST loss level should be 4153, i.e. the number
of MC simulations of 10M multiplied by CM1’s survival probability over 5 years and by CM1’s
RST loss level probability estimated in Table 8as 0.0428%, which is of course far too many to
report. Nonetheless a focus on the 20 worst ones already illustrates the type of information that
can be exploited for such exercises.
304 D. BASTIDE, S. CR´
EPEY, S. DRAPEAU AND M. TADESE
cm id 99.9% 1.5×99.9% RST scenario probability
0 4.9022 (-0.9%, 0.9%) 7.3534 0.0387% (5%)
1 6.7852 (-0.9%, 1%) 10.1778 0.0428% (4.4%)
2 5.4194 (-1%, 1%) 8.1291 0.0435% (4.4%)
3 4.3402 (-0.9%, 0.9%) 6.5103 0.0437% (4.3%)
4 3.3843 (-0.9%, 1%) 5.0764 0.0452% (4.5%)
5 2.7516 (-0.9%, 1%) 4.1274 0.044% (4.6%)
6 2.1804 (-1%, 0.9%) 3.2706 0.0437% (3.9%)
7 1.7481 (-1%, 1%) 2.6221 0.0434% (4.2%)
8 1.3452 (-1%, 1.2%) 2.0178 0.0435% (4.1%)
9 1.0714 (-1%, 1%) 1.6071 0.0433% (4.2%)
10 0.8362 (-1%, 1.1%) 1.2544 0.0429% (4.4%)
11 0.6325 (-0.9%, 1%) 0.9487 0.0427% (4.4%)
12 0.5330 (-1%, 0.9%) 0.7995 0.0431% (4.2%)
13 0.3688 (-0.9%, 0.9%) 0.5533 0.0428% (4.6%)
14 0.3186 (-1%, 1%) 0.4779 0.0429% (4.4%)
15 0.2626 (-0.9%, 1.1%) 0.394 0.0427% (4.3%)
16 0.2037 (-1.1%, 1.2%) 0.3055 0.0427% (4.4%)
17 0.1403 (-1.1%, 1%) 0.2105 0.0434% (4%)
18 0.1458 (-1%, 1%) 0.2187 0.0427% (4.1%)
19 0.0753 (-1%, 0.9%) 0.113 0.0431% (4%)
Table 8. Stress test (ST) extreme quantile, 1.5×ST extreme
quantile and RST probability to breach 1.5 times the 99.9th quan-
tile loss level, for each member, based on 10M simulations (in paren-
theses: corresponding 95% confidence intervals).
Rank Loss n µDefaulters Losses triggered by defaulters
1 17.22 2 0.23 cm0, 7 842.25, 0
2 12.68 6 0.31 cm0, 2, 5, 9, 11, 14 300.22, 0, 92.28, 36.03, 20.83, 0
3 12.16 5 0.29 cm0, 2, 5, 14, 15 335.56, 0, 112.91, 0, 9.67
4 11.82 7 0.33 cm0, 3, 5, 7, 8, 9, 14 394.96, 0, 0, 0, 0, 0.65, 0
5 11.05 5 0.26 cm0, 5, 6, 10, 15 465.41, 0, 0.59, 0, 0.05
6 10.99 1 0.21 cm0 566.93
7 10.74 1 0.19 cm2 608.58
8 9.23 2 0.23 cm0, 7 451.35, 0
9 9.1 1 0.21 cm0 469.81
10 8.83 5 0.31 cm0, 2, 5, 8, 12 300.42, 14.38, 0, 0, 0
11 8.57 3 0.23 cm0, 6, 16 408.40, 1.88, 0
12 8.41 3 0.22 cm0, 16, 17 429.53, 0, 0
13 8.22 12 0.51 cm0, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17 81.53, 0, 32.12, 22.44, 17.66, 0, 0, 9.95, 6.19, 4.84, 2.63, 1.51, 0
14 8.09 2 0.23 cm0, 7 395.87, 0
15 7.87 1 0.21 cm0 406
16 7.86 8 0.31 cm0, 4, 6, 7, 8, 9, 12, 14 275.27, 0, 0, 0, 0, 0, 0, 0
17 7.83 2 0.22 cm0, 9 391.07, 0
18 7.49 12 0.55 cm0, 2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 18 51.85, 0, 27.98, 25.76, 20.82, 0, 0, 8.41, 4.26, 5.33, 3.21, 1.62, 0
19 6.85 7 0.36 cm0, 2, 3, 7, 8, 12, 17 0, 84.82, 70.26, 29.05, 20.74, 0, 2.06
20 6.7 3 0.22 cm0, 10, 11 330.03, 0.39, 0.34
Table 9. Economic Capital 20 worst scenarios details for member
1 in decreasing order of total loss where column with header µ
indicates allocated coefficient loss to member 1 and nis the number
of defaults within the scenario.
