Wrong-Way-Risk Add-ons and Reverse Stress Testing for Clearing House Portfolios

Abstract

A pro-active risk management strategy for Central Counterparties (CCPs) hinges on the timely identification of ”bad-apple” members and on strategies to offset potential losses amplified by Wrong-Way-Risk (WWR). One of such strategy is the consistent imposition of WWR margin add-ons. The challenge to the modeler wanting to find WWR add-ons lies at the intersection between the areas of Value-at-Risk, Counterparty Credit Risk (CCR) and Reverse Stress Testing (RST) analytics.

We consider a single period setup with no aging of trades as in the tradition of VaR models. Similarly to Credit Valuation Adjustment (CVA) models, we simulate the credit of members dynamically and define a WWR add-on as the amount which reduces the CVA to its uncorrelated value. As in RST models, we generate granular credit-market scenarios where members either default or they do not. Unlike most CVA models, we do not model portfolios on a run-off basis and we do not model Probabilities of Default (PD) but instead individual default events. We generate synthetically a large pool of high-quality hypothetical scenarios, we find that 10 milion scenarios are required to have acceptable sampling errors. To insure scenarios are extreme but plausible, we calibrate credit-market models to derivative pricing data using accurate SLVJ models (Stochastic Local Volatility with Jumps).

We discuss the results of a numerical experiment at scale based on a set of CCP portfolios with European and American equity options. We compare the effectiveness of an array of different policies for margins, default fund contributions and risk metrics and analyse the impact of WWR add-ons. RST risk drill-downs are used to achieve a granular and detailed understanding of extreme scenarios.

1 Introduction

In a recent series of articles on CCPs [AD20], [Dic22], [AD19], Andersen and Dickinson argue that ”one bad apple” member affected by wrong way risk can easily cause uncovered losses for the Clearing House. Historical default events recorded at CCPs are rare, but they all support the ”bad-apple” narrative.

The risk manager of a CCP requires a RST toolkit capable of drill-downs to identify ”bad apple” strategies and gauge margin add-ons in line with the over-arching defaulter-pay principle. More broadly, a systematic risk management policy is based on a consistent and combined use of baseline margins, WWR add-ons, default fund contributions and RST monitoring tools.

The challenge to the modeller lies at the intersection between three areas: VaR modeling, CVA modeling and Reverse Stress Testing (RST). The first two areas have developed time-honored but separate sets of conventions and practices. For instance, while VaR analytics are based on a single period framework with no trade aging, CVA models take the opposite view of run-off modeling over the portfolio lifetime. While VaR analytics are based on historical time series, CVA analytics are based on scenarios calibrated risk neutrally.

RST is a more recent area but itself also at odds with the other two. Unlike in VaR models, RST analytics typically include credit risk and default events. However, unlike CVA models, a RST scenario is one where one or a cluster of several counterparties default, while CVA analytics take a more abstract approach and are based on scenarios where the probability of potential defaults (PD) is modeled, as opposed to expressing individual binary default events. Also, RST correlations sometimes express hypothetical cause-and-effect sequences: if entity A defaults and market factor X falls below a certain level, then this will trigger the default of entity B. These correlations between rare events would not spoil much the risk-neutral calibration of CVA models, but is simply not implementable in a setup where scenarios are identified with a PD and, by construction, do not contain the information of default occurrence.

The literature on CCPs is fairly extensive and developed at fast pace in the aftermath of the introduction of mandatory central clearing and margin requirements for non-centrally cleared derivatives under Dodd-Frank and EMIR regulations, see the CPSS-IOSCO document [Bas12].

