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Credit Risk Models I: Default Correlation in Intensity Models
"... This report analyzes the modelling of default intensities and probabilities in singlefirm reducedform models, and reviews the three main approaches to incorporating default dependencies within the framework of reduced models. The first approach, the conditionally independent defaults (CID), introd ..."
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Cited by 13 (5 self)
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This report analyzes the modelling of default intensities and probabilities in singlefirm reducedform models, and reviews the three main approaches to incorporating default dependencies within the framework of reduced models. The first approach, the conditionally independent defaults (CID), introduces credit risk dependence between firms through the dependence of the firms’ intensity processes on a common set of state variables. Contagion models extend the CID approach to account for the empirical observation of default clustering. There exist periods in which the firms ’ credit risk is increased and in which the majority of the defaults take place. Finally, default dependencies can also be accounted for using copula functions. The copula approach takes as given the marginal default probabilities of the different firms and plugs them into a copula function, which provides the model with the default dependence structure. After a description of copulas, we present two different approaches of using copula functions in intensity models, and discuss the issues of the choice and calibration of the copula function.
Simple model for credit migration and spread curves
, 2004
"... We propose and examine a simple model for credit migration and spread curves of a single firm both under realworld and riskneutral measures. This model is a hybrid of a structural and a reducedform model. Default is triggered either by successive downgradings of the firm or an unpredictable jump ..."
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Cited by 12 (5 self)
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We propose and examine a simple model for credit migration and spread curves of a single firm both under realworld and riskneutral measures. This model is a hybrid of a structural and a reducedform model. Default is triggered either by successive downgradings of the firm or an unpredictable jump of the state process. The default time is accordingly decomposed into predictable and totally inaccessible part.
Modelling Bonds and Credit Default Swaps Using a Structural Model with Contagion
 Affine Point Processes and Portfolio Credit Risk. Siam J
"... This paper develops a twodimensional structural framework for valuing credit default swaps and corporate bonds in the presence of default contagion. Modelling the values of related firms as correlated geometric Brownian motions with exponential default barriers, analytical formulae are obtained for ..."
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Cited by 6 (1 self)
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This paper develops a twodimensional structural framework for valuing credit default swaps and corporate bonds in the presence of default contagion. Modelling the values of related firms as correlated geometric Brownian motions with exponential default barriers, analytical formulae are obtained for both credit default swap spreads and corporate bond yields. The credit dependence structure is influenced by both a longerterm correlation structure as well as by the possibility of default contagion. In this way, the model is able to generate a diverse range of shapes for the term structure of credit spreads using realistic values for input parameters. 1
2008: Credit derivatives and risk aversion
 In Advances in Econometrics
"... We discuss the valuation of credit derivatives in extreme regimes such as when the timetomaturity is short, or when payoff is contingent upon a large number of defaults, as with senior tranches of collateralized debt obligations. In these cases, risk aversion may play an important role, especially ..."
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Cited by 5 (1 self)
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We discuss the valuation of credit derivatives in extreme regimes such as when the timetomaturity is short, or when payoff is contingent upon a large number of defaults, as with senior tranches of collateralized debt obligations. In these cases, risk aversion may play an important role, especially when there is little liquidity, and utility indifference valuation may apply. Specifically, we analyze how shortterm yield spreads from defaultable bonds in a structural model may be raised due to investor risk aversion. 1
Exploring the relationship between credit spreads and default probabilities. Working Paper No
, 2004
"... The views in this paper are those of the author and do not necessarily reflect those of the Bank of England. The author is very grateful to Merxe Tudela for provision of default probability data generated by the Bank of England’s Merton model and for other helpful comments. Many thanks also to Peter ..."
