FAILURE OF SADDLE-POINT METHOD IN THE PRESENCE OF DOUBLE DEFAULTS

We show that the saddle-point approximation method to quantify the impact of undiversi?ed idiosyncratic risk in a credit portfolio is inappropriate in the presence of double default effects. Speci?cally, we prove that there does not exist an equivalent formula to the granularity adjustment, that accounts for guarantees, in case of the extended single-factor CreditRisk+ model. Moreover, in case of the model underlying the double default treatment within the internal ratings based (IRB) approach of Basel II, the saddle-point equivalent to the GA is too complex and involved to be competitive to a standard Monte Carlo approach.analytical approximation, Basel II, double default, granularity adjustment, IRB approach, saddle- point approximation

An asymptotic expansion for a Black–Scholes type model

AbstractWe consider the Black–Scholes model where we add a perturbation term ∑iεiσi to the model with diffusion coefficient σ0(t). Then we derive an asymptotic expansion for the expected value of an European call option at time t. This is done by applying methods of Malliavin calculus. Borel summability of the derived asymptotic expansion is proven

Treatment of Double Default Effects within the Granularity Adjustment for Basel II

Within the Internal Ratings-Based (IRB) approach of Basel II it is assumed that idiosyncratic risk has been fully diversi?ed away. The impact of undiversi?ed idiosyncratic risk on portfolio Value-at-Risk can be quanti?ed via a granularity adjustment (GA). We provide an analytic formula for the GA in an extended single- factor CreditRisk+ setting incorporating double default e?ects. It accounts for guarantees and their e?ect of reducing credit risk in the portfolio. Our general GA very well suits for application under Pillar 2 of Basel II as the data inputs are drawn from quantities already required for the calculation of IRB capital charges.analytic approximation, Basel II, counterparty risk, double default, granularity adjustment, IRB approach, securitization

Funding Liquidity, Debt Tenor Structure, and Creditor's Belief: An Exogenous Dynamic Debt Run Model

We propose a unified structural credit risk model incorporating both insolvency and illiquidity risks, in order to investigate how a firm's default probability depends on the liquidity risk associated with its financing structure. We assume the firm finances its risky assets by mainly issuing short- and long-term debt. Short-term debt can have either a discrete or a more realistic staggered tenor structure. At rollover dates of short-term debt, creditors face a dynamic coordination problem. We show that a unique threshold strategy (i.e., a debt run barrier) exists for short-term creditors to decide when to withdraw their funding, and this strategy is closely related to the solution of a non-standard optimal stopping time problem with control constraints. We decompose the total credit risk into an insolvency component and an illiquidity component based on such an endogenous debt run barrier together with an exogenous insolvency barrier.Comment: 36 pages, 9 figures. The article was previously circulated under the title A Continuous Time Structural Model for Insolvency, Recovery, and Rollover Risks in Mathematics and Financial Economics, 201

Rollover risk and credit risk under time-varying margin

© 2016 Informa UK Limited, trading as Taylor & Francis Group. For a firm financed by a mixture of collateralized (short-term) debt and uncollateralized (long-term) debt, we show that fluctuations in margin requirements, reflecting funding liquidity shocks, lead to increasing the firm’s default risk and credit spreads. The severity with which a firm is hit by increasing margin requirements highly depends on both its financing structure and debt maturity structure. Our results imply that an additional premium should be added when evaluating debt in order to account for rollover risks, especially for short-matured bonds. In terms of policy implications, our results strongly indicate that regulators should intervene fast to curtail margins in crisis periods and maintain a reasonably low margin level in order to effectively prevent creditors’ run on debt

Improved Robust Price Bounds for Multi-Asset Derivatives under Market-Implied Dependence Information

We show how inter-asset dependence information derived from observed market prices of liquidly traded options can lead to improved model-free price bounds for multi-asset derivatives. Depending on the type of the observed liquidly traded option, we either extract correlation information or we derive restrictions on the set of admissible copulas that capture the inter-asset dependencies. To compute the resultant price bounds for some multi-asset options of interest, we apply a modified martingale optimal transport approach. In particular, we derive an adjusted pricing-hedging duality. Several examples based on simulated and real market data illustrate the improvement of the obtained price bounds and thus provide evidence for the relevance and tractability of our approach

Cox-Type regression and transformation models with change-points based on covariate thresholds

In this thesis we consider Cox-type regression models and transformation models for right-censored survival time data with bent-line change-points in the underlying regression functions according to covariate thresholds. We establish the usual asymptotic properties of the estimates such as √(n) consistency and asymptotic normality. Furthermore, we applied the Cox regression model with change-points to different data sets.Die Dissertation beinhaltet Cox Regressionsmodelle und ein lineares Transformationsmodell für rechtszensierte Lebensdauerdaten mit Change-Points in der zugrunde liegenden Regressionsfunktion. Wir zeigen die üblichen asymptotischen Eigenschaften der Schätzer wie √(n) Konsistenz und asymptotische Normalität. Weiterhin wenden wir das Cox Regressionsmodell mit Change-Points auf verschiedene Datensätze an

Numerical equivalence defined on Chow groups of Noetherian local rings

In the present paper, we define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings, which is an analogue of that on smooth projective varieties. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensional ${\Bbb Q}$-vector space, as in the case of smooth projective varieties. Numerical equivalence on local rings is deeply related to that on smooth projective varieties. For example, if Grothendieck's standard conjectures are true, then a vanishing of Chow group (of local rings) modulo numerical equivalence can be proven. Using the theory of numerical equivalence, the notion of numerically Roberts rings is defined. It is proved that a Cohen-Macaulay local ring of positive characteristic is a numerically Roberts ring if and only if the Hilbert-Kunz multiplicity of a maximal primary ideal of finite projective dimension is always equal to its colength. Numerically Roberts rings satisfy the vanishing property of intersection multiplicities. We shall prove another special case of the vanishing of intersection multiplicities using a vanishing of localized Chern characters.Comment: final version, 45 pages, to appear in Invent. Mat

Noetherian approximation of algebraic spaces and stacks

We show that every scheme/algebraic space/stack that is quasi-compact with quasi-finite diagonal can be approximated by a noetherian scheme/algebraic space/stack. More generally, we show that any stack which is etale-locally a global quotient stack can be approximated. Examples of applications are generalizations of Chevalley's, Serre's and Zariski's theorems and Chow's lemma to the non-noetherian setting. We also show that every quasi-compact algebraic stack with quasi-finite diagonal has a finite generically flat cover by a scheme.Comment: 39 pages; complete overhaul of paper; generalized results and simplified proofs (no groupoid-calculations); added more applications and appendices with standard results on constructible properties and limits for stacks; generalized Thm C (no finite presentation hypothesis); some minor changes in 2,1-2.8, 8.2, 8.8 and 8.9; final versio