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• W2046483551 abstract "Abstract □ This paper derives a measure of central counterparty (CCP) clearing-network risk that is based on the probability that the maximum exposure (the N-th order statistic) of a CCP to an individual general clearing member is large. Our analytical derivation of this probability uses the theory of Laplace asymptotics, which is related to the large deviations theory of rare events. The theory of Laplace asymptotics is an area of applied probability that studies the exponential decay rate of certain probabilities and is often used in the analysis of the tails of probability distributions. We show that the maximum-exposure probability depends on the topology, or structure, of the clearing network. We also derive a CCP's Maximum-Exposure-at-Risk, which provides a metric for evaluating the adequacy of the CCP's and general clearing members’ loss-absorbing financial resources during rare but plausible market conditions. Based on our analysis, we provide insight into how clearing-network structure can affect the maximum-exposure risk of a CCP and, thereby, network stability. We show that the rate function (the exponential decay rate) of the maximum-exposure probability is informative and can be used to compare the relative maximum-exposure risks across different network configurations. Keywords: Central counterpartyClearingLarge deviations theoryNetwork analysisSystemic riskMathematics Subject Classification: E58G01G18 ACKNOWLEDGMENT The views expressed here are not necessarily those of the Office of the Comptroller of the Currency or the U.S. Department of the Treasury. We thank an anonymous referee for many comments and suggestions that substantially improved the substance and presentation of the paper. We also thank David Lynch, David Malmquist, Travis Nesmith, Mark Pocock, Doug Robertson, Akhtar Siddique, and Kostas Tzioumis for helpful comments on earlier drafts. The authors are responsible for any errors and omissions. Notes Systemic risk refers to the possibility that the failure of a large interconnected firm could cause a breakdown in the entire financial system. Many types of OTC derivatives are already cleared through CCPs (i.e., centrally cleared), including interest-rate swaps (LCH Clearnet), credit-default swap (CDS) indices and single-name CDS (ICE Trust and ICE Clear Europe), and fixed-income OTC derivatives (FICC). See Tumpel-Gugerell[ Citation 33 ] for an overview of recent advances in modeling systemic risk using network analysis. They also study the impact of network structure on initial margin requirements. We do not address margin requirements in this article. The Laplace principle is a basic theorem in large deviations theory. In probability theory, the theory of large deviations deals with the asymptotic behavior of distant tails of sequences of probability distributions. Large deviations theory formalizes the ideas of concentration of probability measures and generalizes the notion of convergence of probability measures. In sum, large deviations theory deals with the exponential decay of the probability measures of certain kinds of extreme or tail events. According to the 5th CPSS-IOSCO Recommendation for CCPs, “A CCP should maintain sufficient financial resources to withstand [..] a default by the participant to which it has the largest exposure in extreme but plausible market conditions.” See the consultative paper, published in March 2011, at http://www.bis.org/publ/cpss94.pdf for details. Also, see “Capitalization of Bank Exposures to Central Counterparties,” Consultative Document, Bank for International Settlements, October 2011. Haene and Sturm[ Citation 17 ] examine the factors that affect the tradeoff between margins and the default fund in a model of CCP risk management, and find that the optimal balance between the two risk-management instruments depends on collateral costs, participants’ default probability, and the extent to which margin requirements are associated with risk-mitigating incentives. Cohen-Cole et al.[ Citation 8 ] examine the role played by market structure in CCP clearing networks in determining liquidity provision and in understanding shock amplification. Pirrong[ Citation 28 ] provides a comprehensive overview of the benefits and risks of central clearing. We do not consider issues related to margins, collateral, or netting. See Singh[ Citation 31 ] for a discussion of these issues for a CCP network structure. For a detailed discussion of trading, clearing, and settlement in clearing networks, see Hasenpusch[ Citation 18 ]. Note that T m, m = 0 and T m′, m = −T m, m′ (i.e., T is skew-symmetric), so we, in effect, have nomM2 independent standard Gaussian random variables. Moreover, it serves as a useful benchmark that can be used to examine the