From CM1 viewpoint (i.e. with CM1 in the role of the reference clearing mem-
ber), 18 scenarios entail significant losses over the collateral posted by the defaulted
CM0 (positive first entries in the last column of Table 9). CM0 bears a very large
concentrated position compared to other members. Even if CM0 has more IM and
DF requirements than others, this is still not enough: this example highlights that
employed DF allocation rules in this example dilute the DF collateral requirements
for concentrated positions. It also illustrates that scenarios with multiple defaults
DERIVATIVES RISKS AS COSTS 305
do not necessarily lead to extreme losses, due to the fact that members with medium
or small positions have large default fund contributions stemming from others’ con-
centrated positions.
In Table 10, we report, for the example illustrated by Figure 1with 2 CCPs,
the trading loss levels (value-at-risks) at confidence levels 90% and 99.9%, for the
24 common members on the two CCPs. The corresponding numbers in the case
where the two CCPs would be considered separately is reported in the columns
VII and IX. For members with very low size on one of the two CCPs compared
to the other, considering the common memberships or not does not affect the loss
estimates, as expected41 . For other members, however, at 90% confidence levels, the
value-at-risks are significantly higher (compare columns VII and VIII in Table 10)
when the common membership are considered compared to the stand-alone value-
at-risks calculation conducted on each CCP and summed, especially for the first ten
members. On the contrary, at the confidence level 99.9%, the sum of the stand-alone
value-at-risks is well above the value-at-risk when common memberships are taken
into consideration (columns IX and X in Table 10). These two situations illustrate
that a regulator and the board (top management) of the bank could equally and
rightfully criticise a simplistic standalone approach, too aggressive in some cases
(making it unacceptable by the regulator) and over-conservative in others (making
it unacceptable by the board).
7. Optimizing the porting of defaulted client portfolios. In case a clearing
member defaults, the CCP tentatively novates part of the CCP portfolio of the
defaulted member through auctions among the surviving clearing members [33,27],
and it liquidates the residual on the market. A natural baseline is that the CCP
novates (auctions among surviving members) client trades and their mirroring client
account positions, collectively dubbed client positions for brevity hereafter, whereas
house account positions are liquidated.
The liquidation side of the procedure cannot be handled in our modeling setup,
which does not embed the fundamentals of price formation (our MtM processes are
assumed to be exogenously given). On the other hand, an XVA-based procedure
can be used for rendering what would be the output of an idealized, efficient auction,
assuming a large number of clearing members [23, Section 3.3]. Namely, supposing
that the reference clearing member, labeled by 0 in Sections 2-3, defaults at time
0, i.e. just after that all portfolios have been settled, for each surviving member
CM∗successively envisioned as a potential taker of the defaulted (client) positions of
CM0, one computes the incremental (∆) XVAs of porting the defaulted positions to
CM∗, for each surviving member (CM∗included42). The corresponding incremental
XVA numbers are then summed over metrics and survivors, resulting in the funds
transfer price (FTP∗) of porting defaulted client positions to CM∗. The effective
taker is then the surviving member for which the ensuing FTP∗is the smallest43.
See [5, Section 5.2] for more details on such “XVA Pareto optimally driven” novation
procedures.
41as the CCP with the very low size compared to the other should have marginal impact.
42note that all members are impacted by additional margin to fund due to the re-calibration
of their DF by the CCP, whereas only the member taker of the portfolio sees in addition its IM
adjusted.
43or, indifferently in case of multiple minima, one of the minimizing FTP∗members.