A strain of literature on Financial networks compares the benefits of multilateral and bilateral derivative markets. Duffie and Zhu [DZ11] show that netting benefits may be lost in a market with several competing CCPs. Cont and Kokholm [CK14] emphasise that CCPs are efficient at facilitating the absorbtion of WWR losses across the system. Ghamami and Glasserman [GG16] point out that the cost of capital for default fund contributions may penalise clearing even if bilateral trades attract both inital and variation margin. Pirrong [Pir11] discusses the benefits of central clearing in the default resolution process, but points out that reduction in netting efficiency causes systemic risk as multiple CCPs become crucial nodes in a financial network. Barker et al. [BDLV17] develop a simulation framework of the financial network of members and CCPs and conclude that, although there may be significant wrong-way risks between volatility and defaults, the size of CCP-related losses compared to the capital of members is sufficiently small that a domino-effect transmitted through the financial network via the CCPs is unlikely.

A second strain of literature is about the CCP capital structure and risk management strategies. Fenn and Kupiec [FK93] develop a model for optimizing margin levels. Knott and Mills [KM02] note the relevance of extreme value theory (RST in modern parlance) to capture fat tails and the need to balance margin requirements with add-ons and the default fund. Similarly Cumming and Noss [CN13] apply RST analysis to capture the WWR tail dominated by market moves correlated to default events. The sufficiency of the ”Cover-2” principle under different assumptions concerning the distribution of risk among the members is investigated in [PCG$+$18] by Poce et al. and in [MNW14] by Murphy and Nahai-Williamson. Their results indicate the significance of concentration risk within the CCP. Albanese, Armenti and Crepey [AAC22] set out an optimal capital structure framework for CCP based on XVA theory.

In this article, we take the viewpoint of a single CCP and do not consider financial networks, bilateral markets and bank counterparties. In particular, we assume that the entire portfolio of the CCP is known. Our objective is to investigate a set of fundamental principles for the capital structure of a CCP and its risk management across four areas:

• Baseline margin
• WWR add-ons
• Default fund size and contribution amounts
• RST reports and controls

A principled based approach is also informative for other neighboring and related areas such as securities financing, FRTB risk management and bank XVA models.

Our setup of choice is a single period model of the kind used for VaR calculations. The time horizon $\tau$ typically used for clearing varies between 2 and 10 days, the horizon used for VaR models in banks according to FRTB varies between 10 days and 1 year. In our numerical examples we choose a time horizon of 10 days but nothing would impede a different choice. The choice of time horizon has an impact on some of the key numbers which are subject to being rescaled.

In a CCP, default protection is not factored out by an XVA desk as happens for a bank and members are liable to default risk. CCP risk is highly granular: the event of default of even a single member in a scenario so extreme that he is caught under-collateralised is a rare occurrence but typically comes with significant market repercussions. It is thus important to analyse pre-emptively individual events in isolation. We thus refrain from the usual practice in XVA calculations to model the Probability of Default (PD) of a member and then compute the CVA as an average quantity, but instead model default scenarios in isolation. On this basis, a RST drill down analysis will fully resolve the drivers of risky scenarios in their individuality, and not simply reflect the possibility of their occurrence in sample averages. This greater accuracy comes of course at the cost of having to simulate a larger number of scenarios.

Margin add-ons are priced according to the following:

While banks price counterparty credit risk in terms of a cash exchange struck at CVA, according to this principle CCPs can value counterparty credit risk in terms of collateral and create a level playing field among members by adjusting margins so that WWR risk does not increase CVA exposure. In the same spirit, one could also go one step further and recognise that certain portfolio exposures are characterised by Right Way Risk (RWR) which might, by the same token, justify margin discounts. However, we don’t discuss RWR discounts in this article.

The Default Fund size is usually calculated with the Cover-2 rule, according to which the CCP should be capable to absorb the default of its two largest members with only margin and default fund capital. However, this rule is not a proper, scale invariant risk preference: it expresses a degree of risk aversion which is stronger for the case of CCPs with few members than in the case of CCP with numerous members. We thus assess the Default Fund based on the following principle:

If the quantile level for the CES metric is sufficiently large, this metric is more conservative than a Cover-2 rule but is also a more consistent expression of risk preferences across a broader variety of situations.

