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.

Multidimensional Black-Scholes options

In this article we propose an extension of the classical Black-Scholes option in a multidimensional setup. The underlying financial asset is a basket of equity stocks on which a general European type option pay$-$off is considered. Using the distributional Fourier transform, we derive a general formal solution and provide a sufficient condition to construct the former explicitly in a fairly rich set of functions. Finally, we develop two derivative options, which are given in closed$-$form: the first option can be expressed as a linear combination of the classical call/put options, while the second one is a new option with multidimensional underlying, nameley a $\chi^2-$option

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

Multidimensional Black-Scholes options

In this article we propose an extension of the classical Black-Scholes option in a multidimensional setup. The underlying financial asset is a basket of equity stocks on which a general European type option pay$-$off is considered. Using the distributional Fourier transform, we derive a general formal solution and provide a sufficient condition to construct the former explicitly in a fairly rich set of functions. Finally, we develop two derivative options, which are given in closed$-$form: the first option can be expressed as a linear combination of the classical call/put options, while the second one is a new option with multidimensional underlying, nameley a $\chi^2-$option

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

Credit risk tools, (numerical methods for finance, university of Limerick 2011).

In this work, we solve a risk measurement problem, which involves both credit and market risk. Specifically, We deal with the problem of pricing a synthetic CDO tranche and with the assessment of the evolution behavior of value of the net income resulting from the exposure to a single credit derivative of this sort. We cope with the pricing problem by constructing algorithms capable of computing the key variables. The second problem is solved via Monte Carlo simulation. The calculations, which constitute the main input of the simulation engine, can be easily implemented since they only result in the operations of matrix inversion and numerical integration. The flexibility of the risk evaluation method, which has been achieved through stochastic simulation, allows the system to be easily escalated and extended to a collection of basket credit derivatives

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

Multiple hypothesis testing of market risk forecasting models

Extending previous risk model backtesting literature, we construct multiple hypothesis testing (MHT) with the stationary bootstrap. We conduct multiple tests which control for the generalized confidence level and employ the bootstrap MHT to design multiple comparison testing. We consider absolute and relative predictive ability to test a range of competing risk models, focusing on Value-at-Risk (VaR) and Expected Shortfall (ExS). In devising the test for the absolute predictive ability, we take the route of recent literature and construct balanced simultaneous confidence sets that control for the generalized family-wise error rate, which is the joint probability of rejecting true hypotheses. We implement a step-down method which increases the power of the MHT in isolating false discoveries. In testing for the ExS model predictive ability, we design a new simple test to draw inference about recursive model forecasting capability. In the second suite of statistical testing, we develop a novel device for measuring the relative predictive ability in the bootstrap MHT framework. The device, we coin multiple comparison mapping, provides a statistically robust instrument designed to answer the question: ''which model is the best model?''

Filtering and likelihood estimation of latent factor jump-diffusions with an application to stochastic volatility models

In this article we use a partial integral-differential approach to construct and extend a non-linear filter to include jump components in the system state. We employ the enhanced filter to estimate the latent state of multivariate parametric jump-diffusions. The devised procedure is flexible and can be applied to non-affine diffusions as well as to state dependent jump intensities and jump size distributions. The particular design of the system state can also provide an estimate of the jump times and sizes. With the same approch by which the filter has been devised, we implement an approximate likelihood for the parameter estimation of models of the jump-diffusion class. In the development of the estimation function, we take particular care in designing a simplified algorithm for computing. The likelihood function is then characterised in the application to stochastic volatility models with jumps. In the empirical section we validate the proposed approach via Monte Carlo experiments. We deal with the volatility as an intrinsic latent factor, which is partially observable through the integrated variance, a new system state component that is introduced to increase the filtered information content, allowing a closer tracking of the latent volatility factor. Further, we analyse the structure of the measurement error, particularly in relation to the presence of jumps in the system. In connection to this, we detect and address an issue arising in the update equation, improving the system state estimate