Credit Valuation Adjustment Research Articles

Fast and stable second-order credit sensitivities of credit valuation adjustment

Fast and stable second-order credit sensitivities of credit valuation adjustment

Evaluation of counterparty credit risk under netting agreements

We investigate counterparty credit risk and credit valuation adjustments in portfolios including derivatives with early-exercise opportunities, under a netting agreement. We show that credit risk and netting agreements have a significant impact on the way portfolios are managed (that is, on options’ exercise strategies) and, therefore, on the value of the portfolio and on the price of counterparty risk. We derive the value of a netted portfolio as the solution of a zero-sum, finite horizon, discrete-time stochastic game. We show that this dynamic-game interpretation can be used to determine the value of the reglementary capital charges required of financial institutions to cover for counterparty credit risk and we propose a numerical valuation method. Numerical investigations show that currently used numerical approaches can grossly misestimate the value of credit valuation adjustments.

EFFICIENT WRONG-WAY RISK MODELING FOR FUNDING VALUATION ADJUSTMENTS

Wrong-Way Risk (WWR) is an important component in Funding Valuation Adjustment (FVA) modeling. Yet, the standard assumption is independence between market risks and the counterparty defaults and funding costs. This typical industrial setting is our point of departure, where we aim to assess the impact of WWR without running a full Monte Carlo simulation with all credit and funding processes. We propose to split the exposure profile into two parts: an independent and a WWR-driven part. For the former, exposures can be re-used from the standard xVA calculation. We express the second part of the exposure profile in terms of the stochastic drivers and approximate these by a common Gaussian stochastic factor. Within the affine setting, the proposed approximation is generic, is an add-on to the existing xVA calculations and provides an efficient and robust way to include WWR in FVA modeling. Case studies for an interest rate swap and a representative multi-currency portfolio of swaps illustrate that the approximation method is applicable in a practical setting. We analyze the approximation error and use the approximation to compute WWR sensitivities, which are needed for risk management. The approach is equally applicable to other metrics such as Credit Valuation Adjustment.

Bilateral Credit Valuation Adjustment of CDS Under Systemic and Correlated Idiosyncratic Risks

Bilateral Credit Valuation Adjustment of CDS Under Systemic and Correlated Idiosyncratic Risks

A static replication approach for callable interest rate derivatives: mathematical foundations and efficient estimation of SIMM–MVA

The computation of credit risk measures such as exposure and Credit Value Adjustments (CVA) requires the simulation of future portfolio prices. Recent metrics, such as dynamic Initial Margin (IM) and Margin Value Adjustments (MVA) additionally require the simulation of future conditional sensitivities. For portfolios with non-linear instruments that do not admit closed-form valuation formulas, this poses a significant computational challenge. This problem is addressed by proposing a static replication algorithm for interest rate options with early-exercise features under an affine term-structure model. Under the appropriate conditions, we can find an equivalent portfolio of vanilla options that replicate these products. Specifically, we decompose the product into a portfolio of European swaptions. The weights and strikes of the portfolio are obtained by regressing the target option value with interpretable, feed-forward neural networks. Once an equivalent portfolio of European swaptions is determined, we can leverage on closed-form expressions to obtain the conditional prices and sensitivities, which serve as an input to exposure and SIMM-driven MVA quantification. For a consistent forward sensitivity estimation, this involves the differentiation of the portfolio-weights. The accuracy and convergence of the method is demonstrated through several representative numerical examples, benchmarked against the established least-square Monte Carlo method.

CVA with wrong-way risk and correlation between defaults: An application to an interest rate swap

This paper presents a counterparty credit risk adjustment model to value over-the-counter financial derivatives. To do so, a bilateral credit valuation adjustment with wrong-way risk (WWR) and dependency between the defaults of both contract parties is developed in line with the Hull-White model (2012), which calculates default probabilities using a hazard rate modelled as an exponential function dependent on the value of the derivative. The model proposed incorporates a modified hazard rate for each entity, which includes the company’s own exposure to credit risk attributable to the other entity’s default. By so doing, a correlation between the respective defaults of the entities party to the financial derivative is added. The model developed is also applied to obtain the fair value of an interest rate swap and the results obtained, using Monte Carlo simulation, demonstrate that the value of this swap adjusted to the credit risk falls when the dependency between the entities’ defaults is considered.

