Credit Valuation Adjustment Research Articles

LEAST SQUARES MONTE CARLO CREDIT VALUE ADJUSTMENT WITH SMALL AND UNIDIRECTIONAL BIAS

Credit value adjustment (CVA) and related charges have emerged as important risk factors following the Global Financial Crisis. These charges depend on uncertain future values of underlying products, and are usually computed by Monte Carlo simulation. For products that cannot be valued analytically at each simulation step, the standard market practice is to use the regression functions from least squares Monte Carlo method to approximate their values. However, these functions do not necessarily provide accurate approximations to product values over all simulated paths and can result in biases that are difficult to control. Motivated by a novel characterization of the CVA as the value of an option with an early exercise opportunity at a stochastic time, we provide an approximation for CVA and other credit charges that rely only on the sign of the regression functions. The values are determined, instead, by pathwise deflated cash flows. A comparison of CVA for Bermudan swaptions and cancellable swaps shows that the proposed approximation results in much smaller errors than the standard approach of using the regression function values.

RANDOM TIME FORWARD-STARTING OPTIONS

We introduce a natural generalization of the forward-starting options. The main feature of the contract presented here is that the strike-determination time is not fixed ex-ante, but allowed to be random, usually related to the occurrence of some event, either of financial nature or not. We will call these options random time forward-starting (RTFS). We show that, under an appropriate “martingale preserving” hypothesis, we can exhibit arbitrage free prices, which can be explicitly computed in many classical market models, at least under independence and in absence of simultaneous jumps between the random time and the assets' prices. Practical implementations of the pricing methodologies are also provided. Finally, a credit value adjustment (CVA) formula for these over the counter (OTC) options is computed for the unilateral counterparty credit risk.

BOUNDING WRONG‐WAY RISK IN CVA CALCULATION

AbstractA credit valuation adjustment (CVA) is an adjustment applied to the value of a derivative contract or a portfolio of derivatives to account for counterparty credit risk. Measuring CVA requires combining models of market and credit risk to estimate a counterparty's risk of default together with the market value of exposure to the counterparty at default. Wrong‐way risk refers to the possibility that a counterparty's likelihood of default increases with the market value of the exposure. We develop a method for bounding wrong‐way risk, holding fixed marginal models for market and credit risk and varying the dependence between them. Given simulated paths of the two models, a linear program computes the worst‐case CVA. We analyze properties of the solution and prove convergence of the estimated bound as the number of paths increases. The worst case can be overly pessimistic, so we extend the procedure by constraining the deviation of the joint model from a baseline reference model. Measuring the deviation through relative entropy leads to a tractable convex optimization problem that can be solved through the iterative proportional fitting procedure. Here, too, we prove convergence of the resulting estimate of the penalized worst‐case CVA and the joint distribution that attains it. We consider extensions with additional constraints and illustrate the method with examples.

CVA for Pricing: Comparison of CEM, SA-CCR, and Advanced Approach

Credit Valuation Adjustments (CVA) have been common standard since the crisis in the banking sector. Dr. Monika Marquart describes three different methods for calculating exposure at default for CVA: The new regulatory SA-CCR, the widely used CEM and an advanced approach are introduced and compared regarding their quantitative impact on swap pricing. The article also gives an overview on differences between the regulatory approach of the Basel Committee and the best practice assumptions usually made in accounting for estimating IFRS fair value adjustments.

EXTREMAL DEPENDENCE FOR BILATERAL CREDIT VALUATION ADJUSTMENTS

Recognizing counterparty default risk as integral part of the valuation process of financial derivatives has changed the classical view on option pricing. Calculating the bilateral credit valuation adjustment (BCVA) including wrong way risk (WWR) requires a sound model for the dependence structure between three quantities: the default times of the two contractual parties and the derivative/portfolio value at the first of the two default times. There exist various proposals, but no market consensus, on how this dependence structure should be modeled. Moreover, available mathematical tools depend strongly on the marginal models for the default times and the model for the underlying of the derivative. In practice, independence between all (or some) quantities is still a popular (over-)simplification, which completely misses the root of WWR. In any case, specifying the dependence structure imposes one to model risk and even within some parametric model one typically obtains a considerable interval of BCVA values when the parameters are taken to the extremes. In this work, we present a model-free approach to identify the dependence structure that implies the extremes of BCVA. This is achieved by solving a mass-transportation problem using tools from optimization.

On the Correlation Approach and Parametric Approach for CVA Calculation

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, the so-called wrong-way risk (WWR) must be taken into account. The right-way risk (RWR) takes place when the two factors move in the opposite directions. These two effects are also called directional-way risk (DWR). A lot of efforts have been made to reduce the computational burden to calculate 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 study allows us to explain the parameters in the parametric approach. Our results also suggest that the parametric approach can become sensitive when calculating the WWR in certain scenarios. For risk model governance and validation purposes, cautions should be given while using the parametric approach for CVA calculation.

