Credit Valuation Adjustment Research Articles

Credit contingent interest rate swap pricing

Credit value adjustment (CVA) is an adjustment to an existing trading price based on the counterparty-risk premium. Currently, CVA is computed with an implicit assumption that the replacement contract is default-free after the original counterparty defaults, with the assumption that those trades will not re-assigned. In the actual counterparty default settlement, it is the norm that trades will be re-assigned, especially on the buy side. Since the counterparty of the replacement contract could also default within the lifetime of an existing contract, ignoring the possibility of counterparty defaults of replacement contracts will either under or over estimate the cost of the risk. An important practical question is, therefore, how to estimate under/over pricing of CVA under current practice.In this paper, we considered the pricing of credit contingent interest rate swap (CCIRS) or credit contingent default swap (CCDS), which is considered the CVA hedge for interest rate swaps (IRS). We derived partial differential Eqs. (PDEs) satisfied by the approximated CVA with the assumption that the replacement contracts do not default. For comparison purposes, we also derived the PDEs for the cost of CVA by relaxing the assumption of default-free replacement contracts with a finite number of counterparty defaults. It shows that the no-default and two default cases can be derived within the same analytical solution framework, similar to the Funding Valuation Adjustment (FVA) problem where continuous funding is a reasonable assumption. The finite number of default case is non-trivial. The PDE for the two default case is derived in this paper.We calibrate our model based on market data and carry out extensive computations for the purpose of comparing these three CVAs. Our basic finding is that the values of the two CVAs are close for top rated counterparties. On the other hand, for counterparties with lower credit ratings, the difference among the two CVAs can be significant.

An Overview of Collateralization Fundamentals & the ISDA Credit Support Annex

In this paper we discuss the continued transition of financial markets towards trade standardization and the clearing of transactions, outlining the role of central clearing counterparties CCPs in reducing systemic and idiosyncratic risk. We contrast this with a discussion on bespoke non-cleared bilateral trading, the associated credit risk management process and the charging of credit valuation adjustments termed collectively as XVA adjustments, with the principal credit valuation adjustment termed CVA. The majority of trading counterparties require collateral to be posted to cover bilateral trading positions and have extensive legal documentation in place to govern the collateral calculation and transfer process, of which the ISDA Credit Support Annex or CSA agreement is the most common and market standard. We discuss both the collateral posting process in detail and outline all the main important terminology and terms outlined in the ISDA CSA. Finally we outline trading features that require special considerations, namely trade break-clauses and physical and cash settlement features.

Monte Carlo Calculation of Exposure Profiles and Greeks for Bermudan and Barrier Options under the Heston Hull-White Model

Valuation of Credit Valuation Adjustment (CVA) has become an important field as its calculation is required in Basel III, issued in 2010, in the wake of the credit crisis. Exposure, which is defined as the potential future loss of a default event without any recovery, is one of the key elementsfor pricing CVA. This paper provides a backward dynamics framework for assessing exposure profiles of European, Bermudan and barrier options under the Heston and Heston Hull-White asset dynamics. We discuss the potential of an efficient and adaptive Monte Carlo approach, the Stochastic Grid Bundling Method}(SGBM), which employs the techniques of simulation, regression and bundling. Greeks of the exposure profiles can be calculated in the same backward iteration with little extra effort. Assuming independence between default event and exposure profiles, we give examples of calculating exposure, CVA and Greeks for Bermudan and barrier options.

Valuation of CDS counterparty risk under a reduced-form model with regime-switching shot noise default intensities

We study the counterparty risk for a credit default swap (CDS) in a regime-switching market driven by an underlying continuous-time Markov chain. We model the default dependence via some correlated Cox processes with regime-switching shot noise intensities containing common shock. Under the proposed model, the general bilateral counterparty risk pricing formula for CDS contracts with the possibility of joint defaults is presented. Based on some expressions for the conditional Laplace transform of the integrated intensity processes, semi-analytical solution for the bilateral credit valuation adjustment (CVA) is derived. When the model parameters satisfy some conditions, explicit formula for the bilateral CVA at time 0 is also given.

On the correlation and parametric approaches to calculation of credit value adjustment

Credit value adjustment (CVA) is an adjustment added to the fair value of an over-the-counter trade due to the risk of counterparty defaults. When the exposure to the counterparty and the counterparty default risk tend to change in the same direction, so-called wrong-way risk (WWR) must be taken into account. Right-way risk takes place when the two factors move in opposite directions. These two comovement effects are also called directional-way risk (DWR). Many efforts have been made to reduce the computational burden of calculating CVA with DWR. The two most popular approaches are the parametric approach and the correlation approach. In this paper, we develop a connection between these two approaches. In particular, by decomposing the DWR into a robust correlation coefficient and a profile multiplier, we bring the parametric approach into the correlation approach framework. This allows us to explain the parameters in the parametric approach. Our results suggest that the parametric approach can become sensitive when calculating the WWR in certain scenarios. For risk model governance and validation purposes, caution should be exercised when using the parametric approach for CVA calculation.

