Calculation of ruin probabilities for a dense class of heavy tailed distributions

In this paper, we propose a class of infinite-dimensional phase-type distributions with finitely many parameters as models for heavy tailed distributions. The class of finite-dimensional phase-type distributions is dense in the class of distributions on the positive reals and may hence approximate any such distribution. We prove that formulas from renewal theory, and with a particular attention to ruin probabilities, which are true for common phase-type distributions also hold true for the infinite-dimensional case. We provide algorithms for calculating functionals of interest such as the renewal density and the ruin probability. It might be of interest to approximate a given heavy tailed distribution of some other type by a distribution from the class of infinite-dimensional phase-type distributions and to this end we provide a calibration procedure which works for the approximation of distributions with a slowly varying tail. An example from risk theory, comparing ruin probabilities for a classical risk process with Pareto distributed claim sizes, is presented and exact known ruin probabilities for the Pareto case are compared to the ones obtained by approximating by an infinite-dimensional hyper-exponential distribution.     

Predicting loss given default (LGD) for residential mortgage loans: A two-stage model and empirical evidence for UK bank data

With the implementation of the Basel II regulatory framework, it became increasingly important for financial institutions to develop accurate loss models. This work investigates the loss given default (LGD) of mortgage loans using a large set of recovery data of residential mortgage defaults from a major UK bank. A Probability of Repossession Model and a Haircut Model are developed and then combined to give an expected loss percentage. We find that the Probability of Repossession Model should consist of more than just the commonly used loan-to-value ratio, and that the estimation of LGD benefits from the Haircut Model, which predicts the discount which the sale price of a repossessed property may undergo. This two-stage LGD model is shown to perform better than a single-stage LGD model (which models LGD directly from loan and collateral characteristics), as it achieves a better R² value and matches the distribution of the observed LGD more accurately.     

The negative news threshold—An explanation for negative skewness in stock returns

Abstract

A vast literature documents negative skewness in stock index return distributions on several markets. In this paper the issue of negative skewness is approached from a different angle to previous studies by combining the Trueman's 1997 model of management disclosure practices with symmetric market responses in order to explain negative skewness in stock returns. Empirical tests reveal that returns for days when non-scheduled news items are disclosed are the source of negative skewness in stock returns, as predicted. These findings suggest that negative skewness in stock returns is induced by asymmetries in the news disclosure policies of firm management. Furthermore, it is found that the returns are negatively skewed only for non-scheduled firm-specific news disclosures for firms where the management is compensated with stock options.     

On option pricing models in the presence of heavy tails

We propose a modification of the option pricing framework derived by Borland which removes the possibilities for arbitrage within this framework. It turns out that such arbitrage possibilities arise due to an incorrect derivation of the martingale transformation in the non-Gaussian option models which are used in that paper. We show how a similar model can be built for the asset price processes which excludes arbitrage. However, the correction causes the pricing formulas to be less explicit than the ones in the original formulation, since the stock price itself is no longer a Markov process. Practical option pricing algorithms will therefore have to resort to Monte Carlo methods or partial differential equations and we show how these can be implemented. An extra parameter, which needs to be specified before the model can be used, will give market makers some extra freedom when fitting their model to market data.     

Implicit expectiles and measures of implied volatility

It is well known that the normal distribution is inadequate in capturing the skewed and heavy-tailed behaviour of exchange rate returns. To this end, various flexible distributions that are capable of modelling the asymmetric and tailed behaviour of returns have been proposed. In this paper, we investigate the performance of the generalized lambda distribution (GLD) to capture the skewed and leptokurtic behaviour of exchange rate returns. We do this by conducting a comprehensive numerical study to compare the performance of the GLD against the performances of the skewed t distribution, the unbounded Johnson family of distributions and the normal inverse Gaussian (NIG) distribution. Our results suggest that in terms of the value-at-risk and expected shortfall, the GLD shows at least similar performance to the skewed t distribution and the NIG distribution. Considering the ease in GLD’s use for random variate generation in Monte Carlo simulations, we conclude that the GLD can be a good alternative in various financial applications where modelling of the heavy tail behaviour is critical.     

The impact of the choice of VaR models on the level of regulatory capital according to Basel II

The Basel II framework allows the calculation of the capital requirements for market risk with Value-at-Risk models. Since no special model is prescribed in the framework, banks may use simple models with questionable assumptions concerning their underlying distributions. Our numerical analysis reveals that simple VaR models that perform noticeably worse than comparable simple models with more realistic assumptions may lead to a lower level of regulatory capital for banks. For this reason, banks have a major incentive to implement bad models. This is obviously contrary to the interests of regulatory authorities.     

Valuación de un producto estructurado de compra sobre el SX5E cuando la incertidumbre de los rendimientos está modelada con procesos log-estables

We report the participation level, we pricing a first generation's European call options on the Eurostoxx structured product, when returns’ uncertainty is modeled by log-stable processes, we present the basic statistics of the index's returns, we estimate the α-estable parameters, and we compare the structured products pricing by the both log-stable and log-Gaussian models using inputs of the debt markets. We conclude that investors get higher returns than debt markets using both models and returns’ differences depend of the participation level and the maturity.     

L‐moments skewness and kurtosis as measures of regional convergence and cohesion

The paper deals with the statistical modeling of convergence and cohesion over time with the use of kurtosis, skewness and L‐moments. Changes in the shape of the distribution related to the spatial allocation of socio‐economic phenomena are considered as an evidence of global shift, divergence or convergence. Cross‐sectional time‐series statistical modeling of variables of interest is to overpass the minors of econometric theoretical models of convergence and cohesion determinants. L‐moments perform much more stable and interpretable than classical measures. Empirical evidence of panel data proves that one pure pattern (global shift, polarization or cohesion) rarely exists and joint analysis is required.     

Moments expansion densities for quantifying financial risk

We propose a novel semi-nonparametric distribution that is feasibly parameterized to represent the non-Gaussianities of the asset return distributions. Our Moments Expansion (ME) density presents gains in simplicity attributable to its innovative polynomials, which are defined by the difference between the nth power of the random variable and the nth moment of the density used as the basis. We show that the Gram–Charlier distribution is a particular case of the ME-type of densities. The latter being more tractable and easier to implement when quadratic transformations are used to ensure positiveness. In an empirical application to asset returns, the ME model outperforms both standard and non-Gaussian GARCH models along several risk forecasting dimensions.     

Tail-risk protection trading strategies

Starting from well-known empirical stylized facts of financial time series, we develop dynamic portfolio protection trading strategies based on econometric methods. As a criterion for riskiness, we consider the evolution of the value-at-risk spread from a GARCH model with normal innovations relative to a GARCH model with generalized innovations. These generalized innovations may for example follow a Student t, a generalized hyperbolic, an alpha-stable or a Generalized Pareto distribution (GPD). Our results indicate that the GPD distribution provides the strongest signals for avoiding tail risks. This is not surprising as the GPD distribution arises as a limit of tail behaviour in extreme value theory and therefore is especially suited to deal with tail risks. Out-of-sample backtests on 11 years of DAX futures data, indicate that the dynamic tail-risk protection strategy effectively reduces the tail risk while outperforming traditional portfolio protection strategies. The results are further validated by calculating the statistical significance of the results obtained using bootstrap methods. A number of robustness tests including application to other assets further underline the effectiveness of the strategy. Finally, by empirically testing for second-order stochastic dominance, we find that risk averse investors would be willing to pay a positive premium to move from a static buy-and-hold investment in the DAX future to the tail-risk protection strategy.