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Cited by 5 (0 self)
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The views in this paper are those of the author and do not necessarily reflect those of the Bank of England. The author is very grateful to Merxe Tudela for provision of default probability data generated by the Bank of England’s Merton model and for other helpful comments. Many thanks also to Peter Brierley, Ian Marsh, Kamakshya Trivedi, John Whitley and Garry Young for fruitful discussions and advice, and to seminar participants at the Bank of England and two anonymous referees for useful comments. Copies of working papers may be obtained from Publications Group, Bank of England,
Modelling Basket Credit Default Swaps with Default Contagion
 J. Credit Risk
, 2007
"... This work is kindly supported by Nomura and the EPSRC. 1 Executive Summary The specification of a realistic dependence structure is key to the pricing of multiname credit derivatives. We value small k thtodefault CDS baskets in the presence of asset correlation and default contagion. Using a firs ..."
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Cited by 4 (2 self)
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This work is kindly supported by Nomura and the EPSRC. 1 Executive Summary The specification of a realistic dependence structure is key to the pricing of multiname credit derivatives. We value small k thtodefault CDS baskets in the presence of asset correlation and default contagion. Using a firstpassage framework, firm values are modeled as correlated geometric Brownian motions with exponential default thresholds. Idiosyncratic links between companies are incorporated through a contagion mechanism whereby a default event leads to jumps in volatility at related entities. Our framework allows for default causality and is extremely flexible, enabling us to evaluate the spread impact of firm value correlations and credit contagion for symmetric and asymmetric baskets.
Affine Markov chain models of multifirm credit migration
 J. of credit risk
, 2007
"... This paper introduces and explores variations on a natural extension of the intensity based doubly stochastic framework for credit default. The essential addition proposed here is to introduce a Markov chain for the “credit rating ” of each firm, which are independent conditioned on a stochastic tim ..."
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This paper introduces and explores variations on a natural extension of the intensity based doubly stochastic framework for credit default. The essential addition proposed here is to introduce a Markov chain for the “credit rating ” of each firm, which are independent conditioned on a stochastic time change, or equivalently a stochastic intensity. The stochastic time change is then combined with other stochastic factors, here the interest rate and the recovery rate, into a multidimensional affine process. The resulting general framework has the computational effectiveness of the intensity based models. This paper aims to illustrate the potential of the general framework by exploring a minimal implementation which is still capable of combining stochastic interest rates, stochastic recovery rates and the multifirm default process. Already within this minimal version we see very good reproduction of essential features of credit spread curves, default correlations and multifirm default distributions. Increased flexibility can also be achieved with a number of mathematical extensions of the basic framework. In a companion paper, [Hurd and Kuznetsov (2006)] we show how the same framework extends to large scale basket credit derivatives, particularly CDOs (collateralized debt obligations). Key words: Credit risk, stochastic intensity, credit migration, stochastic recovery, default correlation, credit spread 2 1
A Structural Model with Unobserved Default Boundary
, 2006
"... We consider a firmvalue model similar to the one proposed by Black and Cox (1976) where ..."
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Cited by 3 (2 self)
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We consider a firmvalue model similar to the one proposed by Black and Cox (1976) where
Dependence of defaults and recoveries in structural credit risk models
"... 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 distributi ..."
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Cited by 3 (1 self)
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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 default probability. This relation depends on a single parameter only. In Monte Carlo simulations we find that the same functional dependence also holds for jump–diffusion and GARCH processes. We discuss how to incorporate this structural recovery rate into reduced–form models, in order to restore essential structural information which is usually neglected in the reduced–form approach.
An Infinite Factor Model for Credit Risk
, 2004
"... The defaultable term structure is modeled using stochastic differential equations in Hilbert spaces. This leads to an infinite dimensional model, which is free of arbitrage under a certain drift condition. Furthermore, the model is extended to incorporate ratings based on a Markov chain. ..."
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Cited by 2 (2 self)
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The defaultable term structure is modeled using stochastic differential equations in Hilbert spaces. This leads to an infinite dimensional model, which is free of arbitrage under a certain drift condition. Furthermore, the model is extended to incorporate ratings based on a Markov chain.