Wrong Way Risk Modeling and Computation in Credit Valuation Adjustment for European and Bermudan Options

We study the impact of wrong-way-risk (WWR) on credit valuation adjustment (CVA) for European and Bermudan options, based on an intensity model. WWR is modeled by a dependency between the underlying asset and the intensity of the counterparty’s default. We consider three different models. We take the difference between the default-free value and the default-adjusted value for the purpose of the CVA calculations. Two numerical algorithms, the COS method and the Stochastic Grid Bundling Method (SGBM) are generalized and employed for the computations. By varying correlation coefficients, we perform a CVA stress test for European options, show differences in the optimal exercise boundaries, in corresponding Bermudan option values and compute the CVA shortfall.

An Asymptotic Expansion for Forward–Backward SDEs: A Malliavin Calculus Approach

This paper proposes a new analytical approximation scheme for the representation of the forward–backward stochastic differential equations (FBSDEs) of Ma and Zhang (Ann Appl Probab, 2002). In particular, we obtain an error estimate for the scheme applying Malliavin calculus method for the forward SDEs combined with the Picard iteration scheme for the BSDEs. We also show numerical examples for pricing option with counterparty risk under local and stochastic volatility models, where the credit value adjustment is taken into account.

The Basis Goes Stochastic: A Jump-Diffusion Model for Financial Risk Applications

There are several pricing and risk model applications where the assumption of a deterministic LIBOR-OIS basis can lead to severe mispricing. By modeling such a basis using a jump-diffusion process, we show how stochastic basis can impact the valuation of specific deals such as zero-coupon swaps, as well as credit valuation adjustments and gap risk in particular. Explicit formulas are provided and numerical examples showcased.

A Gentle Introduction to Default Risk and Counterparty Credit Modelling

In this paper we introduce the reader to the basic tools for the computation of Counterparty Credit Risk such as Credit Value Adjustment and Debt Value Adjustment. We also present the effect of mitigating clauses, like netting and collateral, in reducing the credit exposure. Detailed numerical examples are presented with reference to commodity derivatives.

Efficient estimation of sensitivities for counterparty credit risk with the finite difference Monte Carlo method

According to Basel III, financial institutions have to charge a credit valuation adjustment (CVA) to account for a possible counterparty default. Calculating this measure and its sensitivities is one of the biggest challenges in risk management. Here, we introduce an efficient method for the estimation of CVA and its sensitivities for a portfolio of financial derivatives. We use the finite difference Monte Carlo (FDMC) method to measure exposure profiles and consider the computationally challenging case of foreign exchange barrier options in the context of the Black-Scholes as well as the Heston stochastic volatility model, with and without stochastic domestic interest rate, for a wide range of parameters. In the case of a fixed domestic interest rate, our results show that FDMC is an accurate method compared with the semi analytic COS method and, advantageously, can compute multiple options on one grid. In the more general case of a stochastic domestic interest rate, we show that we can accurately compute exposures of discontinuous one-touch options by using a linear interpolation technique as well as sensitivities with respect to initial interest rate and variance. This paves the way for real portfolio level risk analysis.

Option-Based Pricing of Wrong Way Risk for CVA

The two main issues for managing wrong way risk (WWR) for the credit valuation adjustment (CVA, i.e. WW-CVA) are calibration and hedging. Hence we start from a novel model-free worst-case approach based on static hedging of counterparty exposure with liquid options. We say "start from" because we demonstrate that a naive worst-case approach contains hidden unrealistic assumptions on the variance of the hazard rate (i.e. that it is infinite). We correct this by making it an explicit (finite) parameter and present an efficient method for solving the parametrized model optimizing the hedges. We also prove that WW-CVA is theoretically, but not practically, unbounded. The option-based hedges serve to significantly reduce (typically halve) practical WW-CVA. Thus we propose a realistic and practical option-based worst case CVA.

Uncollateralised Interest Rate Swap xVA

Derivative valuation adjustments (xVA) have been an evolving topic over the past decade, now giving us a rich theory of what was traditionally referred to as the relatively opaque "credit spread" imposed by banks. As independent risk advisors, we are regularly positioned between clients and their banks, considering the impacts of both parties' xVA when entering into and terminating positions. This paper documents our implementation of xVA modelling at the time of writing, specifically for the derivatives which our clients trade the most: uncollateralised interest rate swaps. It is also one of the first open demonstrations of an independent party modelling banks' xVA at levels actually observed by them. We focus on the most significant credit (CVA), funding (FVA) and capital (KVA) valuation adjustments. Our full modelling framework and its calibration are described and examples are given for swaps under the Standardised Approach (SA-CCR) capital framework, effective 1 January 2017.

Credit Value Adjustment with Market-Implied Recovery

We present a model for CVA calculation in which the recovery rate is inferred from the term structure of CDS spreads. We show that the assumption of constant recovery is inconsistent with observed CDS spreads and leads to substantial underestimation of the CVA. We further document that the degree of misspecification induced by the constant recovery assumption is affected by the ratings of the two counterparties and the phase of the business cycle. Assuming constant recovery biases CVA calculations through two channels: estimating the default probability and measuring the impact of correlation and the associated wrong-way risk.

