### Modeling Financial Risk

Organizer: Bonnie Ray, IBMSession Chair: Galit Shmueli, University of Maryland

**Estimating the Term Structure With a Semiparametric Bayesian Population Model: An Application to Corporate Bonds**

Alejandro Cruz-Marcelo, Katherine B. Ensor, and Gary L. Rosner

Dept. of Statistics, Rice Univ. and M.D. Anderson Cancer Center, Univ. of Texas

We introduce a novel framework to estimate term structures of interest rates. We propose to jointly estimate term structures by means of a Bayesian population model with a non-parametric prior based on Dirichlet process mixtures. Our model is not restricted to a particular type of bonds. Thus, it can be used to jointly estimate any combination of corporate and/or government term structures. The main advantage of our model over current estimation methods is its ability to produce accurate estimators based on small samples of bonds. This feature is particularly relevant for corporate bonds because it allows us to estimate term structures of individual firms. In this talk we describe our novel framework as well as an empirical application using U.S. corporate bonds. We use our model to estimate the term structure of 197 individual companies, 75% of them having at most two bonds. The estimators produced with our method greatly outperform those obtained with the popular approach of grouping the bonds by credit rating and independently estimating the term structure for each credit risk class. Specifically, the mean of the absolute price errors of our model is 76% smaller than the one of the popular approach. Out-of-sample tests based on cross-validation lead to the same conclusion. Across different partitions, the mean absolute prediction error is reduced by at least 53% when using our Bayesian model.

**Identifying Systemic Risk through Model-based Clustering of Multivariate Zero-inflated Time Series of Counts**

Sarah J. Thomas

Department of Statistics, Rice University

The objective of this work is to identify systemic risk through analysis of corporate liquidity measured by bond trading activity. Time series of number of trades and price changes are clustered based on similar model representations and characteristics of the series. Prior to clustering, it is necessary to develop a modeling framework for multivariate count time series. The basic modeling paradigm used is an observation driven Poisson regression in the generalized linear model framework. To handle extra zeros relative to the Poisson, the zero-inflated Poisson (ZIP) is used. Additionally, a new modeling paradigm for multivariate zero-inflated counts is introduced that allows for additional zeros at each time epoch for each individual series as well as all multivariate series. The zero-inflation provides the needed flexibility for modeling series when few counts are observed, when counts occur on an irregular basis due to exogenous influences, and when these irregularities occur according to different processes in the multiple count series. To estimate the model parameters, a new Monte Carlo Estimation Maximization (MCEM) algorithm is developed. A distance metric based on the likelihood of the model for each series is used in a hierarchical clustering algorithm to identify relevant clusters for the time series.

**Recovering the Tail Shape of the Risk Neutral Density from Option Prices with Applications in Risk Management**

Kam Hamediah

Department of Statistics, Rice University

Abstract: This paper derives a closed form pricing formula for the out of the money European style options, and develops a method to recover the tail shape of the risk neutral density from the observed option prices. The pricing formula satisfies well known model-free no-arbitrage bounds for the European style options. The method is quite general, and applies to a large class of risk neutral probability density distributions which includes the lognormal density. The method is original and unique in the sense that the focus is only on the tails of the risk neutral density, and not on the entire body of the density as many works have already done this. Example analysis using the S&P 500 option prices is given. Our method would be useful to risk managers, researchers, and practitioners interested in assessing and quantifying future extreme market conditions based on the observed option prices.