Capturing Model Risk and Rating Momentum in the Estimation of Probabilities of Default and Credit Rating Migrations

We present two methodologies on the estimation of rating transition probabilities within Markov and non-Markov frameworks. We first estimate a continuous-time Markov chain using discrete (missing) data and derive a simpler expression for the Fisher information matrix, reducing the computational time needed for the Wald confidence interval by a factor of a half. We provide an efficient procedure for transferring such uncertainties from the generator matrix of the Markov chain to the corresponding rating migration probabilities and, crucially, default probabilities. For our second contribution, we assume access to the full (continuous) data set and propose a tractable and parsimonious self-exciting marked point processes model able to capture the non-Markovian effect of rating momentum. Compared to the Markov model, the non-Markov model yields higher probabilities of default in the investment grades, but also lower default probabilities in some speculative grades. Both findings agree with empirical observations and have clear practical implications. We illustrate all methods using data from Moody's proprietary corporate credit ratings data set. Implementations are available in the R package ctmcd.Comment: 22 pages, 5 Figures, 4 Tables. To Appear in Quantitative Financ

Likelihood based inference for diffusion driven models

This paper provides methods for carrying out likelihood based inference for diffusion driven models, for example discretely observed multivariate diffusions, continuous time stochastic volatility models and counting process models. The diffusions can potentially be non-stationary. Although our methods are sampling based, making use of Markov chain Monte Carlo methods to sample the posterior distribution of the relevant unknowns, our general strategies and details are different from previous work along these lines. The methods we develop are simple to implement and simulation efficient. Importantly, unlike previous methods, the performance of our technique is not worsened, in fact it improves, as the degree of latent augmentation is increased to reduce the bias of the Euler approximation. In addition, our method is not subject to a degeneracy that afflicts previous techniques when the degree of latent augmentation is increased. We also discuss issues of model choice, model checking and filtering. The techniques and ideas are applied to both simulated and real data.Bayes estimation, Brownian bridge, Non-linear diffusion, Euler approximation, Markov chain Monte Carlo, Metropolis-Hastings algorithm, Missing data, Simulation, Stochastic differential equation.

Parameter inference for multivariate stochastic processes with jumps

This dissertation addresses various aspects of estimation and inference for multivariate stochastic processes with jumps. The first chapter develops an unbiased Monte Carlo estimator of the transition density of a multivariate jump-diffusion process. The drift, volatility, jump intensity, and jump magnitude are allowed to be state-dependent and non-affine. The density estimator proposed enables efficient parametric estimation of multivariate jump-diffusion models based on discretely observed data. Under mild conditions, the resulting parameter estimates have the same asymptotic behavior as maximum likelihood estimators as the number of data points grows, even when the sampling frequency of the data is fixed. In a numerical case study of practical relevance, the density and parameter estimators are shown to be highly accurate and computationally efficient. In the second chapter, I examine continuous-time stochastic volatility models with jumps in returns and volatility in which the parameters governing the jumps are allowed to switch according to a Markov chain. I estimate the parameters and the latent processes using the S&P 500 and Nasdaq indices from 1990 to 2014. The Markov-switching parameters characterize well the periods of market stress, such as those in 1997-1998, 2001 and 2007-2010. Several statistical tests favor the model with Markov-switching jump parameters. These results provide empirical evidence about the state-dependent and time-varying nature of asset price jumps, a feature of asset prices that has recently been documented using high-frequency data. The third chapter considers applying Markov-switching affine stochastic volatility models with jumps in returns and volatility, where the jump parameters are not regime-switching. The estimation is performed via Markov Chain Monte Carlo methods, allowing to obtain the latent processes induced by the structure of the models. Furthermore, I propose some misspecification tests and develop a Markov-switching test based on the odds ratios. The parameters and the latent processes are estimated using the S&P 500 index from 1970 to 2014. I show that the S&P 500 stochastic volatility exhibits a Markov-switching behavior, and that most of the high volatility regimes coincide with the recessions identified ex-post by the National Bureau of Economic Research

Efficient likelihood estimation in state space models

Motivated by studying asymptotic properties of the maximum likelihood estimator (MLE) in stochastic volatility (SV) models, in this paper we investigate likelihood estimation in state space models. We first prove, under some regularity conditions, there is a consistent sequence of roots of the likelihood equation that is asymptotically normal with the inverse of the Fisher information as its variance. With an extra assumption that the likelihood equation has a unique root for each $n$, then there is a consistent sequence of estimators of the unknown parameters. If, in addition, the supremum of the log likelihood function is integrable, the MLE exists and is strongly consistent. Edgeworth expansion of the approximate solution of likelihood equation is also established. Several examples, including Markov switching models, ARMA models, (G)ARCH models and stochastic volatility (SV) models, are given for illustration.Comment: With the comments by Jens Ledet Jensen and reply to the comments. Published at http://dx.doi.org/10.1214/009053606000000614; http://dx.doi.org/10.1214/09-AOS748A; http://dx.doi.org/10.1214/09-AOS748B in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Generalized structured additive regression based on Bayesian P-splines

Generalized additive models (GAM) for modelling nonlinear effects of continuous covariates are now well established tools for the applied statistician. In this paper we develop Bayesian GAM's and extensions to generalized structured additive regression based on one or two dimensional P-splines as the main building block. The approach extends previous work by Lang und Brezger (2003) for Gaussian responses. Inference relies on Markov chain Monte Carlo (MCMC) simulation techniques, and is either based on iteratively weighted least squares (IWLS) proposals or on latent utility representations of (multi)categorical regression models. Our approach covers the most common univariate response distributions, e.g. the Binomial, Poisson or Gamma distribution, as well as multicategorical responses. For the first time, we present Bayesian semiparametric inference for the widely used multinomial logit models. As we will demonstrate through two applications on the forest health status of trees and a space-time analysis of health insurance data, the approach allows realistic modelling of complex problems. We consider the enormous flexibility and extendability of our approach as a main advantage of Bayesian inference based on MCMC techniques compared to more traditional approaches. Software for the methodology presented in the paper is provided within the public domain package BayesX