Absolutely Continuous Compensators

We give sufficient conditions on the underlying filtration such that all totally inaccessible stopping times have compensators which are absolutely continuous. If a semimartingale, strong Markov process X has a representation as a solution of a stochastic differential equation driven by a Wiener process, Lebesgue measure, and a Poisson random measure, then all compensators of totally inaccessible stopping times are absolutely continuous with respect to the minimal filtration generated by X. However Cinlar and Jacod have shown that all semimartingale strong Markov processes, up to a change of time and space, have such a representation

Filtration shrinkage, strict local martingales and the F\"{o}llmer measure

When a strict local martingale is projected onto a subfiltration to which it is not adapted, the local martingale property may be lost, and the finite variation part of the projection may have singular paths. This phenomenon has consequences for arbitrage theory in mathematical finance. In this paper it is shown that the loss of the local martingale property is related to a measure extension problem for the associated F\"{o}llmer measure. When a solution exists, the finite variation part of the projection can be interpreted as the compensator, under the extended measure, of the explosion time of the original local martingale. In a topological setting, this leads to intuitive conditions under which its paths are singular. The measure extension problem is then solved in a Brownian framework, allowing an explicit treatment of several interesting examples.Comment: Published in at http://dx.doi.org/10.1214/13-AAP961 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

On absolutely continuous compensators and nonlinear filtering equations in default risk models

We discuss the pricing of defaultable assets in an incomplete information model where the default time is given by a first hitting time of an unobservable process. We show that in a fairly general Markov setting, the indicator function of the default has an absolutely continuous compensator. Given this compensator we then discuss the optional projection of a class of semimartingales onto the filtration generated by the observation process and the default indicator process. Available formulas for the pricing of defaultable assets are analyzed in this setting and some alternative formulas are suggested

Structural Model For Credit Default In One And Higher Dimensions

In this thesis, we provide a new structural model for default of a single name which is an extension in several directions of Merton's seminal work [41] and also propose a new hierarchical model in higher dimensions in a heterogeneous setting. Our new model takes advantage of the fact that currently much more data is readily available about the equity (stock) markets, and through our analysis, can be translated to the much less transparent credit markets. We show how this can be used to provide volatilities for the default indices in structural models for these same stocks. More importantly, we use the equity data to obtain an implied probability distribution for the firms' liabilities, a quantity that is only reported quarterly, and often with questionable reliability. This completes the structural model for a single firm by specifying (probabilistically) the absorbing default barrier. In particular, we can then obtain the default probability of this firm and capture its Credit Default Swap(CDS) spreads. For several companies selected from different industry sectors, the values that our model obtain are in good agreement with the credit market data. Furthermore, we are able to extend this approach to higher dimensional models (e.g., with 125 firms) where the correlations among the firms are essential. Specifically, we use hierarchical models for which each firm’s default boundary a linear combination of a systematic factor (e.g, the Dow Jones Industrial Average) and an idiosyncratic factor, with firm-to-firm correlations obtained through their correlations with the systemic factor. Once again the parameters for these high dimensional structural models are obtained from equity data and the resulting values for the tranche spreads for the CDX: NAIG Series 17 Collateralized Debt Obligations (CDO) compare favorably with actual market data. In the course of this work we also provide results for the probabilistic inverse first passage problem for a Brownian motion default index: given a default probability, find the probability distribution for linear default barriers (equivalently initial distributions) that reproduce the given default probability

Modeling Multi-name default Correlations

This thesis focuses on the study of credit default dependence and related mathematical and computational issues.Firstly, we derive an integral expression of the joint survival probability for the 2D first-passage-time model with default index being correlated Brownian motion and then apply it to give an alternative derivation (PDE approach) of the classical analytical formula of the default time density distribution which was first derived by Iyengar (Probabilistic approach). Furthermore, we prove that for this model both the coefficients of lower and upper tail dependence are zero.Secondly, we create a new model, the crisis model, which is a generalization of the stress event model. In the study of this model, we provide a novel identification of a set of independence conditions of defaults which enables us to derive a series expansion for the unconditional loss of a portfolio. Contrary to most bottom-up approach dynamic models, the distribution of the independence condition in the crisis model has a closed form expression which speeds up computations. We discover that by using a series expansion the loss distribution of a portfolio under the stress event model, which is a special case of the crisis model, can be computed accurately and extremely efficient. Furthermore, the computational cost for additional common factors to the stress event model is mild. This allows more flexibility for calibrations and opens up the possibility to study the multi-factor default dependence of a portfolio via a bottom-up approach. We demonstrate the effectiveness of our approach by calibrating it to investment grade CDS index tranches

Hierarchical Structural Models of Portfolio Credit Risk

In this thesis, we will study hierarchical structural models of portfolio credit defaults that incorporate cyclical dependence and contagion to capture market phenomena such as multi-humped loss distributions. We will use both analytical methods and Monte Carlo simulations in our study. Some of these new models will be calibrated to standard market models to illustrate their effectiveness in pricing single-name CDS’s and CDO tranches simultaneously

Multivariate First-Passage Models in Credit Risk

This thesis deals with credit risk modeling and related mathematical issues. In particular we study first-passage models for credit risk, where obligors default upon first passage of a ``credit quality" process to zero. The first passage problem for correlated Brownian motion is a mathematical structure which arises quite naturally in such models, in particular the seminal multivariate Black-Cox model. In general this problem is analytically intractable, however in two dimensions analytic results are available. In addition to correcting mistakes in several published formulae, we derive an exact simulation scheme for sampling the passage times. Our algorithm exploits several interesting properties of planar Brownian motion and conformal local martingales. The main contribution of this thesis is the development of a novel multivariate framework for credit risk. We allow for both stochastic trend and volatility in credit qualities, with dependence introduced by letting these quantities be driven by systematic factors common to all obligors. Exploiting a conditional independence structure we are able to express the proportion of defaults in an asymptotically large portfolio as a path functional of the systematic factors. The functional in question returns crossing probabilities of time-changed Brownian motion to continuous barriers, and is typically not available in closed form. As such the distribution of portfolio losses is in general analytically intractable. As such we devise a scheme for simulating approximate losses and demonstrate almost sure convergence of this approximation. We show that the model calibrates well, across both tranches and maturities, to market quotes for CDX index tranches. In particular we are able to calibrate to data from 2006, as well as more recent ``distressed" data from 2008

Information reduction via level crossings in a credit risk model

Reduced form models, Structural models, Credit risk, Information reduction, Diffusion, Level-crossings, Brownian motion with drift, 60G55, 60G60, G13, D82,