Contagion models a la carte: which one to choose?

We propose a copula contagion mixture model for correlated default times. The model includes the well-known factor, copula, and contagion models as its special cases. The key advantage of such a model is that we can study the interaction of different models and their pricing impact. Specifically, we model the default times of the underlying names in a reference portfolio to follow contagion intensity processes with exponential decay coupled with a copula dependence structure. We also model the default time of the counterparty and its dependence structure with the reference portfolio. Numerical tests show that correlation and contagion have an enormous joint impact on the rates of CDO tranches and the corresponding credit value adjustments are extremely high to compensate for the wrong-way risk.     

Estimating Portfolio Credit Losses in Downturns

This paper suggests formulas able to capture potential strong connection among credit losses in downturns without assuming any specific distribution for the variables involved. We first show that the current model adopted by regulators (Basel) is equivalent to a conditional distribution derived from the Gaussian Copula (which does not identify tail dependence). We then use conditional distributions derived from copulas that express tail dependence (stronger dependence across higher losses) to estimate the probability of credit losses in extreme scenarios (crises). Next, we use data on historical credit losses incurred in American banks to compare the suggested approach to the Basel formula with respect to their performance when predicting the extreme losses observed in 2009 and 2010. Our results indicate that, in general, the copula approach outperforms the Basel method in two of the three credit segments investigated. The proposed method is extendable to other differentiable copula families and this gives flexibility to future practical applications of the model.     

Transitional credit modelling and its relationship to market value at risk: an Australian sectoral perspective

Internal credit risk modelling is important for banks for the calculation of capital adequacy in terms of the Basel Accords, and for the management of sectoral exposure. We examine Credit Value at Risk (VaR), Conditional Credit Value at Risk (Credit CVaR) and the relationship between market and credit risk. Significant association is found between different Credit CVaR methods, and between market and credit risk. Simpler Credit CVaR methods are found to be viable alternatives to more complex methodology. The relationship between market and credit risk is used to develop a new model that allows banks to incorporate industry risk into transition modelling, without macroeconomic analysis.     

CDS as insurance: Leaky lifeboats in stormy seas

What market features of financial risk transfer exacerbate counterparty risk? To analyze this, we formulate a model which elucidates important differences between financial risk transfer and traditional insurance, using the example of Credit Default Swaps (CDS). We allow for (heterogeneous) insurer insolvency, which captures the possibility that relatively risky counterparties may exist in the market. Further, we find that stable insurers become less stable as the price of the contract decreases. The analysis includes insured parties that have heterogeneous motivations for purchasing CDS. For example, some may own the underlying asset and purchase CDS for risk management, while others buy these contracts purely for trading purposes. We show that traders will choose to contract with less stable insurers, resulting in higher counterparty risk in this market relative to that of traditional insurance; however, a regulatory policy that removes traders can, perversely, cause stable counterparties to become less stable. We conclude with two extensions of the model that consider a Central Counterparty (CCP) arrangement and the consequences of asymmetric information over insurer type.     

Investigating the dependence structure between credit default swap spreads and the U.S. financial market

Under Basel II framework, credit risk assessment is of high significance in the light of correlation risk. Correlation risk is often envisioned along with business conditions and financial market’s impact. We employ copula methodology to identify the dependence structures that may exist between market risk fundamentals and credit risk fundamentals. Considering credit derivative spreads as credit risk fundamentals and market data as market risk determinants, we describe and quantify the asymmetric link prevailing between credit risk and market risk. Credit risk is negatively linked with market price risk whereas it becomes positively linked with market volatility risk. Such patterns give rise to interesting asymmetric dependence structures between both risk sources. We are then able to balance reliably market price risk with market volatility feedback, the market trend supporting a common correlation between securities. In the light of the previous trade-off, we propose also a simple credit risk management rule.     

Counterparty credit risk and the credit default swap market

Counterparty credit risk has become one of the highest-profile risks facing participants in the financial markets. Despite this, relatively little is known about how counterparty credit risk is actually priced. We examine this issue using an extensive proprietary data set of contemporaneous CDS transaction prices and quotes by 14 different CDS dealers selling credit protection on the same underlying firm. This unique cross-sectional data set allows us to identify directly how dealers' credit risk affects the prices of these controversial credit derivatives. We find that counterparty credit risk is priced in the CDS market. The magnitude of the effect, however, is vanishingly small and is consistent with a market structure in which participants require collateralization of swap liabilities by counterparties.     

