Credit risk tools: an overview

This document presents several Credit Risk tools which have been developed for the Credit Derivatives Risk Management. The models used in this context are suitable for the pricing, sensitivity/scenario analysis and the derivation of risk measures for plain vanilla credit default swaps (CDS), standardized and bespoke collateralized debt obligations (CDO) and, in general, for any credit risk exposed A/L portfolio.\\ In this brief work we compute the market implied probability of default (PD) from market spreads and the theoretical CDS spreads from historical default frequencies. The loss given default (LGD) probability distribution has been constructed for a large pool portfolio of credit obligations exploiting a single-factor gaussian copula with a direct convolution algorithm computed at several default correlation parameters. Theoretical CDO tranche prices have been calculated. We finally design stochastic cash-flow stream model simulations to test fair pricing, compute credit value at risk (CV@R) and to evaluate the one year total future potential exposure (FPE) and derive the value at risk (V@R) for a CDO equity tranche exposure.interest rate swap, spot rate term structure, credit default swap, probability of default, copula function, direct convolution, loss given default, collateralized debt obligation, exposure at default, stochastic cash-flow stream model, value at risk, credit value at risk, future potential exposure, Monte Carlo simulation.

On boosted space-times with cosmological constant and their ultrarelativistic limit

The problem of deriving a shock-wave geometry with cosmological constant by boosting a Schwarzschild-de Sitter (or anti-de Sitter) black hole is re-examined. Unlike previous work in the literature, we deal with the exact Schwarzschild-de Sitter (or anti-de Sitter) metric. By virtue of peculiar cancellations in this exact calculation, where the metric does not depend linearly on the mass parameter, we find a singularity of distributional nature on a null hypersurface, which corresponds to a shock-wave geometry derived in a fully non-perturbative way. The result agrees with previous calculations, where the metric had been linearized in the mass parameter.Comment: 13 pages, Latex file. In the final version, the presentation has been improved, and two references have been adde

Non-commutative Einstein equations and Seiberg-Witten map

The Seiberg--Witten map is a powerful tool in non-commutative field theory, and it has been recently obtained in the literature for gravity itself, to first order in non-commutativity. This paper, relying upon the pure-gravity form of the action functional considered in Ref. 2, studies the expansion to first order of the non-commutative Einstein equations, and whether the Seiberg--Witten map can lead to a solution of such equations when the underlying classical geometry is Schwarzschild.Comment: 6 and 1/2 pages, based on talk prepared for the Friedmann Seminar, May-June 2011. In the final version, the presentation has been improved, including a better notatio

La fine di Orfeo e le matres / nurus Ciconum: tra Virgiio e Ovidio

Virgilio, in georg. IV 520, definisce le artefici dell’uccisione di Orfeo matres Ciconum. Ovidio (met. XI 3), che pure non ne nega la natura speciale di baccanti, le definisce nurus Ciconum e così sottolinea la loro condizione di fanciulle da marito, che hanno subito l’onta del rifiuto da colui al quale intendevano unirsi. Tra una versione e l’altra, cambia la caratterizzazione delle donne punitrici, in Virgilio generiche donne (matrone), in Ovidio nubiles o sponsae. Si tratta, da parte di Ovidio, di un’abile e sottile modifica, che sposta il tono della scena dalla sacralità del resoconto virgiliano al contesto, tutto umano, di una vendetta per un amore non corrisposto. Virgil, at georg. IV 520, defines the murderers of Orpheus as matres Ciconum. Ovid (met. XI 3), which also does not negate their special nature of Bacchantes, calls them nurus Ciconum and so underlines their status as girls of marriageable age, who suffered the shame of rejection by those to whom they intended to join. Between the two versions of what happened, occurs a change concerning to characterizations of women authors of the punishment, which in Virgil are generic women (matronae), while in Ovid become nubiles or sponsae. Ovid, therefore, makes a clever and subtle change, which shifts the tone of the scene, from the sacredness of the story as narrated in Virgil, to a purely human revenge for an unrequited love, which seems to be the keynote of the ovidian tale

Single-tree detection in high-density LiDAR data from UAV-based survey

UAV-based LiDAR survey provides very-high-density point clouds, which involve very rich information about forest detailed structure, allowing for detection of individual trees, as well as demanding high computational load. Single-tree detection is of great interest for forest management and ecology purposes, and the task is relatively well solved for forests made of single or largely dominant species, and trees having a very evident pointed shape in the upper part of the canopy (in particular conifers). Most authors proposed methods based totally or partially on search of local maxima in the canopy, which has poor performance for species that have flat or irregular upper canopy, and for mixed forests, especially where taller trees hide smaller ones. Such considerations apply in particular to Mediterranean hardwood forests. In such context, it is imperative to use the whole volume of the point cloud, however keeping computational load tractable. The authors propose the use of a methodology based on modelling the 3D-shape of the tree, which improves performance w.r.t to maxima-based models. A case study, performed on a hazel grove, is provided to document performance improvement on a relatively simple, but significant, case

The variable X-ray emission of PSR B0943+10

The old pulsar PSR B0943+10 (P=1.1 s, characteristic age tau=5 Myr) is the best example of mode-switching radio pulsar. Its radio emission alternates between a highly organized state with regular drifting subpulses (B mode) and a chaotic emission pattern (Q mode). We present the results of XMM-Newton observations showing that the X-ray properties of PSR B0943+10 depend on its radio state (Hermsen et al. 2013). During the radio fainter state (Q mode) the X-ray flux is more than a factor two larger than during the B-mode and X-ray pulsations with about 50% pulsed fraction are detected. The X-ray emission of PSR B0943+10 in the B-mode is well described by thermal emission with blackbody temperature kT=0.26 keV coming from a small hot spot with luminosity of 7x10^28 erg/s, in good agreement with the prediction of the partially screened gap model, which also explains the properties of the radio emission in this mode. We derived an upper limit of 46% on the X-ray pulsed fraction in the B-mode, consistent with the geometry and viewing angle of PSR B0943+10 inferred from the radio data. The higher flux observed during the Q-mode is consistent with the appearance of an additional component with a power-law spectrum with photon index 2.2. We interpret it as pulsed non-thermal X-rays produced in the star magnetosphere. A small change in the beaming pattern or in the efficiency of acceleration of the particles responsible for the non-thermal emission can explain the reduced flux of this component during the radio B-mode.Comment: Accepted for publication in MNRA

Riemann curvature of a boosted spacetime geometry

The ultrarelativistic boosting procedure had been applied in the literature to map the metric of Schwarzschild-de Sitter spacetime into a metric describing de Sitter spacetime plus a shock-wave singularity located on a null hypersurface. This paper evaluates the Riemann curvature tensor of the boosted Schwarzschild-de Sitter metric by means of numerical calculations, which make it possible to reach the ultrarelativistic regime gradually by letting the boost velocity approach the speed of light. Thus, for the first time in the literature, the singular limit of curvature through Dirac's delta distribution and its derivatives is numerically evaluated for this class of spacetimes. Eventually, the analysis of the Kretschmann invariant and the geodesic equation show that the spacetime possesses a scalar curvature singularity within a 3-sphere and it is possible to define what we here call boosted horizon, a sort of elastic wall where all particles are surprisingly pushed away, as numerical analysis demonstrates. This seems to suggest that boosted geometries are ruled by a sort of antigravity effect since all geodesics seem to refuse to enter the boosted horizon, even though their initial conditions are aimed at driving the particles towards the boosted horizon.Comment: 33 pages, 8 figures. In the new version, a section and new references have been added, and the presentation has been amended and improve