effects of relaxing the assumption and allowing for non-Gaussian, heterogeneous, and correlated trading exposures as done by Cont and Kokholm[ Citation 9 ] in a different modeling context. In future work, we plan to relax Assumption 3.1 to allow for non-Gaussian, heterogeneous, and correlated trading exposures. We also intend to use dynamically evolving exposures, i.e., a random time series of matrices in 𝒮 M which would represent the evolution of the trading exposures from one day to the next. We will return to this issue in another paper. The model of clearing here uses the accounting approach of account pooling. According to this approach, a clearer pools, or sums, exposures from its own proprietary trading accounts with its clients’ trading exposures. We do not consider the effect of collateral on the credit risk of exposures. See Siddique[ Citation 30 ] for a discussion of the many different exposure measures used in modeling counterparty risk. Thus, we can replace (Equation2) with A CCP's maximum exposure against network GCMs is referred to as the CME in Galbiati and Soramäki[ Citation 14 ]. In large deviations theory, a rate function is used to quantify the probabilities of rare or atypical events. A rate function is also called a Cramer function. These are simplifying assumptions for a number of reasons. For example, the trading exposures between different counterparties are probably not independent; traders are not ex-ante identical; the trading exposure distributions could be non-Gaussian (i.e., be fat-tailed or skewed); trading exposures may depend importantly on the topology of the clearing network itself; and the zero-mean assumption would not apply to trading strategies that are not market-neutral. Nevertheless, our simplifying assumptions provide a useful starting point for our analysis. Moreover, even with the simplest set of assumptions, we show that it is possible for the CCP to confront extremely large exposures. Equations (Equation10) and (Equation11) still hold when the zero-mean condition fails, i.e., when is nonzero. We then have that Since there is a γ > 0 such that for all y ∈ ℝ G (which uses the fact that the μ i 's are bounded), the asymptotics of (Equation10) again holds. There could be more than one GCM that minimizes (Equation12). These probabilities are calculated with the software package R. Galbiati and Soramäki[ Citation 14 ] define tiering as the absolute number of clients, i.e., M − 1 − G = C. By defining tiering as a fraction, we produce results that are invariant to the absolute size of the network. There are several different measures of concentration that could be used. Here, we use a measure that is based on a piece-wise differentiable cumulative distribution function; whereas, Galbiati and Soramäki[ Citation 14 ] use a measure that is based on a discrete probability function. As a result, the calculated level of network concentration could vary across the different measures. On pp. 15–16 of the BCBS Consultative Document, the formulas for a CCP's hypothetical capital requirement due to its CCR exposures to all of its clearing members and the capital requirement for clearing members are presented. The CCP's capital requirement varies negatively with the variation margin, initial margin, and the pre-funded default contribution of the i’th clearing member, but positively with the exposure value to the i’th clearing member before risk mitigation is taken into account. The aggregate capital requirement for all clearing members is positively related to the CCP's hypothetical capital requirement. For α ≈ 1. The final expression in (Equation16) is derived by solving for L, and using the result that −ln (a) = ln (1/a). Importance sampling is a variance reduction method that is very useful when simulating rare events or sampling from the tails of a distribution. It involves distorting the probability measure in order to sample from the tail region (see Brandimarte[ Citation 5 ], pp. 261–267). In choosing between analytical and numerical methods, Sokal[ Citation 32 ] notes that “Monte Carlo is an extremely bad method; it should be used only when all alternative methods are worse.” The author goes on to say that Monte Carlo methods should be used only when neither analytic nor deterministic methods are workable (or efficient). The Sherman-Morrison formula states that for any n × n invertible matrix A and any n-dimensional column vectors u and v,(A + uv T )−1 = A −1 − (1 + v T Au)−1 A −1 uv T A −1, if 1 + v T A −1 u is not zero. Color versions of one or more of the figures can be found online at www.tandfonline.com/lstm." @default.
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• W2046483551 title "The Topology of Central Counterparty Clearing Networks and Network Stability" @default.
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