To support RST analytics, we pay particular attention at modeling the portfolio dynamics accurately with high-quality, globally calibrated models. Even the most extreme scenarios generated by these models are guaranteed to be arbitrage free and consistent with current market expectation, and thus provides valuable insights into the risk management process.

The combination of RST analytics to identify risky scenarios and WWR add-ons calculated on the basis of accurate portfolio modelling are a highly effective risk management tool at the condition that the correlation model is correctly identified. Model risk is coalesced in the identification of driving risk factors and corresponding regression betas in the correlation model. For these inputs there is no substitute to the experienced judgement of a risk manager, while all the rest can be automated.

To summarise, the key technical difficulties posed by CCP modelling are as follows:

• Each risk factor should be modeled with high-quality, globally calibrated pricing models which are robust and yields meaningful information, in line with market expectations, also in extreme but plausible hypothetical scenarios.
• In our experience, at least ten million granular credit-market scenarios with binary default events (not stochastic PDs) are required to generate sufficient statistics for WWR add-ons and default fund contributions
• An RST framework is required to resolve and analyse in full detail the market conditions that can potentially drive credit market scenarios that would give rise to extreme losses to the CCP.
• Implement a correlation framework aligned with end-user intuition and formulated in terms of simple concepts such as market drivers, regression betas and idiosyncratic risk.

The paper is organised as follows. In Section 2, we describe our SLVJ model and method to calculate margins, CVA, WWR add-ons and Default Fund Contributions. In Section 3 we review numerical results providing links to an extensive report detailing our numerical experiments. Section 4 contains conclusive remarks. (This article contains embedded hyperlinks that point to auxiliary files. If the links do not work, please contact the author.)

2 Model Assumptions

Since the introduction of VaR in the early 1990s, pricing models have progressed at different speed in the trading and risk management contexts. While in risk management locally calibrated models remained in use, trading quality systems have increasingly privileged more elaborate pricing models which eliminated arbitrage opportunities among contracts written on the same underlying asset. Progress included the Dupire local volatility model [Dup94], the Heston stochastic volatility model [Hes93], the Madan variance-gamma model with jumps [MS94] and the more elaborate CGMY model [CGMY02]. All these models have special mathematical properties which allows one to design a special purpose solver. A further and final advance is in the direction of Stochastic and Local Volatility Models with Jumps (SLVJ) which are expressed through generic Markov processes with hidden variables and which require a brute-force numerical solver since they do not possess any distinguishing mathematical property. All models introduced for trading and listed above are special instances in the SLVJ class.

While the requirement of arbitrage freedom has obvious benefits for trading, the same condition is also of interest to risk management. If a Montecarlo simulation which is either historical or hypothetical, generates extreme scenarios which are arbitrageable and have an impact on the relevant metrics for risk, one can argue that these scenarios are not plausible and of no value to risk managers. It can be safely assumed that the action of arbitrageurs will prevent their occurrence. High quality pricing models are particularly relevant in our case also because the day-to-day use of RST analytics and drill downs is essential for CCP risk management.

In our numerical experiments, the CCP has 100 members. Member portfolios are generated synthetically at random, distinguishing several trading strategies such as all-long, all-short, random, hedged and insurance portfolios containing only positions in short out-of-the-money put options. The total number of trades in our synthetic portfolios is about 500 thousand. We assign at random a CDS spread to each member. Portfolio allocations reflect 5 possible strategies: random, long-only, short-only, hedged and insurance (i.e. with only at and out of the money short put positions). The 100 portfolios are designed in such a way that they clear, i.e. for each long position at one member there is another member with the corresponding short position of opposite sign.

We postulate a simple one-factor correlation model whereby all underlying and member equities regress with a beta of 50% against a common factor, while they also have a 50% of idiosynratic risk. Their CDS spreads are also generated at random. We document the results from 5 separate datasets with different synthetic portfolios.

