June 25-26-27 2008, Evry, (France)
Université d'Evry Val d´Essonne
Laboratoire d'Analyse et Probabilités
Département de Mathématiques
Abstracts:
- Y. Braouzec: "How the corporate liquidity process affects the value of the firm", joint work with C. A. Lehalle. (slides)
- J. Brun: "Credit volatility - options and beyond". (slides)
In a highly volatile credit market, credit spread volatility products allow (i) to hedge against spread moves, and (ii) to take positions on the volatility itself. The credit indices (iTraxx and CDX) being the most liquid underlying credit assets, we will focus on the index options, extending the framework developped by Schonbucher on the single-name options. We then introduce models that attempt to naturally reproduce the Black volatility smile and/or term-structure ; we finish by describing exotic volatility products that require such models.
Plan:
* CDS options, and Black's formula
* Credit index options
- payoff specificities
- back to Black
- models without smile ?
* trading strategies involving index options
* new products on credit volatility
- L.Campi: "Kyle-Back equilibrium model with default and initial information asymmetry". (slides)
In this talk, we will present a Kyle-Back equilibrium model, where a risk-neutral insider has an initial additional information about the default time. We will characterize and construct explicitly an equilibrium via Jeulin's decomposition in a Brownian filtration enlarged with the first hitting time of the default barrier. Even in this framework, we will see that an incospicuous insider's trade theorem holds true. Moreover, this model is also able to describe a transition from reduced to structural credit risk models via insider's different behaviours. We will compare the original Kyle-Back's model and our variant including credit risk.
- R. Carmona: "Monte Carlo Computation of Small Loss
Probabilities", joint work with S. Crépey. (slides)
- U. Cetin: "Insider trading in credit markets with dynamic information asymmetry", joint with L. Campi and A. Danilova. (slides)
We study an equilibrium model for a defaultable bond in the setting of Back. The market consists of noise trader, an insider and a risk-neutral market maker. Under the assumption that the insider observes the firm value continuously in time we study the optimal strategies for the insider and the optimal pricing rules for the market maker. We show that there exists an equilibrium where the insider's trades are inconspicuous. In this equilibrium the insider drives the total demand to a certain level at the default time. In particular the nature of the modelling changes from reduced form to structural in the equilibrium.
- R. Cont: "Non-parametric estimation of default intensities from CDO spreads:
solution of an ill-posed problem via intensity control", joint work with Andreea MINCA (Ecole Polytechnique).(slides)
- E. Eberlein: "Advanced credit portfolio modeling and CDO pricing", joint work with R. Frey and
E. A. von Hammerstein, published in Mathematics - Key Technology for the Future. W. Jäger, H.-J. Krebs (eds.),
Springer Verlag (2008), 253-280, 2008. (slides)
Modeling dependence is a key issue when one derives the loss distribution of a portfolio of credit instruments. We extend the factor model approach of Vasicek by using more sophisticated distributions for the factors. Completely different distributions from the class of generalized hyperbolic distributions and their limits can be chosen for the systematic and the idiosyncratic factor in this approach. As a result an almost perfect fit to market quotes of DJ iTraxx Europe standard tranches is achieved. The correlation structure remains flat over all CDO tranches and maturities. No base correlation framework is needed.
References:
E. Eberlein, R. Frey, and E. A. v. Hammerstein: Advanced credit portfolio modeling and CDO pricing. To appear in: H.-J. Krebs (Ed.), Mathematics: Key Technology for the Future, Springer Verlag (2008)
E. Eberlein, E. A. v. Hammerstein: Generalized hyperbolic and inverse Gaussian distributions: limiting cases and approximation of processes. In: R. C. Dalang, M. Dozzi, and F. Russo (Eds.), Seminar on Stochastic Analysis, Random Fields and Applications IV, Progress in Probability 58, Birkhauser Verlag (2004) 221-264
E. Eberlein: Application of generalized hyperbolic Levy motions to finance. In: O. E. Barndorff-Nielsen, T. Mikosch, and S. Resnick (Eds.), Levy Processes: Theory and Applications, Birkhauser Verlag (2001) 319-337
E. Eberlein, U. Keller: Hyperbolic distributions in finance. Bernoulli 1 (1995) 281-299
- N. El Karoui: "Modelling of Successive Defaults Events
with density", joint work with M. Jeanblanc and Y. Jiao. (slides)
- R. Frey: "Dynamic Hedging of Synthetic CDO Tranches
with Spread Risk and Default Contagion". (slides)
We study the hedging of synthetic CDO tranches in a dynamic portfolio credit risk model which incorporates spread risk and default contagion. The model is constructed and studied via Markov-chain techniques. We discuss the immunization of a CDO tranche against spread- and event risk in the Markov-chain model and compare the results with hedge ratios obtained in the standard Gauss copula model. Moreover, we derive model-based dynamic hedging strategies using the concept of risk minimization. Numerical experiments are used to illustrate some of the properties of the risk-minimizing hedging strategies.