Pricing counterparty default risks: Applications to FRNs and vulnerable options

This paper provides simple closed-form pricing models for floating-rate notes and vulnerable options under the counterparty risk framework of [Jarrow, R., Yu, F., 2001. Counterparty risk and the pricing of default risk. Journal of Finance 56, 1765-1799]. After deriving closed-form pricing models for them, this paper illustrates the impact of the default intensity of counterparty on the prices of floating-rate notes and vulnerable options. Numerical examples show that the default risk of counterparty is an important factor of the value of floating-rate notes and vulnerable options.     

Loan pricing under Basel capital requirements

We analyze the loan pricing implications of the reform of bank capital regulation known as Basel II. We consider a perfectly competitive market for business loans where, as in the model underlying the internal ratings based (IRB) approach of Basel II, a single risk factor explains the correlation in defaults across firms. Our loan pricing equation implies that low risk firms will achieve reductions in their loan rates by borrowing from banks adopting the IRB approach, while high risk firms will avoid increases in their loan rates by borrowing from banks that adopt the less risk-sensitive standardized approach of Basel II. We also show that only a very high social cost of bank failure might justify the proposed IRB capital charges, partly because the net interest income from performing loans is not counted as a buffer against credit losses. A net interest income correction for IRB capital requirements is proposed.     

后危机时代信用衍生市场的发展方向

信用衍生市场能够促进信用风险的定价、分散和转移,并有助于金融稳定。遗憾的是,此次金融危机的发生,导致各方对信用衍生品产生很多误解。本文试图对此作一澄清,通过分析后危机时代信用衍生市场的新特点和改革路径,找出其未来的发展方向,并对发展中国的信用衍生市场提出建议。     

A Lévy HJM multiple-curve model with application to CVA computation

We consider the problem of valuation of interest rate derivatives in the post-crisis set-up. We develop a multiple-curve model, set in the HJM framework and driven by a Lévy process. We proceed with joint calibration to OTM swaptions and co-terminal ATM swaptions of different tenors, the calibration to OTM swaptions guaranteeing that the model correctly captures volatility smile effects and the calibration to co-terminal ATM swaptions ensuring an appropriate term structure of the volatility in the model. To account for counterparty risk and funding issues, we use the calibrated multiple-curve model as an underlying model for CVA computation. We follow a reduced-form methodology through which the problem of pricing the counterparty risk and funding costs can be reduced to a pre-default Markovian BSDE, or an equivalent semi-linear PDE. As an illustration, we study the case of a basis swap and a related swaption, for which we compute the counterparty risk and funding adjustments.     

A framework to measure integrated risk

A framework underlying various models that measure the credit risk of a portfolio is extended in this paper to allow the integration of credit risk with a range of market risks using Monte Carlo simulation. A structural model is proposed that allows interest rates to be stochastic and provides closed-form expressions for the market value of a firm's equity and its probability of default. This model is embedded within the integrated framework and the general approach illustrated by measuring the risk of a foreign exchange forward when there is a significant probability of default by the counterparty. For this example moving from a market risk calculation to an integrated risk calculation reduces the expected future value of the instrument by an amount that could not be calculated using the common pre-settlement exposure technique for estimating the credit risk of a derivative.     

Pricing CDOs with state-dependent stochastic recovery rates

Up to the 2007 crisis, research within bottom-up CDO models mainly concentrated on the dependence between defaults. Since then, due to substantial increases in market prices of systemic credit risk protection, more attention has been paid to recovery rate assumptions. In this paper, we use stochastic orders theory to assess the impact of recovery on CDOs and show that, in a factor copula framework, a decrease of recovery rates leads to an increase of the expected loss on senior tranches, even though the expected loss on the portfolio is kept fixed. This result applies to a wide range of latent factor models and is not specific to the Gaussian copula model. We then suggest introducing stochastic recovery rates in such a way that the conditional on the factor expected loss (or, equivalently, the large portfolio approximation) is the same as in the recovery markdown case. However, granular portfolios behave differently. We show that a markdown is associated with riskier portfolios than when using the stochastic recovery rate framework. As a consequence, the expected loss on a senior tranche is larger in the former case, whatever the attachment point. We also deal with implementation and numerical issues related to the pricing of CDOs within the stochastic recovery rate framework. Due to differences across names regarding the conditional (on the factor) losses given default, the standard recursion approach becomes problematic. We suggest approximating the conditional on the factor loss distributions, through expansions around some base distribution. Finally, we show that the independence and comonotonic cases provide some easy to compute bounds on expected losses of senior or equity tranches.     

