XVA metrics for CCP optimization
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Claudio Albanese
Abstract
Based on an XVA analysis of centrally cleared derivative portfolios, we consider two capital and funding issues pertaining to the efficiency of the design of central counterparties (CCPs). First, we consider an organization of a clearing framework, whereby a CCP would also play the role of a centralized XVA calculator and management center. The default fund contributions would become pure capital at risk of the clearing members, remunerated as such at some hurdle rate, i.e. return-on-equity. Moreover, we challenge the current default fund Cover 2 EMIR sizing rule with a broader risk based approach, relying on a suitable notion of economic capital of a CCP. Second, we compare the margin valuation adjustments (MVAs) resulting from two different initial margin raising strategies. The first one is unsecured borrowing by the clearing member. As an alternative, the clearing member delegates the posting of its initial margin to a so-called specialist lender, which, in case of default of the clearing member, receives back from the CCP the portion of IM unused to cover losses. The alternative strategy results in a significant MVA compression. A numerical case study shows that the volatility swings of the IM funding expenses can even be the main contributor to an economic capital based default fund of a CCP. This is an illustration of the transfer of counterparty risk into liquidity risk triggered by extensive collateralization.
Funding statement: The research of Stéphane Crépey benefited from the support EIF grant “Collateral management in centrally cleared trading” (before 2019) and (since 2019) of the Chair Stress Test, RISK Management and Financial Steering, led by the French Ecole polytechnique and its Foundation and sponsored by BNP Paribas.
A CCP toy model
When a systematically important financial institution defaults, the impact on interest rates and foreign exchange rates is bound to be major. In the XVA analysis of centrally cleared derivatives, a model of joint defaults and a granular simulation of the latter is necessary if one wants to be able to account for the corresponding “hard wrong-way risk” issue. A credit portfolio model with particularly good calibration and defaults simulation properties is the common shock or dynamic Marshall–Olkin copula model of [15, Chapters 8–10] and [16] (see also [18, 19]).
In this section, we describe the corresponding CCP simulation setup, which is used in the numerics of Section 7.
In particular,
A.1 Market model
As common asset driving all our clearing member portfolios, we consider a stylized swap with strike rate
by the martingale property of the Black process
The following numerical parameters are used:
The nominal (Nom) of the swap is set so that each leg has a time-0 mark-to-market of one (i.e.
Mean and 2.5 % and 97.5 % quantiles, in basis points as a function of time, of the process
A.2 Credit model
For the default times
Example A.1.
A shock
As demonstrated numerically in [15, Section 8.4], a few common shocks are typically enough to ensure a good calibration of the model to market data regarding the credit risk of the clearing members and their default dependence (or to expert views about these).
Given a family
(we recall that the default time τ of the reference clearing member corresponds to
Example A.2.
Consider a family of shocks
| 0 | 1 | 2 | 3⃝ | 4 | 5 | |||
| 0 | 1 | 2 | 3 | 4 | 5⃝ | |||
| 0 | 1⃝ | 2 | 3 | 4 | 5 | |||
| 0⃝ | 1 | 2⃝ | 3 | 4⃝ | 5 |
At time
We consider a CCP with
(sign-wise, the processes
(Top) Average 3 and 5 year CDS spread
| 45 | 52 | 56 | 61 | 73 | 108 | 176 | 367 | 1053 | |
| 9.20 | (1.80) | (4.60) | 1.00 | (6.80) | 0.80 | (13.80) | 8.80 | 7.20 |
Hereafter, we denote by Φ and ϕ the standard normal cumulative distribution and density functions.
A.3 Initial margins
We assume that the margins and default fund contribution of each clearing member are updated in continuous time[3] while the member is non-default and are stopped before its default time, until the liquidation of its portfolio occurs after a period of length
for some IM quantile level
with
Remark A.1.
At least, (A.5) holds whenever there is no coupon date between t and
between the left-hand side and the right-hand side in (A.5).
As δ is of the order of a few days, a coupon between t and
Lemma A.1.
For
where
A.4 CVA of the CCP
Lemma A.2.
We have, for
where
Proof.
In view of (A.4) and (A.5), the conditional version of the identity
The result follows in view of the Black model used for
Proposition A.1.
We have, for
where, setting
we denote
Proof.
We have (cf. (4.2))
by virtue of (A.5) and of the conditional distribution properties of the common shock model provided in [15, Section 8.2.1]. We conclude the proof by an application of Lemma A.2 to the first line in (A.11) and of the Black formula to the second line. ∎
A.5 Unsecured borrowing vs. margin lending MVAs
In the setup of our case study, the generic expressions in (5.7) for the unsecured borrowing vs. margin lending MVAs reduce to deterministic time integrals.
Let
Proposition A.2.
The unsecured borrowing
Proof.
We set
Hence the result follows from Lemma A.1. ∎
Lemma A.3.
We have, for
where
Proof.
In view of (A.5), it comes
Hence the result follows from the Black formula. ∎
Proposition A.3.
The margin lending
The blending ratio in (5.6) is constant and given by
Proof.
By the last identity in (5.7) and the immersion properties of the common shock model, we have
where, for
The blending ratio in (5.6) is given by
where the
Acknowledgements
Yannick Armenti contributed to this article in his personal capacity. The views expressed are his own and do not represent the views of BNP PARIBAS or its subsidiaries.
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