Impact of multiple curve dynamics in credit valuation adjustments under collateralization

We present a detailed analysis of interest rate derivatives valuation under credit risk and collateral modeling. We show how the credit and collateral extended valuation framework presented in Pallavicini et al. [Funding valuation adjustment: FVA consistent with CVA, DVA, WWR, collateral, netting and re-hyphotecation, 2011], and the related collateralized valuation measure, can be helpful in defining the key market rates underlying the multiple interest rate curves that characterize current interest rate markets. A key point is that spot Libor rates are to be treated as market primitives rather than being defined by no-arbitrage relationships. We formulate a consistent realistic dynamics for the different rates emerging from our analysis and compare the resulting model performances to simpler models used in the industry. We include the often neglected margin period of risk, showing how this feature may increase the impact of different rates dynamics on valuation. We point out limitations of multiple curve models with deterministic basis considering valuation of particularly sensitive products such as basis swaps. We stress that a proper wrong way risk analysis for such products requires a model with a stochastic basis and we show numerical results confirming this fact.

Evaluation of credit value adjustment in K-forward

We model and quantify counterparty credit risk for K-forward, a newly proposed longevity-linked security. We focus on the evaluation of credit value adjustment (CVA) from the longevity risk hedger’s perspective. The modelling involves two folds. First, we use a vector autoregressive integrated moving-average process to model the time series of mortality indexes that is obtained by applying the original Cairns–Blake–Dowd model. Then, the risk-neutral default probability of the hedge provider is obtained by calibrating a reduced-form default model on the market price of bonds issued by the hedge provider. We calculate and compare CVA in K-forwards for different combinations of hedger provider, reference year and recovery rate.

Liquidity risk in derivatives valuation: an improved credit proxy method

The models used to calculate post-crisis valuation adjustments, market risk and capital measures for derivatives are subject to liquidity risk due to severe lack of available information to obtain market implied model parameters. The European Banking Authority has proposed an intersection methodology to calculate a proxy CDS or Bond spread. Due to practical issues of this method, Chourdakis et al. introduce a cross-section approach. In this paper, we extend the cross-section methodology using equity returns, and show that our methodology is significantly more accurate compared to both existing methodologies, and produces more reliable, stable and robust market risk and capital measures, and credit valuation adjustment.

Liquidity Risk in Derivatives Valuation: An Improved Credit Proxy Method

The models used to calculate post-crisis valuation adjustments, market risk and capital measures for derivatives are subject to liquidity risk due to severe lack of available information to obtain market implied model parameters. The European Banking Authority has proposed an intersection methodology to calculate a proxy CDS or Bond spread. Due to practical issues of this method, Chourdakis et al. introduce a cross-section approach. In this paper, we extend the cross-section methodology using equity returns, and show that our methodology is significantly more accurate compared to both existing methodologies, and produces more reliable, stable and robust market risk and capital measures, and credit valuation adjustment.

Using Statistical Estimators to Gain Much Improved Convergence of Nested Monte-Carlo Simulations

The problem of estimating expectations of functions of conditional expectations using nested Monte Carlo simulation is studied. It is shown that typically the bias arising from non-linearity is in leading order inversely proportional to the number of sub-simulation paths when using a naive estimate. Various improved statistical estimators are introduced for the inner simulation. Applications to pricing of VIX derivatives, the computation of credit valuation adjustments and computation of Value-At-Risk are presented. It is shown that only small numbers of sub-paths are necessary for high accuracy.

A New Model for Pricing Collateralized Financial Derivatives

In recent years, collateralization of derivative contracts has extended from the standard mark-to-market and margining systems for exchange-traded contracts to the over-the-counter (OTC) market. This expansion of credit enhancement in OTC contracts has been underway at least since the 1990s, but it was formalized in the Dodd–Frank Act. The great majority of derivatives are now subject to collateralization requirements that are specified in a Credit Support Annex to the counterparties’ ISDA agreement. Derivatives counterparties must make a credit value adjustment (CVA) to the value of a contract on their books to account for the effect of counterparty credit risk. But the pricing models generally do not take the interaction between market prices and collateral values into account. This article develops a valuation approach that incorporates the counterparty’s credit quality and the effect of collateral, given market practices on how collateral is handled both under normal conditions and in a bankruptcy. Empirical comparison of the pricing of matched swaps with and without collateral support shows that the market does adjust for the mitigation of counterparty risk by the use of collateral, and that both factors are important. <b>TOPICS:</b>Counterparty risk, legal/regulatory/public policy, risk management

