Kernel Based Goodness-of-Fit Tests for Copulas with Fixed Smoothing Parameters

We study a test statistic on the integrated squared difference between a kernel estimator of the copula density and a kernel smoothed estimator of the parametric copula density. We show for fixed smoothing parameters that the test is consistent and that the asymptotic properties are driven by a U-statistic of order 4 with degeneracy of order 3. For practical implementation we suggest to compute the critical values through a semiparametric bootstrap. Monte Carlo results show that the bootstrap procedure performs well in small samples. In particular size and power are less sensitive to smoothing parameter choice than they are under the asymptotic approximation obtained for a vanishing bandwidth.Nonparametric; Copula density; Goodness-of-fit test; U-statistic.

A Kolmogorov-Smirnov Type Test for Positive Quadrant Dependence

We consider a consistent test, that is similar to a Kolmogorov-Smirnov test, of the complete set of restrictions that relate to the copula representation of positive quadrant dependence. For such a test we propose and justify inference relying on a simulation based multiplier method and a bootstrap method. We also explore the finite sample behaviour of both methods with Monte Carlo experiments. A first empirical illustration is given for US insurance claim data. A second one examines the presence of positive quadrant dependence in life expectancies at birth of males and females among countries.Nonparametric; Positive Quadrant Dependence; Copula; Risk Management; Loss Severity Distribution; Bootstrap; Multiplier Method; Empirical Process

Local Multiplicative Bias Correction for Asymmetric Kernel Density Estimators

We consider semiparametric asymmetric kernel density estimators when the unknown density has support on [0, ¥). We provide a unifying framework which contains asymmetric kernel versions of several semiparametric density estimators considered previously in the literature. This framework allows us to use popular parametric models in a nonparametric fashion and yields estimators which are robust to misspecification. We further develop a specification test to determine if a density belongs to a particular parametric family. The proposed estimators outperform rival non- and semiparametric estimators in finite samples and are simple to implement. We provide applications to loss data from a large Swiss health insurer and Brazilian income data.semiparametric density estimation; asymmetric kernel; income distribution; loss distribution; health insurance; specification testing

On the Way to Recovery: A Nonparametric Bias Free Estimation of Recovery Rate Densities

In this paper we analyse recovery rates on defaulted bonds using the Standard and Poor’s/PMD database for the years 1981-1999. Due to the specific nature of the data (observations lie within 0 and 1), we must rely on nonstandard econometric techniques. The recovery rate density is estimated nonparametrically using a beta kernel method. This method is free of boundary bias, and Monte Carlo comparison with competing nonparametric estimators show that the beta kernel density estimator is particularly well suited for density estimation on the unit interval. We challenge the usual market practice to model parametrically recovery rates using a beta distribution calibrated on the empirical mean and variance. This assumption is unable to replicate multimodal distributions or concentration of data at total recovery and total loss. We evaluate the impact of choosing the beta distribution on the estimation of credit Value-at-Risk.default, recovery, kernel estimation, credit risk

Linear-Quadratic Jump-Diffusion Modeling with Application to Stochastic Volatility

We aim at accommodating the existing affine jump-diffusion and quadratic models under the same roof, namely the linear-quadratic jump-diffusion (LQJD) class. We give a complete characterization of the dynamics underlying this class of models as well as identification constraints, and compute standard and extended transforms relevant to asset pricing. We also show that the LQJD class can be embedded into the affine class through use of an augmented state vector. We further establish that an equivalence relationship holds between both classes in terms of transform analysis. An option pricing application to multifactor stochastic volatility models reveals that adding nonlinearity into the model significantly reduces pricing errors, and further addition of a jump component in the stock price largely improves goodness-of-fit for in-the-money calls but less for out-of-the-money ones.Linear-quadratic models; affine models; jump-diffusions; generalized Fourier transform; option pricing; stochastic volatility

Weak Convergence of Hedging Strategies of Contingent Claims

This paper presents results on the convergence for hedging strategies in the setting of incomplete financial markets. We examine the convergence of the so-called locally risk-minimizing strategy. It is proved that such a choice for the trading strategy, when perfect hedging of contingent claims is infeasible, is robust under weak convergence. Several fundamental examples, such as trinomial trees and stochastic volatility models, extracted from the financial modeling literature illustrate this property for both deterministic and random time intervals shrinking to zero.Weak convergence; Incomplete financial markets; Locally risk-minimizing strategy; Hedging strategy; Minimal martingale measure

SOME STATISTICAL PITFALLS IN COPULA MODELING FOR FINANCIAL APPLICATIONS

In this paper we discuss some statistical pitfalls that may occur in modeling cross-dependences with copulas in financial applications. In particular we focus on issues arising in the estimation and the empirical choice of copulas as well as in the design of time-dependent copulas.Copulas; Dependence Measures; Risk Management

Sensitivity Analysis of VaR Expected Shortfall for Portfolios Under Netting Agreements

In this paper, we characterize explicitly the first derivative of the Value at Risk and the Expected Shortfall with respect to portfolio allocation when netting between positions exists. As a particular case, we examine a simple Gaussian example in order to illustrate the impact of netting agreements in credit risk management. We further provide nonpara-metric estimators for sensitivities and derive their asymptotic distributions. An empirical application on a typical banking portfolio is finally provided.Value at Risk, Expected Shortfall, Sensitivity, Risk Management, Credit Risk, Netting.

Sensitivity Analysis of Values at Risk

The aim of this paper is to analyze the sensitivity of Value at Risk (VaR) with respect to portfolio allocation. We derive analytical expressions for the first and second derivatives of the Value at Risk, and explain how they can be used to simplify statistical inference and to perform a local analysis of the Value at Risk. An empirical illustration of such an analysis is given for an portfolio of French stocks.Value at Risk; risk management; VaR efficient portfolio; iso VaR; kernel estimators; quantile