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Defaultable bonds with an infinite number of Lévy factors

Jacek Jakubowski, Mariusz Niewęgłowski (2010)

Applicationes Mathematicae

A market with defaultable bonds where the bond dynamics is in a Heath-Jarrow-Morton setting and the forward rates are driven by an infinite number of Lévy factors is considered. The setting includes rating migrations driven by a Markov chain. All basic types of recovery are investigated. We formulate necessary and sufficient conditions (generalized HJM conditions) under which the market is arbitrage-free. Connections with consistency conditions are discussed.

Dynamic term structure modelling with default and mortality risk: new results on existence and monotonicity

Thorsten Schmidt, Stefan Tappe (2015)

Banach Center Publications

This paper considers dynamic term structure models like the ones appearing in portfolio credit risk modelling or life insurance. We study general forward rate curves driven by infinitely many Brownian motions and an integer-valued random measure, generalizing existing approaches in the literature. A precise characterization of absence of arbitrage in such markets is given in terms of a suitable criterion for no asymptotic free lunch (NAFL). From this, we obtain drift conditions which are equivalent...

Generalized CreditRisk+ model and applications

Jakub Szotek (2015)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

In the paper we give a mathematical overview of the CreditRisk+ model as a tool used for calculating credit risk in a portfolio of debts and suggest some other applications of the same method of analysis.

On Conditional Value at Risk (CoVaR) for tail-dependent copulas

Piotr Jaworski (2017)

Dependence Modeling

The paper deals with Conditional Value at Risk (CoVaR) for copulas with nontrivial tail dependence. We show that both in the standard and the modified settings, the tail dependence function determines the limiting properties of CoVaR as the conditioning event becomes more extreme. The results are illustrated with examples using the extreme value, conic and truncation invariant families of bivariate tail-dependent copulas.

Optional splitting formula in a progressively enlarged filtration

Shiqi Song (2014)

ESAIM: Probability and Statistics

Let 𝔽 F be a filtration andτbe a random time. Let 𝔾 G be the progressive enlargement of 𝔽 F withτ. We study the following formula, called the optional splitting formula: For any 𝔾 G-optional processY, there exists an 𝔽 F-optional processY′ and a function Y′′ defined on [0,∞] × (ℝ+ × Ω) being [ 0 , ] 𝒪 ( 𝔽 ) ℬ[0,∞]⊗x1d4aa;(F) measurable, such that Y = Y ' 1 [ 0 , τ ) + Y ' ' ( τ ) 1 [ τ , ) . Y=Y′1[0,τ)+Y′′(τ)1[τ,∞). (This formula can also be formulated for multiple random timesτ1,...,τk). We are interested in this formula because of its fundamental role in many...

Pricing bonds and CDS in the model with rating migration induced by a Cox process

Jacek Jakubowski, Mariusz Niewęgłowski (2008)

Banach Center Publications

We investigate the properties of a rating migration process assuming that it is given by subordination of a discrete time Markov chain and a Cox process. The problem of pricing of defaultable bonds with fractional recovery of par value with rating migration and credit default swaps is considered. As an example of applications of our results, we give an explicit solution to the pricing problem in a model with short rate and intensity processes given by the solution of a two-dimensional Ornstein-Uhlenbeck...

Quantifying the impact of different copulas in a generalized CreditRisk + framework An empirical study

Kevin Jakob, Matthias Fischer (2014)

Dependence Modeling

Without any doubt, credit risk is one of the most important risk types in the classical banking industry. Consequently, banks are required by supervisory audits to allocate economic capital to cover unexpected future credit losses. Typically, the amount of economical capital is determined with a credit portfolio model, e.g. using the popular CreditRisk+ framework (1997) or one of its recent generalizations (e.g. [8] or [15]). Relying on specific distributional assumptions, the credit loss distribution...

Small perturbations with large effects on value-at-risk

Manuel L. Esquível, Luís Dimas, João Tiago Mexia, Philippe Didier (2013)

Discussiones Mathematicae Probability and Statistics

We show that in the delta-normal model there exist perturbations of the Gaussian multivariate distribution of the returns of a portfolio such that the initial marginal distributions of the returns are statistically undistinguishable from the perturbed ones and such that the perturbed V@R is close to the worst possible V@R which, under some reasonable assumptions, is the sum of the V@Rs of each of the portfolio assets.

The pricing of credit risky securities under stochastic interest rate model with default correlation

Anjiao Wang, Zhong Xing Ye (2013)

Applications of Mathematics

In this paper, we study the pricing of credit risky securities under a three-firms contagion model. The interacting default intensities not only depend on the defaults of other firms in the system, but also depend on the default-free interest rate which follows jump diffusion stochastic differential equation, which extends the previous three-firms models (see R. A. Jarrow and F. Yu (2001), S. Y. Leung and Y. K. Kwok (2005), A. Wang and Z. Ye (2011)). By using the method of change of measure and...

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