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Pricing and Hedging of Portfolio Credit Derivatives with Interacting Default Intensites
, 2007
"... We consider reducedform models for portfolio credit risk with interacting default intensities. In this class of models default intensities are modelled as functions of time and of the default state of the entire portfolio, so that phenomena such as default contagion or counterparty risk can be mode ..."
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Cited by 31 (1 self)
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We consider reducedform models for portfolio credit risk with interacting default intensities. In this class of models default intensities are modelled as functions of time and of the default state of the entire portfolio, so that phenomena such as default contagion or counterparty risk can be modelled explicitly. In the present paper this class of models is analyzed by Markov process techniques. We study in detail the pricing and the hedging of portfoliorelated credit derivatives such as basket default swaps and collaterized debt obligations (CDOs) and discuss the calibration to market data.
Pricing and hedging of credit derivatives via the innovations approach to nonlinear filtering
, 2010
"... Abstract In this paper we propose a new, informationbased approach for modelling the dynamic evolution of a portfolio of credit risky securities. In our setup market prices of traded credit derivatives are given by the solution of a nonlinear filtering problem. The innovations approach to nonlinear ..."
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Cited by 19 (5 self)
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Abstract In this paper we propose a new, informationbased approach for modelling the dynamic evolution of a portfolio of credit risky securities. In our setup market prices of traded credit derivatives are given by the solution of a nonlinear filtering problem. The innovations approach to nonlinear filtering is used to solve this problem and to derive the dynamics of market prices. Moreover, the practical application of the model is discussed: we analyze model calibration, the pricing of exotic credit derivatives and the computation of riskminimizing hedging strategies. The paper closes with a small numerical case study.
Hedging default risks of CDOs in Markovian contagion models. Working paper
, 2008
"... Abstract We describe a hedging strategy of CDO tranches based upon dynamic trading of the corresponding credit default swap index. We rely upon a homogeneous Markovian contagion framework, where only single defaults occur. In our framework, a CDO tranche can be perfectly replicated by dynamically t ..."
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Cited by 17 (4 self)
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Abstract We describe a hedging strategy of CDO tranches based upon dynamic trading of the corresponding credit default swap index. We rely upon a homogeneous Markovian contagion framework, where only single defaults occur. In our framework, a CDO tranche can be perfectly replicated by dynamically trading the credit default swap index and a riskfree asset. Default intensities of the names only depend upon the number of defaults and are calibrated onto an input loss surface. Numerical implementation can be carried out fairly easily thanks to a recombining tree describing the dynamics of the aggregate loss. Both continuous time market and its discrete approximation are complete. The computed credit deltas can be seen as a credit default hedge and may also be used as a benchmark to be compared with the market credit deltas. Though the model is quite simple, it provides some meaningful results which are discussed in detail. We study the robustness of the hedging strategies with respect to recovery rate and examine how input loss distributions drive the credit deltas. Using market inputs, we find that the deltas of the equity tranche are lower than those computed in the standard base correlation framework and relate this to the dynamics of dependence between defaults.
2009, Pricing Credit Derivatives under Incomplete Information: a NonlinearFiltering Approach
"... Abstract This paper considers a general reduced form pricing model for credit derivatives where default intensities are driven by some factor process X. The process X is not directly observable for investors in secondary markets; rather, their information set consists of the default history and of n ..."
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Cited by 17 (5 self)
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Abstract This paper considers a general reduced form pricing model for credit derivatives where default intensities are driven by some factor process X. The process X is not directly observable for investors in secondary markets; rather, their information set consists of the default history and of noisy price observation for traded credit products. In this context the pricing of credit derivatives leads to a challenging nonlinear filtering problem. We provide recursive updating rules for the filter, derive a finite dimensional filter for the case where X follows a finite state Markov chain and propose a novel particle filtering algorithm. A numerical case study illustrates the properties of the proposed algorithms.
UP AND DOWN CREDIT RISK
, 2008
"... This paper discusses the main modeling approaches that have been developed so far for handling portfolio credit derivatives. In particular the so called top, top down and bottom up approaches are considered. We first provide an overview of these approaches. Then we give some mathematical insights to ..."
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Cited by 10 (7 self)
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This paper discusses the main modeling approaches that have been developed so far for handling portfolio credit derivatives. In particular the so called top, top down and bottom up approaches are considered. We first provide an overview of these approaches. Then we give some mathematical insights to the fact that information, namely, the choice of a relevant model filtration, is the major modeling issue. In this regard, we examine the notion of thinning that was recently advocated for the purpose of hedging a multiname derivative by singlename derivatives. We then give a further analysis of the various approaches using simple models, discussing in each case the issue of possibility of hedging. Finally we explain by means of numerical simulations (semistatic hedging experiments) why and when the portfolio loss process may not
A BottomUp Dynamic Model of Portfolio Credit Risk. Part I: Markov Copula Perspective
, 2013
"... We consider a bottomup Markovian copula model of portfolio credit risk where instantaneous contagion is possible in the form of simultaneous defaults. Due to the Markovian copula nature of the model, calibration of marginals and dependence parameters can be performed separately using a twosteps pr ..."