306 D. BASTIDE, S. CR´
EPEY, S. DRAPEAU AND M. TADESE
I II III IV V VI VII VIII IX X
3 0.1 19.9 21 -97.48 23 0.0755 (-1.5%, 1.5%) 0.0794 (-1.1%, 1.1%) 4.0067 (-1.8%, 1.7%) 3.6185 (-1.8%, 1.8%)
4 0.1 80.79 24 -18.79 22 0.0687 (-1.5%, 1.4%) 0.0739 (-0.9%, 1%) 3.4324 (-2%, 2%) 3.0625 (-1.5%, 1.9%)
9 3.1 -31.58 29 17.74 23 0.0478 (-1.3%, 1.3%) 0.0624 (-0.7%, 0.7%) 1.9749 (-1.8%, 2.3%) 1.6109 (-2%, 2.1%)
12 0.1 17.97 21 -16.75 24 0.0351 (-0.9%, 0.9%) 0.0423 (-0.6%, 0.6%) 1.0592 (-1.8%, 1.9%) 0.8008 (-2%, 2.3%)
13 0.1 -14.9 22 15.81 25 0.0337 (-0.9%, 0.9%) 0.0403 (-0.6%, 0.6%) 0.9738 (-1.8%, 1.9%) 0.7319 (-2%, 2.2%)
14 0.2 12.34 23 -14.93 26 0.0323 (-0.9%, 0.9%) 0.0385 (-0.6%, 0.6%) 0.8805 (-1.7%, 1.9%) 0.6518 (-1.5%, 1.5%)
15 0.1 -10.23 24 14.09 27 0.0311 (-0.8%, 0.9%) 0.0364 (-0.6%, 0.6%) 0.8245 (-1.8%, 1.8%) 0.6175 (-2.1%, 1.8%)
17 0.3 -7.03 26 -13.3 28 0.0295 (-0.8%, 0.8%) 0.0337 (-0.5%, 0.6%) 0.6991 (-1.8%, 1.8%) 0.5259 (-2.1%, 1.7%)
19 0.2 -4.83 28 12.56 29 0.0278 (-0.7%, 0.8%) 0.0309 (-0.5%, 0.5%) 0.622 (-1.8%, 1.8%) 0.4808 (-1.8%, 2%)
22 3.9 2.75 20 -11.86 30 0.0301 (-0.7%, 0.7%) 0.0315 (-0.6%, 0.6%) 0.524 (-1.9%, 2%) 0.4439 (-1.6%, 2.2%)
26 0.1 1.3 24 11.2 20 0.0159 (-0.7%, 0.7%) 0.0164 (-0.6%, 0.6%) 0.3061 (-1.8%, 1.8%) 0.2695 (-2%, 2.1%)
27 0.1 1.07 25 -10.57 21 0.0157 (-0.7%, 0.7%) 0.016 (-0.7%, 0.6%) 0.2958 (-1.7%, 1.7%) 0.2649 (-2%, 2%)
28 1.5 0.89 26 9.98 22 0.0163 (-0.7%, 0.7%) 0.0166 (-0.7%, 0.6%) 0.2831 (-1.8%, 2.1%) 0.2533 (-2.1%, 2%)
31 0.1 -0.51 29 -9.42 23 0.0151 (-0.7%, 0.7%) 0.0151 (-0.6%, 0.7%) 0.2723 (-1.7%, 1.8%) 0.2547 (-1.9%, 1.8%)
34 0.1 0.29 21 8.89 24 0.0147 (-0.6%, 0.7%) 0.0147 (-0.6%, 0.6%) 0.2539 (-1.7%, 1.8%) 0.2469 (-2%, 1.8%)
35 0.1 -0.24 22 -8.4 25 0.0145 (-0.6%, 0.7%) 0.0144 (-0.6%, 0.6%) 0.2482 (-1.8%, 1.7%) 0.242 (-2%, 1.8%)
36 0.1 0.2 23 7.93 26 0.0142 (-0.6%, 0.6%) 0.0141 (-0.7%, 0.6%) 0.2427 (-1.8%, 1.6%) 0.2375 (-2%, 1.8%)
39 0.1 -0.11 26 -7.48 27 0.0138 (-0.6%, 0.6%) 0.0138 (-0.6%, 0.6%) 0.2361 (-1.8%, 1.7%) 0.2328 (-1.8%, 1.7%)
40 0.5 0.09 27 7.07 28 0.0138 (-0.7%, 0.6%) 0.0138 (-0.7%, 0.7%) 0.225 (-2%, 1.6%) 0.2218 (-2%, 1.4%)
44 0.1 0.04 20 -6.67 29 0.0132 (-0.6%, 0.6%) 0.0131 (-0.6%, 0.6%) 0.2224 (-1.8%, 1.7%) 0.2217 (-1.8%, 1.7%)
49 0.1 -0.02 25 6.3 30 0.0129 (-0.6%, 0.6%) 0.0129 (-0.6%, 0.7%) 0.2168 (-1.8%, 1.7%) 0.2163 (-1.8%, 1.7%)
50 0.1 0.01 26 -5.95 20 0.0081 (-0.6%, 0.6%) 0.0081 (-0.6%, 0.6%) 0.1366 (-1.7%, 1.7%) 0.1364 (-1.7%, 1.7%)
51 0.1 -0.01 27 5.61 21 0.008 (-0.6%, 0.6%) 0.008 (-0.6%, 0.6%) 0.1355 (-1.8%, 1.7%) 0.1352 (-1.8%, 1.7%)