We assume the Clearing House clears European and American style equity options. We price the options by means of a carefully calibrated Stochastic Local Volatility model with Jumps (SLVJ) which can also be seen as a 2-factor GARCH model. Jumps occur in two sizes, small and large, the smaller ones are more frequent than the larger ones. The SLVJs are defaultable equity models also suitable to model credit risk dynamically. We thus model the credit of each member with a model in the same class. This sort of model shares features that are best practice for both pricing theory and buy-side investment analytics.

Our SLVJ model is specified as follows:

$$S_t=e^{-{\int \; <!--\; nolimits\; -->}_0^{t}r\left(s\right)\mathit{ds}}{X}_t$$

$$d{X}_t={\mu }_t\mathit{dt}+{\sigma }_t{X}_t^{\beta }d{W}_t+d{J}^s+d{J}^l$$

$${\sigma }_t={\varphi }_t({\mathit{vol}}_{\infty }+({\mathit{vol}}_0-{\mathit{vol}}_{\infty }\left)e^{-t∕\tau }\right)$$

where

• $\mathit{dW}$ is a Wiener processes
• $r\left(t\right)$ is the future short rate as a function of time assumed to be deterministic and calibrated as the instantaneous forward rate
• $d{J}^s$ and $d{J}^l$ are two jump processes distributed according to a Student-t distribution with 4 degrees of freedom and with relative jump size and probability intensity of occurrence in unit time (measured in years) as listed in the table
• ${\varphi }_t$ is a stochastic process taking values in the lattice $\{e^{-3\alpha },e^{-2\alpha },e^{-\alpha },1,e^{\alpha },e^{2\alpha },e^{3\alpha }\}$ where $\alpha$ is the parameter denoted as vol scale in the table
• ${\mu }_t$ is an adjustment to the dividend rate deriving from corporate announcement and calibrated in such a way to achieve consistency with observed option and forward prices.

We calibrate SLVJ models to European Index Options and to American single stock options for each underlying separately and then use them to simulate equity option prices.

The credit process of all members is also modeled with an SLVJ model. To keep things simple and stylised, all members are modeled with an SLVJ specified with the parameters of the equity index, except that we add a lower absorbing barrier struck at a level that implies the corresponding CDS spread, assuiming a recovery rate of 40%.

We postulate a simple correlation model with a single factor and idiosyncratic risk. The vector of all equity processes $S_t^j$, including the underlying stocks and the equity of the members, are obtained by applying a copula transformation to a family of standard Wiener processes $W_t^j$. The latter are all correlated to a single process $W_t^I$ in such a way that

$$d{W}_t^j={\beta }_{j}d{W}_t^I+d{𝜖}_t^j$$

whereby the processes ${𝜖}_t^j$ are mutually independent standard Wiener processes.

Numerically, there would be no difficulty in extending the correlation model to include multiple factors. In practice, a risk manager would do just that and identify factors which are of particular relevance to specific members.

The PFMI regulatory document [Bas12] published by CPSS-IOSCO in 2012 draws a distinction between two kinds of Wrong-Way Risk:

• General Wrong-Way risk arises when the likelihood of default by counterparties is positively correlated with general market risk factors
• Specific Wrong-Way risk arises when future exposure to a specific counterparty is positively correlated with the counterparty’s PD due to the nature of the transactions with the counterparty. An institution shall be considered to be exposed to Specific Wrong-Way risk if the future exposure to a specific counterparty is expected to be high when the counterparty’s probability of a default is also high.

In our case, since we model portfolios directly and from the ground up, one could argue that correlations give rise to Specific Wrong Way Risk. However, since we introduce market-wide factors in the correlation model, one could also argue that we are modeling General Wrong Way Risk. As a matter of fact, both facets are intertwined at the modelling level in our context and cannot be easily disentangled and separated.