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Cited by 8 (8 self)
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We consider a bottomup Markovian copula model of portfolio credit risk where instantaneous contagion is possible in the form of simultaneous defaults. Due to the Markovian copula nature of the model, calibration of marginals and dependence parameters can be performed separately using a twosteps procedure, much like in a standard static copula setup. In this sense this model solves the bottomup topdown puzzle which the CDO industry had been trying to do for a long time. It can be applied to any dynamic credit issue like consistent valuation and hedging of CDSs, CDOs and counterparty risk on credit portfolios.
Derivatives and Credit Contagion in Interconnected Networks
, 1202
"... 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 macroeconomic conditions, but also by directly triggering each other through contagion. Although credit default swap ..."
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Cited by 7 (0 self)
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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 macroeconomic conditions, 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 system using a stochastic setting, which allows us to exploit limit theorems to exactly solve the contagion dynamics for the entire system. Our analysis shows that, by creating additional contagion channels, CDS can actually lead to greater instability of the entire network in times of economic stress. This is particularly pronounced when CDS are used by banks to expand their loan books (arguing that CDS would offload the additional risks from their balance sheets). Thus, even with complete hedging through CDS, a significant loan book expansion can lead to considerably enhanced probabilities for the occurrence of very large losses and very high default rates in the system. Our approach adds a new dimension to research on credit contagion, and could feed into a rational underpinning of an improved regulatory framework for credit derivatives. 1
About the pricing equations in finance
 ParisPrinceton Lectures on Mathematical Finance
, 2009
"... 1We derive the pricing equation of a general (American or Game) Contingent Claim in the setup of a rather generic Markovian factor process. As an aside we establish the convergence of stable, monotone and consistent approximation schemes to the pricing function (solution to the pricing equation). 2 ..."
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Cited by 7 (4 self)
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1We derive the pricing equation of a general (American or Game) Contingent Claim in the setup of a rather generic Markovian factor process. As an aside we establish the convergence of stable, monotone and consistent approximation schemes to the pricing function (solution to the pricing equation). 21 Model We first introduce a flexible Markovian model X = (X,N) made of a jump diffusionX interacting with a pure jump processN (which in the simplest case reduces to a Markov chain in continuous time). The state space is E = [0, T]×Rd × I with I = {1, · · · , k}, where T> 0 is a finite horizon and d and k denote positive integers. Note that a function u = u(t, x, i) on [0, T] × Rd × I may equivalently be considered as a system u = (ui)i∈I of functions ui = ui(t, x) on [0, T]×Rd. Likewise we denote u(t, x, i, j), or ui,j(t, x) for a function u of (t, x, i, j). Let us be given a stochastic basis (Ω,F,P) on [0, T] endowed with a ddimensional Brownian motion B and an integervalued random measure χ. Our model consists of an Rd × Ivalued Markov càdlàg process X = (X,N) on [0, T] with initial condition (x, i) at time 0, such that the Rdvalued process 3X satisfies: dXt = b(t,Xt, Nt) dt+ σ(t,Xt, Nt) dBt +
A simple dynamic model for pricing and hedging heterogenous CDOs
, 2008
"... We present a simple bottomup dynamic credit model that can be calibrated simultaneously to the market quotes on CDO tranches and individual CDSs constituting the credit portfolio. The model is most suitable for the purpose of evaluating the hedge ratios of CDO tranches with respect to the underlyin ..."
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Cited by 6 (0 self)
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We present a simple bottomup dynamic credit model that can be calibrated simultaneously to the market quotes on CDO tranches and individual CDSs constituting the credit portfolio. The model is most suitable for the purpose of evaluating the hedge ratios of CDO tranches with respect to the underlying credit names. Default intensities of individual assets are modeled as deterministic functions of time and the total number of defaults accumulated in the portfolio. To overcome numerical difficulties, we suggest a semianalytic approximation that is justified by the large number of portfolio members. We calibrate the model to the recent market quotes on CDO tranches and individual CDSs and find the hedge ratios of tranches. Results are compared with those obtained within the static Gaussian Copula model.
Dynamic hedging of portfolio credit derivatives
, 2011
"... As shown by the recent turmoil in credit markets, much remains to be done for the proper risk management of credit derivatives. In particular, the static copulabased models commonly used for pricing portfolio credit derivatives appear to be inappropriate for hedging and risk management. We study he ..."
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Cited by 6 (1 self)
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As shown by the recent turmoil in credit markets, much remains to be done for the proper risk management of credit derivatives. In particular, the static copulabased models commonly used for pricing portfolio credit derivatives appear to be inappropriate for hedging and risk management. We study hedging of index CDO tranches with the underlying index default swap using various portfolio loss models which account for default contagion and spread risk. Numerical results obtained from models calibrated to iTraxx Europe data reveal significant differences in hedge ratios across models and show, unlike what had been previously suggested in the literature by comparing copulabased models, that hedging strategies are subject to substantial model risk. An empirical analysis based on recent market data shows that strategies based on deltahedging of spread movements have poorly performed during the 20072008 subprime crisis, while varianceminimizing hedges led to significantly smaller losses. Our empirical study also reveals that, while sudden large moves do occur in index spreads, these jumps do not necessarily occur on default dates of index constituents, an observation which contradicts the intuition conveyed by some recently proposed credit risk models.