55 0.1 -0.01 20 -5.3 22 0.008 (-0.6%, 0.6%) 0.008 (-0.6%, 0.6%) 0.134 (-1.7%, 1.7%) 0.1338 (-1.8%, 1.8%)
Table 10. Quantile loss levels (confidence errors) for 90% and 99.9% confidence levels across members for the
example with 2 CCPs and 155 members including 24 common members. Legend for column headers: I. Member
Id, II. DP (%), III. Size on CCP1, IV. Volatility on CCP1, V. Size on CCP2, VI. Volatility on CCP2, VII. 90th
Perc. stand-alone, VIII. 90th Perc., IX. 99.9th Perc. stand-alone, X. 99.9th Perc.
In what follows, based on the example of Table 5(which only involves client
positions), we analyze from this perspective a first scenario of a single default on
the CCP.
Taking the first case with a single default, we first assume the scenario whereby
CM0 defaults at time 0. Table 11 summarizes the total ∆XVA∗aggregated over
survivors, across members ∗from 1 to 19, in increasing order of the FTP∗indicated
DERIVATIVES RISKS AS COSTS 307
Surv. member ∗Total ∆CMVA∗Total ∆CCVA∗Total ∆KVA∗Total FTP∗
1 0.0768 (0.0295) -0.0511 (-0.0038) -0.9182 (-0.1709) -0.8926 (-0.1452)
2 0.0921 (0.0428) -0.047 (0.0028) -0.8262 (-0.0866) -0.7811 (-0.0411)
3 0.1054 (0.0576) -0.0394 (0.0088) -0.7307 (-0.0185) -0.6647 (0.0479)
19 0.1298 (0.0818) -0.0253 (0.039) -0.6785 (0.2818) -0.574 (0.4026)
18 0.1417 (0.0939) -0.0192 (0.0379) -0.5995 (0.2693) -0.477 (0.401)
17 0.1549 (0.107) -0.0138 (0.0377) -0.5547 (0.2702) -0.4137 (0.4149)
16 0.1688 (0.1208) -0.0088 (0.037) -0.4435 (0.2665) -0.2835 (0.4243)
15 0.1814 (0.1334) -0.0032 (0.0363) -0.3841 (0.2622) -0.2059 (0.4319)
4 0.1525 (0.1022) -0.0364 (0.0159) -0.2671 (0.0427) -0.151 (0.1608)
14 0.1903 (0.1426) 0.0035 (0.0353) -0.3146 (0.253) -0.1208 (0.4309)
13 0.2061 (0.1582) 0.008 (0.035) -0.2108 (0.2583) 0.0033 (0.4515)
12 0.2171 (0.1692) 0.0101 (0.0335) -0.1767 (0.2458) 0.0506 (0.4485)
8 0.234 (0.1881) 0.02 (0.0264) -0.1305 (0.1778) 0.1235 (0.3924)
11 0.2285 (0.1807) 0.0147 (0.0326) -0.101 (0.2415) 0.1422 (0.4548)
7 0.2327 (0.1876) 0.02 (0.0235) -0.0949 (0.1501) 0.1578 (0.3612)
10 0.2385 (0.1908) 0.0188 (0.0311) -0.0259 (0.2316) 0.2314 (0.4535)
9 0.2478 (0.2003) 0.0205 (0.029) 0.0442 (0.2158) 0.3125 (0.4451)
6 0.2687 (0.2225) 0.0215 (0.0205) 0.2937 (0.1399) 0.5839 (0.3828)
5 0.2728 (0.2274) 0.0189 (0.0163) 0.3527 (0.0922) 0.6444 (0.3359)
Table 11. Total ∆XVA∗aggregated over survivors corresponding
to the different surviving CM∗, i.e. for ∗other than 0, assuming an
instant default of CM0 at time 0. In parenthesis, the contributions
to ∆XVA∗of CM∗itself.