- A. Herbertsson: "Default contagion in large homogeneous portfolios". (slides)
- C. Jessen: "Constant Proportion Debt Obligations", joint work with R. Cont. (slides)
- Y.H. Kan: "Dynamic hedging of portfolio credit derivatives", joint work with R. Cont. (slides)
- T. Kokholm: "Sato Processes in Default Modelling", joint work with E. Nicolato. (slides)
- J. P. Laurent: "Comparison results for exchangeable credit risk
portfolios", joint work with Areski Cousin. (slides)
This paper is dedicated to the risk analysis of credit portfolios. Assuming that default indicators form an exchangeable sequence of Bernoulli random variables and as a consequence of de Finetti's theorem, default indicators are Binomial mixtures. We can characterize the supermodular order between two exchangeable Bernoulli random vectors in terms of the convex ordering of their corresponding mixture distributions. Thus we can proceed to some comparisons between stoploss premiums, CDO tranche premiums and convex risk measures on aggregate losses. This methodology provides a unified analysis of dependence for a number of CDO pricing models based on factor copulas, multivariate Poisson and structural approaches.
- P. Del Moral: "Particle rare event simulation". (slides)
- F. Patras: "A mixed dynamic model for multiname credit derivatives", joint work with Pierre Cohort. (slides)
It is well-known that, in spite of all their successes in the pricing of CDO tranches, copula models fail to capture the dynamics of credit spreads. There have been various attempts to build new generations models able to overcome these insufficiencies. Whether bottom-up or top-down, they usually depart from the Gaussian copula picture, both on the technical and the modelling side. One of their common shortcomings is the lack of transparency on the financial hypothesis underlying the chosen mathematical model. We introduce a new approach to the subject that combines the standard copula approach with dynamic features. The numerical performance and enhancement potential of the new class of models will be discussed in detail.
- W. Runggaldier: "Credit risk and incomplete information: linear filtering and EM parameter estimation", joint work with Claudio
Fontana.(slides)
We consider a reduced-form credit risk model where default intensity and interest rate are linear functions of a not fully observable Markovian factor process. Our main goal is to determine arbitrage-free prices of OTC products based on and in line with information coming from the financial market, in particular yields and credit spreads. In our context this can be accomplished via a linear filtering approach, formulated under a martingale/pricing measure and coupled with an EM-algorithm for parameter estimation in lieu of the more traditional calibration. Shifting to a formulation under the physical measure, a further goal is to determine quantities related to risk management, such as default probabilities, based on information deriving from both within and outside the financial market, in the latter case the rating score.
- A. Sbuelz: "Systematic Equity-Based Credit Risk: A CEV
Model With Jump to Default", joint work with Luciano Campi and
Simon Polbennikov. (slides)
We use equity as the traded primitive for a detailed analysis of systematic default risk. Default is parsimoniously represented by equity value hitting the zero barrier so that, unlike in reduced-form models, the explicit linkage to the firm's capital structure is preserved, but, unlike in structural models, restrictive assumptions on the structure are avoided. Default risk is either jump-like or diffusive. The equity price can jump to default: In line with recent empirical evidence on the jump-to-default risk price, we highlight how reasonable choices of the pricing kernel can imply remarkable differences in the equity-price-dependent status between the objective default intensity and the risk-neutral intensity. As equity returns experience negative diffusive shocks, their CEV-type local variance increases and boosts the objective and risk-neutral probabilities of diffusive default. A parsimonious version of our general model simultaneously enables analytical credit-risk management and analytical pricing of credit-sensitive instruments. Easy cross-asset hedging ensues.
- T. Schmidt: "Pricing and Hedging of Credit Derivatives via Nonlinear
Filtering", joint work with R. Frey and A. Gabih. (slides)
In this work a new, information-based approach for modelling the dynamic evolution of a portfolio of credit risky securities is proposed. In this context market prices of liquidly traded derivatives are given by the solution of a nonlinear filtering problem. This problem is solved via the innovations approach to nonlinear filtering. Moreover, we derive the ensuing asset price dynamics and compute risk-minimizing hedging strategies. In the last part we discuss a numerical approach - based on particle filtering - to some of the arising filtering problems.
- U. Schmock: "A Generalization of Panjer's Recursion and Numerically Stable Risk
Aggregation", joint work with S. Gerhold and R. Warnung. (slides)
Portfolio credit risk models as well as models for operational risk can often be treated analogously to the collective risk model coming from insurance. Applying the classical Panjer recursion in the collective risk model can lead to numerical instabilities, for instance if the claim number distribution is extended negative binomial or extended logarithmic. We present a generalization of Panjer's recursion that leads to numerically stable algorithms. The algorithm can be applied to the collective risk model, where the claim number follows, for example, a Poisson distribution mixed over a tempered stable distribution with exponent in (0,1). DePril's recursion can be generalized in the same vein. Time permitting, we also mention an analogue of our method for the collective model with a severity distribution having mixed support.
- T. Vargiolu: "Optimal prepayment rule for mortgage-backed securities", joint work with G. De Rossi. (slides)