Modelling Counterparty Credit Risk in Czech Interest Rate Swaps

According to the Basel Committee’s estimate, three quarters of counterparty credit risk losses during the financial crisis in 2008 originate from credit valuation adjustment’s losses and not from actual defaults. Therefore, from 2015, the Third Basel Accord (EU, 2013a) and (EU, 2013b) instructed banks to calculate the capital requirement for the risk of credit valuation adjustment (CVA). Banks are trying to model CVA to hold the prescribed standards and also reach the lowest possible impact on their profit. In this paper, we try to model CVA using methods that are in compliance with the prescribed standards and also achieve the smallest possible impact on the bank’s earnings. To do so, a data set of interest rate swaps from 2015 is used. The interest rate term structure is simulated using the Hull-White one-factor model and Monte Carlo methods. Then, the probability of default for each counterparty is constructed. A safe level of CVA is reached in spite of the calculated the CVA achieving a lower level than CVA previously used by the bank. This allows a reduction of capital requirements for banks.

PDE models and numerical methods for total value adjustment in European and American options with counterparty risk

Since the last financial crisis, a relevant effort in quantitative finance research concerns the consideration of counterparty risk in financial contracts, specially in the pricing of derivatives. As a consequence of this new ingredient, new models, mathematical tools and numerical methods are required. In the present paper, we mainly consider the problem formulation in terms of partial differential equations (PDEs) models to price the total credit value adjustment (XVA) to be added to the price of the derivative without counterparty risk. Thus, in the case of European options and forward contracts different linear and nonlinear PDEs arise. In the present paper we propose suitable boundary conditions and original numerical methods to solve these PDEs problems. Moreover, for the first time in the literature, we consider XVA associated to American options by the introduction of complementarity problems associated to PDEs, as well as numerical methods to be added in order to solve them. Finally, numerical examples are presented to illustrate the behavior of the models and numerical method to recover the expected qualitative and quantitative properties of the XVA adjustments in different cases. Also, the first order convergence of the numerical method is illustrated when applied to particular cases in which the analytical expression for the XVA is available.

Internal Valuation of Assets with Liquidity Risk

FVA, funding valuation adjustment, is a new and controversial adjustment to the value of a firm’s derivatives position, joining CVA (credit valuation adjustment) and DVA (debt valuation adjustment), both of which attempt to account for the effects of counterparty credit risk on the value of a contract on a firm’s books. FVA adjusts for the fact that an investor who holds a security but finances it in the money market will have to pay a premium over the riskless rate in order to do so. This premium reduces the investment value of the position by an amount that is supposed to be captured by the FVA. The problem that Nauta raises is that this treatment does not distinguish between a security that is traded in a liquid market and can be sold for its fair value at any time, meaning overnight funding is OK, versus an illiquid security that must be financed through maturity. The article shows how this key difference translates into a liquidity-based FVA and an optimal funding horizon for less than perfectly liquid assets. <b>TOPICS:</b>Counterparty risk, quantitative methods, credit risk management

Partial differential equation pricing method for double-name credit-linked notes with counterparty risk in a reduced-form model with common shocks

In this study, we propose a partial differential equation method for pricing a double-name credit-linked note (CLN) with counterparty risk in a reduced-form framework. The default correlation among the CLN issuer and the reference entities of the double-name CLN is related to the occurrence of “common shocks,” which can simultaneously trigger defaults in some pre-specified groups of the aforementioned credit entities. By assuming that the stochastic intensities of common shocks are directly and inversely proportional to the spot interest rate, which follows a Cox–Ingersoll–Ross process, we deduce the corresponding explicit formulae for double-name CLN values. We also conducted some numerical experiments to examine how the correlated default risks among the credit entities affect the double-name CLN values and their credit valuation adjustment.

CVA under Bates Model with Stochastic Default Intensity

Counterparty credit risk has received increasing attention and become a topical issue since 2007 credit crisis, particularly for its impact on the valuation of the OTC derivatives. Credit Value Adjustment (CVA) has become an import field and it is required in Basel III. This paper studies CVA for European options under Bates model with stochastic default intensity. We develop a Monte Carlo and finite difference method framework for assessing exposure profiles and impact of counterparty credit risk in pricing. The exposures are computed by solving a partial integro-differential Equation (PIDE) using implicit-explicit (IMEX) time discretization schemes. CVA in presence of wrong way risk (WWR) is embedded in the correlation between risk factor and default intensity. Meanwhile, the jump-at-default feature of the models offers an effective means to assess WWR. Our results show that both jump and WWR play an important role in evaluating CVA and exposures. The impact is significant and it is crucial for risk management purpose.