in the last column. Based on the results of Table 11, CM1 appears to be the
potential taker leading to the least overall FTP costs across all surviving members.
This is understandable as this member’s portfolio size (184 in Table 5) nets the
most the defaulted member’s portfolio size (-242), with volatility and credit default
probability similar to44 the ones of the defaulted member.
As CM1 concentrates more risks due in particular to non-perfect offset45 between
its prior positions and the defaulting one, there is an increase of its IM reflected
through an increase of CMVA. But the new risk of CM1 is less than the sum of
the former risks of CM0 and CM1, hence the ∆CCVA aggregated across surviving
members is reduced. This only happens when CM1 takes over the defaulting port-
folio, other potential takers leading to an overall increase of the CCVA. As for the
KVA, there is a reduction effect for CM1 when CM1 is the taker (see the term in
parentheses in Table 11), and an overall decrease in the total KVA (aggregated over
all surviving members), which is also the case for most members. Having CM1 as
the taker allows to obtain the most significant decrease in ∆KVA.
As expected, among the three XVA components, KVA is the main determinant
of the optimal taker: see Table 12.
Once the CCP has re-allocated all defaulted client positions, the resulting finan-
cial network formerly depicted in Figure 3becomes the network with 19 members
44in particular, not significantly higher than.
45By offset we refer to risk reduction when taking over some additional position. The effect of
correlation is such that an opposite sign in portfolio size does not imply an equal offset of the risk
of the aggregated positions. For instance, even with opposite sizes and same volatilities but for
ρmkt ∈(0,1/2), the member ends up with more risk.
308 D. BASTIDE, S. CR´
EPEY, S. DRAPEAU AND M. TADESE
∆CMVA ∆CCVA ∆KVA
0.0593 0.0251 0.3557
Table 12. Standard deviation across surviving members ∗of the
∆XVA∗for the example with 1 CCP and 20 members, assuming
an instant default of CM0 at time 0.
shown in Figure 8. The thick lines represent the new portfolio exposures for CM1
and the pale dashed lines show the defaulted CM0 positions.
CCP
B0
B1
B2
B3 B4 B5 B6 B7
B8
B9
B10
B11
B12
B13
B14
B15
B16
B17
B18
B19
C15
C5
C4
C3
C2
C1
C0
C6
C7
C8
C9
C10
C14
C13
C12
C11
C16
C17
C18
C19
Figure 8. The 1-CCP, former 20-member financial network with
19 members post CM0 default. Defaulted CM0, labeled “B0” in
the presented network, is represented as pale dashed node with pale
dashed links to reflect former exposures to its client and toward
the CCP. The optimal porting of CM0 portfolio with CM1, labeled
“B1”, is outlined with bold links to reflect the new exposures for
CM1.
