The impact of a stress scenario of default events on the loss distribution of a credit portfolio can be assessed by determining the loss distribution conditional on these events. While it is conceptually easy to estimate loss distributions conditional on default events by means of Monte Carlo simulation, it becomes impractical for two or more simultaneous defaults as then the conditioning event is extremely rare. We provide an analytical approach to the calculation of the conditional loss distribution for the CreditRisk+ portfolio model with independent random loss given default distributions. The analytical solution for this case can be used to check the accuracy of an approximation to the conditional loss distribution whereby the unconditional model is run with stressed input probabilities of default (PDs). It turns out that this approximation is unbiased. Numerical examples, however, suggest that the approximation may be seriously inaccurate but that the inaccuracy leads to overestimation of tail losses and hence the approach errs on the conservative side.
We propose a novel credit default model that takes into account the impact of macroeconomic information and contagion effect on the defaults of obligors. We use a set-valued Markov chain to model the default process, which is the set of all defaulted obligors in the group. We obtain analytic characterizations for the default process, and use them to derive pricing formulas in explicit forms for synthetic collateralized debt obligations (CDOs). Furthermore, we use market data to calibrate the model and conduct numerical studies on the tranche spreads of CDOs. We find evidence to support that sy…
We compare the five major sources of corporate Credit Default Swap prices: GFI, Fenics, Reuters, CMA, and Markit, using the most liquid single name 5-year CDS in the iTraxx and CDX indexes from 2004 to 2010. Deviations from the common trend among prices in the different databases are not random but are explained by idiosyncratic factors, financing costs, global risk, and other trading factors. The CMA quotes lead the price discovery process. Moreover, we find that there is not a full agreement among databases in the results of the price discovery analysis between stock and CDS returns.
The current research on credit risk is primarily focused on modeling default probabilities. Recovery rates are often treated as an afterthought; they are modeled independently, in many cases they are even assumed constant. This is despite of their pronounced effect on the tail of the loss distribution. Here, we take a step back, historically, and start again from the Merton model, where defaults and recoveries are both determined by an underlying process. Hence, they are intrinsically connected. For the diffusion process, we can derive the functional relation between expected recovery rate and…
A multi-dimensional extension of the structural default model with firms' values driven by diffusion processes with Marshall-Olkin-inspired correlation structure is presented. Semi-analytical methods for solving the forward calibration problem and backward pricing problem in three dimensions are developed. The model is used to analyze bilateral counterparty risk for credit default swaps and evaluate the corresponding credit and debt value adjustments.
We study insolvency cascades in an interbank system when banks are allowed to insure their loans with credit default swaps (CDS) sold by other banks. We show that, by properly shifting financial exposures from one institution to another, a CDS market can be designed to rewire the network of interbank exposures in a way that makes it more resilient to insolvency cascades. A regulator can use information about the topology of the interbank network to devise a systemic insurance surcharge that is added to the CDS spread. CDS contracts are thus effectively penalized according to how much they cont…
We compare two models of corporate default by calculating the Jeffreys-Kullback-Leibler divergence between their predicted default probabilities when asset correlations are either high or low. Our main results show that the divergence between the two models increases in highly correlated, volatile, and large markets, but that it is closer to zero in small markets, when asset correlations are low and firms are highly leveraged. These findings suggest that during periods of financial instability the single-and multi-factor models of corporate default will generate increasingly inconsistent predi…
Credit risk management in Italy is characterized, in the period June 2008 to June 2012, by frequent (frequency=0.5 cycles per year) and intense (peak amplitude: mean=39.2 billion Euros, s.e.=2.83 billion Euros) quarterly contractions and expansions around the mean (915.4 billion Euros, s.e.=3.59 billion Euros) of the nominal total credit used by non-financial corporations. Such frequent and intense fluctuations are frequently ascribed to exogenous Basel II procyclical effects on credit flow into the economy and, consequently, Basel III output based point in time Credit to GDP countercyclical b…
Banks and financial institutions all over the world manage portfolios containing tens of thousands of customers. Not all customers are high credit-worthy, and many possess varying degrees of risk to the Bank or financial institutions that lend money to these customers. Hence assessment of credit risk is paramount in the field of credit risk management. This paper discusses the use of Bayesian principles and simulation-techniques to estimate and calibrate the default probability of credit ratings. The methodology is a two-phase approach where, in the first phase, a posterior density of default …
This article presents a new model for valuing a credit default swap (CDS) contract that is affected by multiple credit risks of the buyer, seller and reference entity. We show that default dependency has a significant impact on asset pricing. In fact, correlated default risk is one of the most pervasive threats in financial markets. We also show that a fully collateralized CDS is not equivalent to a risk-free one. In other words, full collateralization cannot eliminate counterparty risk completely in the CDS market.