We consider a simple one-period setup according to the usual conventions for VaR models. The time horizon of VaR models for clearing typically ranges from 2 to 10 business days. Time horizons for VaR models for banks trading books range from 10 business to as long as one year. As a compromise, in our numerical experiment we choose a 10 business days time horizon, but any other time horizon would be computable with the same effort.

Following the usual convention for VaR models, we also assume that the portfolio is not subject to aging. Namely, the portfolio held at the end of the time horizon has exactly the same composition as today’s portfolio except that all payments are shifted back in time.

In the references [AD20], [Dic22], [AD19] the authors put forward a more elaborate approach based on a run-off setup similarly to how portfolios are modeled for XVA applications. Since defaults are modeled in terms of a PD, it is possible that members default and are revived numerous times at each time step. In principle, we could tweak our setup to also model portfolios on a run-off basis, although defaults would occur just once in our context and be final. However, if we went down this path we would still be confronted with the need of projecting out base margin, WWR add-ons and Default Fund contributions for all members. This would then require a nested simulation in which the internal loop evaluates WWR add-ons, presumably with a one-step VaR-like model like the one we are discussing. Although this analysis would be very useful to cross-validate a policy for WWR add-ons and would be particularly important for applications such as CVA and MVA estimations from a bank member standpoint, it cannot be a substitute for a simpler rule to actually calculate WWR add-ons from the CCP standpoint.

We compare a variety of collateral policies across a range of parameter choices, such as:

• Three quantile specifications: 97.5%, 99% and 99.7%
• Three risk measures: Value-At-Risk (VaR), Expected Shortfall (ES) and a more conservative Wighted Expected Shortfall (WES).
• Three rules for calculating the Default Fund Contribution (DFC) based on Conditional Expected Shortfall (CES) of quantiles 95%, 97.5% and 99%. Here, the conditioning is with respect to events whereby a under-collateralised member defaults. These three policies are demonstrated to all be more conservative than the Cover-2 rule.

For each combination of quantile choices and risk measures above, we calculate collateral requirements and default fund contributions with and without WWR add-ons and assess the risk-mitigation impact of WWR add-ons.

In accordance with the principle in the previous section, the WWR add-on is calculated as the amount which, added to base margin for the member, is such that expected losses to the Clearing House due to the default of this member (also known as Credit Valuation Adjustment or CVA) equals the CVA calculated assuming that:

• the credit spread of the member is 50bp
• the credit risk of the member is uncorrelated to market risk.

We consider 5 separate datasets. The tradable instruments include about 42,000 American equity options and 3,000 European equity options. Options are written on 63 single-name underlying stocks and one equity index. Each dataset corresponds to a different day in the year 2022. Models have been calibrated to market data and then paramters have been slightly perturbed at random. On each day, we generate random portfolios for a total of 100 members. Member portfolios are subdivided into 5 separate categories:

• Random allocations
• Hedged allocations whereby out of the money options are either long calls or long puts.
• Long-only allocations whereby all calls are long and all puts are short
• Short-only allocations whereby all calls are short and all puts are long
• Insurance portfolios containing only short positions in out of the money or slightly in the money put options

In the next section we discuss our findings.

3 Numerical experiment

We have generated 5 separate datasets, each corresponding to a different set of synthetic portfolios and calibrated models to a different market-data input. The reports are at the following links:

The summary page of each report gives an overview of 18 different margin policies spanning

• three risk measures: VaR, ES and WES (a weighted variant of ES with over-weighting by a factor 5 of the most extreme 20% of scenarios)
• three quantile levels for baseline margin: 97.5%, 99% and 99.7%
• the cases where WWR margin add-ons are included and excluded are both treated separately for comparison purposes.

For comparison purposes, it is useful to examine two cases: ES at 97.5% with WWR add-on and WES at 99.7% without WWR add-on. An extract of the corresponding table is reported in Figure 1. The policy based on ES[97.5%]+WWR attracts a grand total including margin, WWR add-on and default fund contribution of €565,079,140. In the policy based on WES[99.7%] without WWR, the grand total stands slightly lower at €534,707,650. The two grand totals are thus quite close to one another. The relative allocation of baseline margin posted by the members is also quite similar and is given by the pie-charts in Figure 2.