8. Conclusion. We have proposed a fully integrated risk management framework
that can be used for stress test analysis, including reverse stress test in line with
regulatory requirements, or for optimizing the porting of defaulted portfolios, in a
setup encompassing all the trades (bilateral as centrally cleared and their hedges)
of a reference bank. The framework includes dependence between financial partic-
ipants portfolios, joint defaults, and a configurable wrong-way risk feature. This
is done in a numerically tractable static setup (although already quite demanding
on large financial networks)46. A possible improvement would be to incorporate
regulatory constraints such as minimum regulatory capital requirements and liq-
uidity leverage ratios. More fundamentally, in this paper, we tackle the derivatives
46The dynamic extension considered in [11, Section 6] is only workable at a much higher
computational burden, using the simulation and learning techniques of [1].
DERIVATIVES RISKS AS COSTS 309
risk problem from a pure counterparty credit risk viewpoint: if members, clients
and counterparties are all default free, then in view of Proposition 3.9 all consid-
ered XVAs are zero, so that our setup becomes trivial. Another dimension to the
problem is liquidity [8,14]. Depending on the considered applications47 , credit or
liquidity is the main force at hand. A challenging research project would be to
integrate both in a common setup.
Appendix A. Proofs.
A.1. Proof of Lemma 3.4.On the bilateral trades of the bank and their hedges,
the Treasury of the bank receives PbVMbof variation margin from its counterpar-
ties and has to post an aggregated amount PbMtMbof variation margin. Assump-
tion 3.3 then leads to (6).
A.2. Proof of Lemma 3.5.In view of Lemma 3.4 and Assumption 3.3, the (risky)
funding expenses of the bank correspond to the formula (9) for F. Regarding C,
On the CCP survival event {J = 1}, the CCP receives, by Assumption 3.2,
X
iJi(Pi+Pi) + (1 −Ji)Pi∧(MtMi+ IMi) + Pi∧(MtMi+ IMi)
+(Pi−(MtMi+ IMi))++ (Pi−(MtMi+ IMi))+∧DFi.
(30)
By the CCP clearing condition in Assumption 3.1,
0 = X
i
(Pi+Pi) = X
iJi(Pi+Pi) + (1 −Ji)(Pi+Pi).
Hence (30) is equal to
−X
i
(1 −Ji)(Pi−MtMi−IMi)++ (Pi−MtMi−IMi)+−DFi+=−L,(31)
by definition (8) of L.
On the bank survival event {J= 1}(⊆ {J = 1}), by the respective Assumptions
3.2 and 3.1, the bank receives from its clients and counterparties
X
cJcPc+ (1 −Jc)Pc∧(MtMc+ IMc)
+X
bJbPb+ (1 −Jb)Pb∧(VMb+ IMb),
(32)
respectively pays to the CCP
X
cPc+X
bPb=X
cJcPc+ (1 −Jc)Pc+X
bJbPb+ (1 −Jb)Pb.(33)
Subtracting (32) from (33), we obtain
X
c
(1 −Jc)(Pc−MtMc−IMc)++X
b
(1 −Jb)(Pb−VMb−IMb)+.
On top of this comes the participation µLof the bank to the CCP default losses,
which yields the formula (7) for C.
47see e.g. the beginning of Section 7.
310 D. BASTIDE, S. CR´
EPEY, S. DRAPEAU AND M. TADESE
A.3. Proof of Theorem 3.7.By the result recalled after (12), EC is nonnegative
as an expected shortfall under Rof the random variable J(C+F − CA), which is
centered under R?and therefore under R, by (10). The first four formulas in Table
2directly follow from the definitions of Table 1and Lemma 3.6, which also implies
that KVA = Eh(EC −KVA)+=h(EC −KVA)+. As his nonnegative, this KVA
semilinear equation is equivalent to
(KVA >EC and KVA = 0) or (KVA ≤EC and KVA = h
1 + hEC),
where (KVA >EC and KVA = 0) contradicts the nonnegativity of EC, whereas,
for h∈[0,1] as assumed and EC ≥0, KVA = h
1+hEC implies KVA ≤EC,
i.e. max(EC,KVA) = EC. This and Lemma 3.6 yield
FVA = EγX
b
(MtMb−VMb)−CA−EC+=γX
b
(MtMb−VMb)−CA−EC+.
As CA = CCVA + CMVA + BCVA + BMVA + FVA, this is an FVA semilinear
equation, which, as γis nonnegative, is equivalent to the FVA formula
FVA = γ
1 + γX
b
(MtMb−VMb)−(CCVA +