Modeling and Quantifying of the Global Wrong Way Risk

The counterparty risk issue has become increasingly important in the world of finance. This risk is defined as the loss due to the counterparty default. The regulator uses the Credit Value Adjustment (CVA) to measure this risk. However, there is the independency assumption between the default and the exposure behind the CVA computation and it is not verified on the financial market. This paper presents two mathematical models for the assessment and the quantification of the counterparty risk without this assumption. This kind of risk is known as Wrong Way Risk (WWR). This study focuses on three approaches: empirical, copula and mixed model. The first one is based on the hazard rate modelling to express the correlation between the probability of default and the exposure. The second one is about calculating the WWR effect using copulas. The last one is a combination of both. There is another assumption that makes easier the CVA computation: The constant of the loss given default (LGD). As we know this assumption is not verified because the LGD could be deterministic or stochastic. Otherwise, it could lead to a correlation effect between the LGD, the exposure and the default, and we then obtain a Global Wrong Way Risk (GWWR). Indeed, we propose a model allowing the CVA quantification without these assumptions.

Derivatives pricing under bilateral counterparty risk

We consider risk-neutral valuation of a contingent claim under bilateral counterparty risk in a reduced-form setting similar to that of Duffie and Huang [1996] and Duffie and Singleton [1999]. The probabilistic valuation formulas derived under this framework cannot be usually used for practical pricing due to their recursive path-dependencies. Instead, finite-difference methods are used to solve the quasi-linear partial differential equations that equivalently represent the claim value function. By imposing restrictions on the dynamics of the risk-free rate and the stochastic intensities of the counterparties’ default times, we develop path-independent probabilistic valuation formulas that have closed-form solution or can lead to computationally efficient pricing schemes. Our framework incorporates the so-called wrong way risk (WWR) as the two counterparty default intensities can depend on the derivatives values. Inspired by the work of Ghamami and Goldberg [2014] on the impact of WWR on credit value adjustment (CVA), we derive calibration-implied formulas that enable us to mathematically compare the derivatives values in the presence and absence of WWR. We illustrate that derivatives values under unilateral WWR need not be less than the derivatives values in the absence of WWR. A sufficient condition under which this inequality holds is that the price process follows a semimartingale with independent increments.

Revisiting Interest Rate Swap Valuation with Counterparty Risk, Wrong-Way Risk, and OIS Discounting

This article extends extant valuation models of interest rate swaps with counterparty credit risk by accounting for wrong-way risk and overnight index swap (OIS) discounting. The proposed model extends the models of previous researchers by capturing wrong-way risk in the credit value adjustment (CVA) calculation by way of the correlation between the intensity of default of the counterparty and the market interest rate. Under the proposed no-arbitrage pricing model, cash flows are discounted using OIS rates (mostly used by market practitioners following the 2007–2009 credit crisis), a proxy for risk-free rates. The authors thus propose a unified framework that captures under one umbrella: CVA, wrong-way risk, and OIS discounting. The model parameters are estimated using real market data. Their findings indicate that it is important to account for both counterparty and wrong-way risk in interest rate swap valuation since the two phenomena have nonnegligible impacts on CVA value. Additionally, using OIS rates as risk-free discount rates, the model yields adjustment values higher than those obtained with traditional LIBOR discount rates.

Counterparty Credit Risk in OTC Derivatives under Basel III

Recent financial crises were the root of many changes in regulatory implementations in the banking sector. Basel previously covered the default capital charge for counterparty exposures however, the crisis showed that more than two third of the losses related to this risk emerged from the exposure to the movement of the counterparty’s credit quality and not its actual default therefore, Basel III divided the required counterparty risk capital into two categories: The traditional default capital charge and an additional counter-party credit valuation adjustment (CVA) capital charge. In this article, we explain the new methodologies to compute these capital charges on the OTC market: The standardized approach for default capital charge (SA-CCR) and the basic approach for CVA (BA-CVA). Based on historical calibration and future estimations, we built internal models in order to compare them with the amended standardized approach. Up till June 2015, interest rate and FX derivatives constituted more than 90% of the traded total OTC notional amount; we constructed our application on such portfolios containing and computed their total counterparty capital charge. The analysis reflected different impacts of the netting and collateral agreements on the regulatory capital depending on the instruments’ typologies. Moreover, results showed an important increase in the capital charge due to the CVA addition doubling it in some cases.