We introduce the general arbitrage-free valuation framework for counterparty risk adjustments in presence of bilateral default risk, including default of the investor. We illustrate the symmetry in the valuation and show that the adjustment involves a long position in a put option plus a short position in a call option, both with zero strike and written on the residual net value of the contract at the relevant default times. We allow for correlation between the default times of the investor, counterparty and underlying portfolio risk factors. We use arbitrage-free stochastic dynamical models. …
The importance of adequately modeling credit risk has once again been highlighted in the recent financial crisis. Defaults tend to cluster around times of economic stress due to poor macro-economic conditions, {\em but also} by directly triggering each other through contagion. Although credit default swaps have radically altered the dynamics of contagion for more than a decade, models quantifying their impact on systemic risk are still missing. Here, we examine contagion through credit default swaps in a stylized economic network of corporates and financial institutions. We analyse such a syst…
The recent "correlation breakdown" in the modeling of credit default swaps, in which model correlations had to exceed 100% in order to reproduce market prices of supersenior tranches, is analyzed and argued to be a fundamental market inconsistency rather than an inadequacy of the specific model. As a consequence, markets under such conditions are exposed to the possibility of arbitrage. The general construction of arbitrage portfolios under specific conditions is presented.
We introduce an innovative theoretical framework to model derivative transactions between defaultable entities based on the principle of arbitrage freedom. Our framework extends the traditional formulations based on Credit and Debit Valuation Adjustments (CVA and DVA). Depending on how the default contingency is accounted for, we list a total of ten different structuring styles. These include bipartite structures between a bank and a counterparty, tri-partite structures with one margin lender in addition, quadri-partite structures with two margin lenders and, most importantly, configurations w…
We study dynamic hedging of counterparty risk for a portfolio of credit derivatives. Our empirically driven credit model consists of interacting default intensities which ramp up and then decay after the occurrence of credit events. Using the Galtchouk-Kunita-Watanabe decomposition of the counterparty risk price payment stream, we recover a closed-form representation for the risk minimizing strategy in terms of classical solutions to nonlinear recursive systems of Cauchy problems. We discuss applications of our framework to the most prominent class of credit derivatives, including credit swap …
The classical reduced-form and filtration expansion framework in credit risk is extended to the case of multiple, non-ordered defaults, assuming that conditional densities of the default times exist. Intensities and pricing formulas are derived, revealing how information driven default contagion arises in these models. We then analyze the impact of ordering the default times before expanding the filtration. While not important for pricing, the effect is significant in the context of risk management, and becomes even more pronounced for highly correlated and asymmetrically distributed defaults.…
In this paper, we performs a credit risk analysis, on the data of past loan applicants of a company named Lending Club. The calculation required the use of exploratory data analysis and machine learning classification algorithms, namely, Logistic Regression and Random Forest Algorithm. We further used the calculated probability of default to design a credit derivative based on the idea of a Credit Default Swap, to hedge against an event of default. The results on the test set are presented using various performance measures.
A new procedure is presented for the objective comparison and evaluation of default definitions. This allows the lender to find a default threshold at which the financial loss of a loan portfolio is minimised, in accordance with Basel II. Alternative delinquency measures, other than simply measuring payments in arrears, can also be evaluated using this optimisation procedure. Furthermore, a simulation study is performed in testing the procedure from `first principles' across a wide range of credit risk scenarios. Specifically, three probabilistic techniques are used to generate cash flows, whi…
For credit risk management purposes in general, and for allocation of regulatory capital by banks in particular (Basel II), numerical assessments of the credit-worthiness of borrowers are indispensable. These assessments are expressed in terms of probabilities of default (PD) that should incorporate a certain degree of conservatism in order to reflect the prudential risk management style banks are required to apply. In case of credit portfolios that did not at all suffer defaults, or very few defaults only over years, the resulting naive zero or close to zero estimates would clearly not involv…
NEC Corporation（日本电气公司）；Institute for International Socio-Economic Studies（国际社会经济研究所）
The European sovereign debt crisis has impaired many European banks. The distress on the European banks may transmit worldwide, and result in a large-scale knock-on default of financial institutions. This study presents a computer simulation model to analyze the risk of insolvency of banks and defaults in a bank credit network. Simulation experiments reproduce the knock-on default, and quantify the impact which is imposed on the number of bank defaults by heterogeneity of the bank credit network, the equity capital ratio of banks, and the capital surcharge on big banks.