Wrong Way Risk corrections to CVA in CIR reduced-form models

In this paper we provide an efficient methodology to compute the credit value adjustment of a European contingent claim subject to some default event concerning the issuer solvability, when the underlying and the default event are correlated. In particular, in a Black and Scholes market/CIR intensity-default model, we consider a second order expansion around the origin of a vulnerable call option with respect to a correlation parameter rho, which may be used to describe the wrong way risk of the contract, measuring the dependence between the underlying asset price and the option’s issuer default intensity. Numerical implementations of this approach are compared with the benchmark Monte Carlo simulations.

Extending the Merton model with applications to credit value adjustment

Extending the Merton model with applications to credit value adjustment

On Wasserstein distances, barycenters, and the cross-section methodology for proxy credit curves

The credit default swap (CDS) market plays an important role for financial institutions. This is not only for their trading activities, but also as it provides a source of information to extract default probabilities to be used for (counterparty) credit risk purposes, as for instance in credit valuation adjustment calculations. Nonetheless, the number of entities for which liquid single-name CDSs are traded is of the order of a few thousands. This requires financial institutions to employ proxy methodologies to estimate the credit risk they face when trading with counterparties for which no (liquid) CDSs are available in the market. In this paper, we propose and compare different approaches to take into account counterparty-specific information in terms of rating, region, sector, etc. at cross-sectional level to strip risk-neutral default probabilities from CDSs. This is achieved by taking into account the intrinsic probabilistic information characterizing each CDS by means of suitably-defined Wasserstein distances and barycenters. The results suggest that default probabilities are likely to be overestimated if the construction of the proxy credit curves overlooks the probability structure underlying the CDS market, potentially resulting in a too conservative counterparty credit risk pricing framework.

Credit Valuation Adjustment in Financial Networks

Credit Valuation Adjustment in Financial Networks

Evaluating credit valuation adjustment with wrong-way risk for Bermudan options

Evaluating credit valuation adjustment with wrong-way risk for Bermudan options

An Integrated Approach to Credit Valuation Adjustment

An Integrated Approach to Credit Valuation Adjustment

Interactions of Logistic Distribution to Credit Valuation Adjustment: A Study on the Associated Expected Exposure and the Conditional Value at Risk

In Basel III, the credit valuation adjustment (CVA) was given, and it was discussed that a bank covers mark-to-market losses for expected counterparty risk with a CVA capital charge. The purpose of this study is threefold. Using the logistic distribution, it is shown how the expected exposure can be derived for an interest rate swap. Secondly, the risk measure of VaR is contributed for the CVA under this distribution. Thirdly, generalizations for the CVA VaR and CVA CVaR are given by considering both the credit spread and the expected positive exposure to follow the logistic distributions with different parameters. Finally, several simulations are provided to uphold the theoretical discussions.

Structural estimation of counterparty credit risk under recovery risk

Counterparty Credit Risk (CCR) represents one of the major sources of uncertainty in many financial contracts. The role of credit value adjustment (CVA) is, in fact, that of rewarding the parties for the exposure to such risk. A key driver of CVA is the recovery risk, generated by the variability of recovery rates. In this paper, we develop a framework to assess the CCR accounting for the recovery risk that arises from the introduction of stochastic recovery rates. Adopting the structural model for the time to default that exploits a time-changed Lévy process for the risk driver of the equity value, we provide a complete picture to monitor the CCR and gauge the effects of the stochastic recovery rates. The model extracts information on the creditworthiness of the parties in the OTC contract combining Fourier Cosine Expansion and Monte Carlo simulations methods to price CDS spreads, the related underlying, and to retrieve the default barrier. We apply the model proposed to a business case analyzing the CCR of two parties involved in the OTC contract with underlying energy commodities. Low average recovery rates reveal to be associated with high implied volatility and depart from the fixed value of 40%, especially during periods of market distress.

Quantum algorithm for credit valuation adjustments

Quantum mechanics is well known to accelerate statistical sampling processes over classical techniques. In quantitative finance, statistical samplings arise broadly in many use cases. Here we focus on a particular one of such use cases, credit valuation adjustment (CVA), and identify opportunities and challenges towards quantum advantage for practical instances. To build a NISQ-friendly quantum circuit able to solve such problem, we draw on various heuristics that indicate the potential for significant improvement over well-known techniques such as reversible logical circuit synthesis. In minimizing the resource requirements for amplitude amplification while maximizing the speedup gained from the quantum coherence of a noisy device, we adopt a recently developed Bayesian variant of quantum amplitude estimation using engineered likelihood functions. We perform numerical analyses to characterize the prospect of quantum speedup in concrete CVA instances over classical Monte Carlo simulations.