Credit Derivatives in an Affine Framework

An efficient method for valuing credit derivatives based on three entities is developed in an affine framework. This includes interdependence of market and credit risk, joint credit migration and counterparty default risk of three firms. As an application we provide closed form expressions for the joint distribution of default times, default correlations, and default swap spreads in the presence of counterparty default risk. Vienna Institute of Finance is funded by WWTF (Vienna Science and Technology Fund).     

创新信用衍生工具 提升风险管理能力——巴塞尔协议Ⅲ背景下的CRM

最新公布的巴塞尔协议Ⅲ,要求银行业充分吸取金融危机教训,切实加强信用风险管理。中国银行间市场推出的信用风险缓释工具,具有多项创新价值,发挥了本土市场成功经验。从微观角度,该工具有助于提高信用风险动态管理能力、丰富资本管理手段、支持逆周期管理和灵活平衡集中度;从宏观角度,该工具有助于打造渐进式风险管理机制、强化资本约束并带动市场基础建设。为进一步推进中国信用衍生产品市场发展,建议发挥商业银行的主导作用和非银行机构的辅助作用,完善交易机制、定价机制和会计处理,创新清算机制,并纳入银行自身风险管理流程。     

Correlation smile matching for collateralized debt obligation tranches with α-stable distributions and fitted Archimedean copula models

As an extension of the standard Gaussian copula model to price collateralized debt obligation (CDO) tranche swaps we present a generalization of a one-factor copula model based on stable distributions. For special parameter values these distributions coincide with Gaussian or Cauchy distributions, but changing the parameters allows a continuous deformation away from the Gaussian copula. All these factor copulas are embedded in a framework of stochastic correlations. We furthermore generalize the linear dependence in the usual factor approach to a more general Archimedean copula dependence between the individual trigger variable and the common latent factor. Our analysis is carried out on a non-homogeneous correlation structure of the underlying portfolio. CDO tranche market premia, even throughout the correlation crisis in May 2005, can be reproduced by certain models. From a numerical perspective, all these models are simple, since calculations can be reduced to one-dimensional numerical integrals.     

A dynamic programming approach for pricing CDS and CDS options

We propose a flexible framework for pricing single-name knock-out credit derivatives. Examples include Credit Default Swaps (CDSs) and European, American and Bermudan CDS options. The default of the underlying reference entity is modelled within a doubly stochastic framework where the default intensity follows a CIR++ process. We estimate the model parameters through a combination of a cross sectional calibration-based method and a historical estimation approach. We propose a numerical procedure based on dynamic programming and a piecewise linear approximation to price American-style knock-out credit options. Our numerical investigation shows consistency, convergence and efficiency. We find that American-style CDS options can complete the credit derivatives market by allowing the investor to focus on spread movements rather than on the default event.     

The determinants of CDS spreads: evidence from the model space

We apply Bayesian model averaging and a frequentistic model space analysis to assess the pricing determinants of credit default swaps (CDSs). Our study focuses on the complete model space of plausible models and thus supports ultimate robustness. Using a large dataset of CDS contracts we find that CDS price dynamics can be mainly explained by factors describing firms’ sensitivity to extreme market movements. More precisely, our results suggest that dynamic copula based measures of tail dependence incorporate most essential pricing information, making other potential determinants such as Merton-type factors or linear variables measuring the systematic market evolution negligible.     

基于藤结构Copula的多元信用风险相关性度量模型及其比较

在信贷组合管理的框架下,以行业信用风险为基础,构建了基于藤结构Copula的多元信用风险相关性度量模型,并以2006年6月～2010年12月中国上市公司数据对模型进行了参数估计,发现Canoni-cal藤结构更适合度量行业信用风险相关性,以电力煤气及水的生产为代表的强周期性行业对信用风险起着引导作用。     

Diversification evidence from international equity markets using extreme values and stochastic copulas

Tail dependence plays an important role in financial risk management and determination of whether two markets crash or boom together. However, the linear correlation is unable to capture the dependence structure among financial data. Moreover, given the reality of fat-tail or skewed distribution of financial data, normality assumption for risk measure may be misleading in portfolio development. This paper proposes the use of conditional extreme value theory and time-varying copula to capture the tail dependence between the Australian financial market and other selected international stock markets. Conditional extreme value theory enables the model adequacy and the tail behavior of individual financial variable, while the time-varying copula can fully disclose the changes of dependence structure over time. The combination of both proved to be useful in determining the tail dependence. The empirical results show an outperformance of the model in the analysis of tail dependence, which has an important implication in cross-market diversification and asset pricing allocation.