The existence of asymmetric information has always been a major concern for financial institutions. Financial intermediaries such as commercial banks need to study the quality of potential borrowers in order to make their decision on corporate loans. Classical methods model the default probability by financial ratios using the logistic regression. As one of the major commercial banks in France, we have access to the the account activities of corporate clients. We show that this transactional data outperforms classical financial ratios in predicting the default event. As the new data reflects t…
The modeling of the probability of joint default or total number of defaults among the firms is one of the crucial problems to mitigate the credit risk since the default correlations significantly affect the portfolio loss distribution and hence play a significant role in allocating capital for solvency purposes. In this article, we derive a closed-form expression for the probability of default of a single firm and the probability of the total number of defaults by any time $t$ in a homogeneous portfolio of firms. We use a contagion process to model the arrival of credit events that causes the…
There is empirical evidence that recovery rates tend to go down just when the number of defaults goes up in economic downturns. This has to be taken into account in estimation of the capital against credit risk required by Basel II to cover losses during the adverse economic downturns; the so-called "downturn LGD" requirement. This paper presents estimation of the LGD credit risk model with default and recovery dependent via the latent systematic risk factor using Bayesian inference approach and Markov chain Monte Carlo method. This approach allows joint estimation of all model parameters and …
We introduce a novel class of credit risk models in which the drift of the survival process of a firm is a linear function of the factors. The prices of defaultable bonds and credit default swaps (CDS) are linear-rational in the factors. The price of a CDS option can be uniformly approximated by polynomials in the factors. Multi-name models can produce simultaneous defaults, generate positively as well as negatively correlated default intensities, and accommodate stochastic interest rates. A calibration study illustrates the versatility of these models by fitting CDS spread time series. A nume…
We propose two structural models for stochastic losses given default which allow to model the credit losses of a portfolio of defaultable financial instruments. The credit losses are integrated into a structural model of default events accounting for correlations between the default events and the associated losses. We show how the models can be calibrated and analyze the impact of correlations between the occurrences of defaults and recoveries by testing our models for a representative sample portfolio.
It had been believed in the conventional practice that the risk of a bank going bankrupt is lessened in a straightforward manner by transferring the risk of loan defaults. But the failure of American International Group in 2008 posed a more complex aspect of financial contagion. This study presents an extension of the asset network systemic risk model (ANWSER) to investigate whether credit default swaps mitigate or intensify the severity of financial contagion. A protection buyer bank transfers the risk of every possible debtor bank default to protection seller banks. The empirical distributio…
We propose a unified framework for equity and credit risk modeling, where the default time is a doubly stochastic random time with intensity driven by an underlying affine factor process. This approach allows for flexible interactions between the defaultable stock price, its stochastic volatility and the default intensity, while maintaining full analytical tractability. We characterise all risk-neutral measures which preserve the affine structure of the model and show that risk management as well as pricing problems can be dealt with efficiently by shifting to suitable survival measures. As an…
We present two methodologies on the estimation of rating transition probabilities within Markov and non-Markov frameworks. We first estimate a continuous-time Markov chain using discrete (missing) data and derive a simpler expression for the Fisher information matrix, reducing the computational time needed for the Wald confidence interval by a factor of a half. We provide an efficient procedure for transferring such uncertainties from the generator matrix of the Markov chain to the corresponding rating migration probabilities and, crucially, default probabilities. For our second contribution, …
A standard quantitative method to access credit risk employs a factor model based on joint multivariate normal distribution properties. By extending a one-factor Gaussian copula model to make a more accurate default forecast, this paper proposes to incorporate a state-dependent recovery rate into the conditional factor loading, and model them by sharing a unique common factor. The common factor governs the default rate and recovery rate simultaneously and creates their association implicitly. In accordance with Basel III, this paper shows that the tendency of default is more governed by system…
Since the 1990s, there have been significant advances in the technology space and the e-Commerce area, leading to an exponential increase in demand for cashless payment solutions. This has led to increased demand for credit cards, bringing along with it the possibility of higher credit defaults and hence higher delinquency rates, over a period of time. The purpose of this research paper is to build a contemporary credit scoring model to forecast credit defaults for unsecured lending (credit cards), by employing machine learning techniques. As much of the customer payments data available to len…
In this paper we develop structural first passage models (AT1P and SBTV) with time-varying volatility and characterized by high tractability, moving from the original work of Brigo and Tarenghi (2004, 2005) [19] [20] and Brigo and Morini (2006)[15]. The models can be calibrated exactly to credit spreads using efficient closed-form formulas for default probabilities. Default events are caused by the value of the firm assets hitting a safety threshold, which depends on the financial situation of the company and on market conditions. In AT1P this default barrier is deterministic. Instead SBTV ass…