Affine term structure models: A time‐change approach with perfect fit to market curves

AbstractWe address the so‐called calibration problem, which consists of fitting in a tractable way a given model to a specified term structure such as yield, prepayment or default probability curves. Time‐homogeneous affine jump diffusions (HAJD) are tractable processes but have limited flexibility; they fail to perfectly replicate actual market curves. Applying a deterministic shift to the latter is a simple but efficient solution that is widely used by both academics and practitioners. However, the shift approach may not be appropriate when positivity is required, a common constraint when dealing with credit spreads or default intensities. In this paper, we address this problem by adopting a time‐change technique. Specific attention is paid to the Cox–Ingersoll–Ross model with compound Poisson jumps (JCIR), which remains standard for modeling intensities. Our time‐changed JCIR (TC‐JCIR) is compared to the shifted JCIR (JCIR++) in various credit applications such as credit default swap (CDS), credit default swaption, and credit valuation adjustment (CVA) under wrong‐way risk (WWR). The TC‐JCIR model is able to generate much larger implied volatilities and covariance effects than JCIR++ under positivity constraints and represents an appealing alternative to the latter.

Credit Valuation Adjustment with Replacement Closeout: Theory and Algorithms

The replacement closeout convention has drawn more and more attention since the 2008 financial crisis. Compared with the conventional risk-free closeout, the replacement closeout convention incorporates the creditworthiness of the counterparty and thus providing a more accurate estimate of the Mark-to-market value of a financial claim. In contrast to the risk-free closeout, the replacement closeout renders a nonlinear valuation system, which constitutes the major difficulty in the valuation of the counterparty credit risk. In this paper, we show how to address the nonlinearity attributed to the replacement closeout in the theoretical and computational analysis. In the theoretical part, we prove the unique solvability of the nonlinear valuation system and study the impact of the replacement closeout on the credit valuation adjustment. In the computational part, we propose a neural network-based algorithm for solving the (high dimensional) nonlinear valuation system and effectively alleviating the curse of dimensionality. We numerically compare the computational cost for the valuations with risk-free and replacement closeouts. The numerical tests confirm both the accuracy and the computational efficiency of our proposed algorithm for the valuation of the replacement closeout.

The implications of dependence, tail dependence, and bounds’ measures for counterparty credit risk pricing

This paper investigates the counterparty credit risk of interest rate swaps positions using the credit valuation adjustment (CVA) measure, and examines the potential dependence relationships between the probability of default (PD) and exposure at default (EAD). We empirically tested, using interest rate swaption implied market volatilities, six tail dependence models: a Basel III Committee independent model, a Gaussian, Student’s t, Gumbel, Clayton, and a wrong way risk (WWR) copula with dependence approach. The results show that the CVA underestimation when using a Clayton copula for modelling the dependence of PD and EAD is about 10%–856% compared to using WWR, and the underestimation between using the standardised Basel independent model and using the Gaussian copula is about 552%–2621%, including the period of the 2007/2008 crisis, signalling that dependence is more important than tail dependence for CVA measurement. This has important implications for regulators, financial institutions, and credit risk managers when calculating counterparty risk.

American Monte Carlo by Stochastic Separation of the Expected Value

American Monte Carlo is a solution to the puzzle of calculating the value of derivatives with the right to an early exercise, when using Monte Carlo simulation. One of the technique uses regression of some suitable basis functions, which is a bit arbitrary, and could if made wrong render in expectation bias. It is also useful when simulating nested simulations, like pricing Credit Value Adjustment (CVA) that is when the portfolio includes derivatives that need simulation to be priced. Here, we use a simple way of calculating the price when using simulation, based on the simple fact that the expectation integral could be divided by the random numbers of the simulation. This method works well even for a small sample of simulations and it also applies to the the case of nested simulations, also in the case of having different measures of the simulations. One example of usage is to calculate different XVA's under the lifespan of the portfolio, especially focusing on Initial Margin (IM) and Marginal Value Adjustment (MVA).

Cash CVA -- Credit Valuation Adjustment in the Cash Form

Credit default swaps (CDS) are unfunded, or the synthetic form of credit exposure, while bonds are fully funded, thus the cash form. Borrowing this industry jargon, credit valuation adjustment (CVA) would be seen synthetic, because it is defined as the present value of buying a default protection on counterparty exposure through CDS. This article presents the cash form of counterparty risk, Cash CVA. It links directly to counterparties’ senior unsecured bond curves, thus no longer requiring recovery rate and CDS curve inputs. Replacing CVA with cash CVA is necessitated, because single name CDS is no longer liquid following massive dealer-bank exit from CDS trading business post the Financial Crisis of 2007-2008, while bond liquidity has significantly improved due to advances in corporate bond electronic trading.