A note on CVA and wrong way risk

Hull and White approach to Wrong Way Risk in the computation of Credit Value Adjustment (CVA) is considered the most straightforward generalization of the standard Basel approach. The model is financially intuitive and it can be implemented by a slight modification of existing algorithms for CVA calculation. However, path dependency in the key quantities has non-elementary consequences in the calibration of model parameters. We propose a simple and fast approach for computing these quantities via a recursion formula. We show in detail the calibration methodology on market data and CVA computations in two relevant cases: a FX forward and an interest rate swap.

Smile and default: the role of stochastic volatility and interest rates in counterparty credit risk

In this research, we investigate the impact of stochastic volatility and interest rates on counterparty credit risk (CCR) for FX derivatives. To achieve this we analyse two real-life cases in which the market conditions are different, namely during the 2008 credit crisis where risks are high and a period after the crisis in 2014, where volatility levels are low. The Heston model is extended by adding two Hull–White components which are calibrated to fit the EURUSD volatility surfaces. We then present future exposure profiles and credit value adjustments (CVAs) for plain vanilla cross-currency swaps (CCYS), barrier and American options and compare the different results when Heston-Hull–White-Hull–White or Black–Scholes dynamics are assumed. It is observed that the stochastic volatility has a significant impact on all the derivatives. For CCYS, some of the impact can be reduced by allowing for time-dependent variance. We further confirmed that Barrier options exposure and CVA is highly sensitive to volatility dynamics and that American options’ risk dynamics are significantly affected by the uncertainty in the interest rates.

BCVA with Stochastic Credit Models

It is common for practitioners to neglect correlation effects when reporting bilateral credit value adjustments (BCVA) to derivatives by using deterministic credit models. In this white paper we consider the addition of stochastic credit models to our interest rate BCVA modelling framework. The models chosen for this experiment are described and results are demonstrated for a 10 year uncollateralised GBP payer swap. We conclude that the effect of this enhancement on BCVA is not significant enough to justify it in our standard framework. The Numerix CrossAsset library is used throughout.

Efficient Calibration for CVA Using Multi-Level Monte Carlo

The evaluation of credit value adjustments (CVA) usually intensive computations basing on Monte Carlo simulations. In practice, often the CVA itself is not the quantity we are looking for, but a parameter which gives a certain CVA level or a worst-case CVA. Concrete examples are the translation of an upfront CVA payment into running spread (RS) payments or worst-case bounds for wrong-way risk (WWR) effects. We introduce Multi-Level Monte Carlo (MLMC) in the context of CVA. In particular, we demonstrate how to apply parametric integration using MLMC to significantly reduce computation time in for CVA calibration, such as RS and worst-case bounds for WWR.

Counterparty Credit Exposures for Interest Rate Derivatives using the Stochastic Grid Bundling Method

ABSTRACTThe regulatory credit value adjustment (CVA) for an outstanding over-the-counter (OTC) derivative portfolio is computed based on the portfolio exposure over its lifetime. Usually, the future portfolio exposure is approximated using the Monte Carlo simulation, as the portfolio value can be driven by several market risk-factors. For derivatives, such as Bermudan swaptions, that do not have an analytical approximation for their Mark-to-Market (MtM) value, the standard market practice is to use the regression functions from the least squares Monte Carlo method to approximate their MtM along simulated scenarios. However, such approximations have significant bias and noise, resulting in inaccurate CVA charge. In this paper, we extend the Stochastic Grid Bundling Method (SGBM) for the one-factor Gaussian short rate model, to efficiently and accurately compute Expected Exposure, Potential Future exposure and CVA for Bermudan swaptions. A novel contribution of the paper is that it demonstrates how different measures, for instance spot and terminal measure, can simultaneously be employed in the SGBM framework, to significantly reduce the variance and bias of the solution.

FVA and CVA for Collateralized Trades with Re-hypothecation

Through this article we join the ongoing “FVA debate” with a strategy of bilateral replication, which enables us to price derivatives trades and identify XVAs, including the FVA. We let the issuer and buyer replicate the derivative's payoffs and the loss given the issuer's default, respectively, while taking into account the costs of funding for replication and collateral posting in the replicating portfolios. The strategy leads to a PDE for the economic value of the derivative, and the PDE also implies the unique risk-neutral pricing measure. A proper solution of the PDE gives rise to a decomposition of the derivatives economic value into the risk-free value of the derivative, the credit valuation adjustment, and the funding valuation adjustment. We show that in general the economic values to the two trading parties are different, due to the different funding needs and funding costs, and both parties may have to bear some funding costs in order to strike a deal. When the premium for buying a derivatives asset is also funded, then to the buyer the trade actually consists of two transactions: borrowing and purchasing, and the FVA for the trade can be obtained by separately evaluating the FVAs of the two transactions.