We compare observed corporate cumulative default probabilities to those calculated using a stochastic model based on an extension of the work of Black and Cox and find that corporations default as if via diffusive dynamics. The model, based on a contingent-claims analysis of corporate capital structure, is easily calibrated with readily available historical default probabilities and fits observed default data published by Standard and Poor's. Applying this model to the Standard and Poor's default data we find that the difference in default behavior between credit ratings can be explained large…
We provide analytical pricing formula of corporate defaultable bond with both expected and unexpected default in the case with stochastic default intensity. In the case with constant short rate and exogenous default recovery using PDE method, we gave some pricing formula of the defaultable bond under the conditions that 1)expected default recovery is the same with unexpected default recovery; 2) default intensity follows one of 3 special cases of Willmott model; 3) default intensity is uncorrelated with firm value. Then we derived a pricing formula of a credit default swap. And in the case of …
We review different approaches for measuring the impact of liquidity on CDS prices. We start with reduced form models incorporating liquidity as an additional discount rate. We review Chen, Fabozzi and Sverdlove (2008) and Buhler and Trapp (2006, 2008), adopting different assumptions on how liquidity rates enter the CDS premium rate formula, about the dynamics of liquidity rate processes and about the credit-liquidity correlation. Buhler and Trapp (2008) provides the most general and realistic framework, incorporating correlation between liquidity and credit, liquidity spillover effects betwee…
This paper provides an alternative approach to Duffie and Lando [Econometrica 69 (2001) 633-664] for obtaining a reduced form credit risk model from a structural model. Duffie and Lando obtain a reduced form model by constructing an economy where the market sees the manager's information set plus noise. The noise makes default a surprise to the market. In contrast, we obtain a reduced form model by constructing an economy where the market sees a reduction of the manager's information set. The reduced information makes default a surprise to the market. We provide an explicit formula for the def…
We analyse time series of CDS spreads for a set of major US and European institutions on a pe- riod overlapping the recent financial crisis. We extend the existing methodology of ε-drawdowns to the one of joint ε-drawups, in order to estimate the conditional probabilities of abrupt co-movements among spreads. We correct for randomness and for finite size effects and we find significant prob- ability of joint drawups for certain pairs of CDS. We also find significant probability of trend rein- forcement, i.e. drawups in a given CDS followed by drawups in the same CDS. Finally, we take the matri…
We develop a generalization of the Black-Cox structural model of default risk. The extended model captures uncertainty related to firm's ability to avoid default even if company's liabilities momentarily exceeding its assets. Diffusion in a linear potential with the radiation boundary condition is used to mimic a company's default process. The exact solution of the corresponding Fokker-Planck equation allows for derivation of analytical expressions for the cumulative probability of default and the relevant hazard rate. Obtained closed formulas fit well the historical data on global corporate d…
We consider the pricing of European-style structured credit payoff in a static framework, where the underlying default times are independent given a common factor. A practical application would consist of the pricing of nth-to-default baskets under the Gaussian copula model (GCM). We provide necessary and sufficient conditions so that the corresponding asset prices are martingales and introduce the concept of "break-even" correlation matrix. When no sudden jump-to-default events occur, we show that the perfect replication of these payoffs under the GCM is obtained if and only if the underlying…
Recently, incomplete-market techniques have been used to develop a model applicable to credit default swaps (CDSs) with results obtained that are quite different from those obtained using the market-standard model. This article makes use of the new incomplete-market model to further study CDS hedging and extends the model so that it is capable treating single-name CDS portfolios. Also, a hedge called the vanilla hedge is described, and with it, analytic results are obtained explaining the striking features of the plot of no-arbitrage bounds versus CDS maturity for illiquid CDSs. The valuation …
Credit Spreads' Term Structure: Stochastic Modeling with CIR++ Intensity
LMRS, UMR 6085 CNRS（法国国家科学研究中心第6085号联合研究实验室）；Université de Rouen Normandie（诺曼底鲁昂大学）；LAMME, UMR 8071（法国国家科学研究中心第8071号联合研究实验室）；Université d’Evry（埃夫里大学）；LAGA, UMR 7539 CNRS（法国国家科学研究中心第7539号联合研究实验室）；Université Sorbonne Paris Nord（索邦巴黎北大学）；FBH Associés（FBH合伙公司）
Corporate defaults may be triggered by some major market news or events such as financial crises or collapses of major banks or financial institutions. With a view to develop a more realistic model for credit risk analysis, we introduce a new type of reduced-form intensity-based model that can incorporate the impacts of both observable "trigger" events and economic environment on corporate defaults. The key idea of the model is to augment a Cox process with trigger events. Both single-default and multiple-default cases are considered in this paper. In the former case, a simple expression for t…
The 2008 financial crisis has been attributed to "excessive complexity" of the financial system due to financial innovation. We employ computational complexity theory to make this notion precise. Specifically, we consider the problem of clearing a financial network after a shock. Prior work has shown that when banks can only enter into simple debt contracts with each other, then this problem can be solved in polynomial time. In contrast, if they can also enter into credit default swaps (CDSs), i.e., financial derivative contracts that depend on the default of another bank, a solution may not e…