WWR add-ons are included only in the first case. As the pie-chart in Figure 3 shows, most of the WWR add-ons is posted by members with insurance portfolios (i.e. with only short at- and out-of-the-money put options). Also long and hedged portfolios strategies attract WWR add-ons while short-only strategies do not. The reason why short-only strategies are immune from WWR add-ons is related to our correlation model: since we have a single factor and all members have the same beta with respect to this factor, a member with a short-only portfolio stands a material chance to default in case equity markets have a sharp fall. However, in this situation its portfolio will appreciate and there is thus negligible probability that this member ends up both under-collateralised and in a state of default. Actually, a short-only member could potentially benefit of RWR margin discounts.

Potential CCP losses that are either uncovered or shared across members through the Default Fund roughly reflect the allocation of Default Fund contributions, as shown in Figure 5.

As the table in Figure 6 indicates, the size of the default fund contribution is quite different between the two cases and it is about 80% higher for the policy ES[97.5%]+WWR.

The distribution of CCP losses in Figures 7-8 and 9-10 tell an even more interesting story. Not only expectations are larger, but the tail of the loss distribution for the strategy WES[99.7%] without WWR add-ons is far more leptokurtotic. There are far more events where the CCP suffers either of Default Fund depletion or even uncovered losses in the latter strategy and the largest loss amounts are four times larger.

Not only the distribution of CCP losses has different kurtosis between the two cases with and without WWR add-ons, but it is also populated by quite different scenarios, as an inspection of the tables in the dataset for ES[97.5%]+WWR and the dataset for WES[99.7%] without WWR indicate.

The pie-charts in Figures 11 and 12 show the P&L attribution by underlying risk factor for the single top loss scenarios for the two strategies in our scenario sample. One can notice an entirely different pattern as indeed, the WWR add-on has significantly altered the risk landscape.

4 Conclusions

We have extended the VaR framework for CCP portfolios to include the calculation of WWR add-ons. The principle upon which we base the calculation aims to ensure that the CVA for any member calculated including WWR effects and accounting for the margin add-on matches the CVA calculated assuming no correlation but with no WWR add-on to margin.

We have run extensive numerical experiments to assess the impact of WWR add-ons on the economics of CCPs. We find that WWR add-on trigger a reallocation of collateral. We compared two cases where baseline margin is calculated with two very differ quantile levels of 97.5% and 99.7% respectively and found that, if one implements WWR add-ons to the former but not the latter, the grand-total posted by all members under the two policies differs by only about 5%. However, the allocation of collateral is vastly different according to member’s trading strategies. The WWR add-on approximately doubles the collateral requirements for the members which are most affected by WWR and reduces collateral posting obligations for others. Default fund contributions are also lower and, perhaps more interestingly, are allocated quite differently across members as a function of their trading strategy and nature of their exposure.

WWR add-ons have a major impact on the risk exposure of the CCP. We find that the tail of potential losses for the CCP, including both covered and uncovered credit losses, is far longer and heavier without WWR add-ons than it is with add-ons. Interestingly, RST drill downs show that there is hardly any overlap between the two sets of critical scenarios. We conclude that WWR add-ons with our definition succeed to generate capital cushions that effectively absorb the greatest losses. The number of critical scenarios with CCP losses is also reduced from 287 to 4 out of a sample of 10 milion. The probability of default of the CCP is also drastically reduced from 8.24bp to 0.13bp. Finally, a loss attribution analysis under the critical scenarios shows that the dependencies on underlying risk factors of losses is materially altered by the introduction of WWR add-ons.

We conclude that WWR add-on are an efficient tool for the CCP risk manager that allows to offset CCP risk without a material increase of total costs to members. The tool is particularly useful when combined with RST analytics and drill downs to identify critical scenarios.

References