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Mixed Tempered Stable distribution

In this paper, we introduce a new parametric distribution, the mixed tempered stable. It has the same structure of the normal variance–mean mixtures but the normality assumption gives way to a semi-heavy tailed distribution. We show that, by choosing appropriately the parameters of the distribution and under the concrete specification of the mixing random variable, it is possible to obtain some well-known distributions as special cases. We employ the mixed tempered stable distribution which has many attractive features for modelling univariate returns. Our results suggest that it is flexible enough to accommodate different density shapes. Furthermore, the analysis applied to statistical time series shows that our approach provides a better fit than competing distributions that are common in the practice of finance.

Option pricing in an exponential MixedTS Lévy process

In this paper we present an option pricing model based on the assumption that the underlying asset price is an exponential Mixed Tempered Stable Lévy process. We also introduce a new R package called PricingMixedTS that allows the user to calibrate this model using procedures based on loss or likelihood functions.

Risk parity for Mixed Tempered Stable distributed sources of risk

In this paper we discuss a detailed methodology for dealing with Risk parity in a parametric context. In particular, we use the Independent Component Analysis for a linear decomposition of portfolio risk factors. Each Independent Component is modeled with the Mixed Tempered Stable distribution. Risk parity optimal portfolio weights are calculated for three risk measures: Volatility, modified Value At Risk and modified Expected Shortfall. Empirical analysis is discussed in terms of out-of-sample performance and portfolio diversification.

Risk Measurement Using the Mixed Tempered Stable Distribution

The Mixed Tempered Stable distribution (MixedTS) recently introduced has as special cases parametric distributions used in asset return modelling such as the Variance Gamma (VG) and Tempered Stable. In this paper, we start from this flexible distribution and compare the historical estimates for the two homogeneous risk measures with the quantities obtained using direct numerical integration and the saddle-point approximation. The homogeneity property enables us to go further and look for the most important sources of risk. Although risk decomposition in a parametric context is not straightforward, modified versions of VaR and ES based on asymptotic expansions simplify the problem.

Implicit expectiles and measures of implied volatility

We show how to compute the expectiles of the risk-neutral distribution from the prices of European call and put options. Empirical properties of these implicit expectiles are studied on a data-set of closing daily prices of FTSE MIB index options. We introduce the interexpectile difference , for , and suggest that it is a natural measure of the variability of the risk-neutral distribution. We investigate its theoretical and empirical properties and compare it with the VIX index computed by CBOE. We also discuss a theoretical comparison with implicit VaR and CVaR introduced in Barone Adesi [J. Risk Financ. Manage., 2016, 9].

Some Empirical Evidence on the Need of More Advanced Approaches in Mortality Modeling

Recent literature on mortality modeling suggests to include in the dynamics of mortality rates the effect of time, age, the interaction of the latter two terms and finally a term for possible shocks that introduce additional uncertainty. We consider for our analysis models that use Legendre polynomials, for the inclusion of age and cohort effects, and investigate the dynamics of the residuals that we get from fitted models. Obviously, we expect the effect of shocks to be included in the residual term of the basic model.

Discrete-Time Approximation of a Cogarch(p,q) Model and its Estimation: DISCRETE-TIME APPROXIMATION OF A COGARCH(p,q)

In this paper, we construct a sequence of discrete time stochastic processes that converges in probability and in the Skorokhod metric to a COGARCH(p,q) model. The result is useful for the estimation of the continuous model defined for irregularly spaced time series data. The estimation procedure is based on the maximization of a pseudo log-likelihood function and is implemented in the yuima package.

Sensitivity analysis of Mixed Tempered Stable parameters with implications in portfolio optimization

This paper investigates the use, in practical financial problems, of the Mixed Tempered Stable distribution both in its univariate and multivariate formulation. In the univariate context, we study the dependence of a given coherent risk measure on the distribution parameters. The latter allows to identify the parameters that seem to have a greater influence on the given measure of risk. The multivariate Mixed Tempered Stable distribution enters in a portfolio optimization problem built considering a real market dataset of seventeen hedge fund indexes. We combine the flexibility of the multivariate Mixed Tempered Stable distribution, in capturing different tail behaviors, with the ability of the ARMA-GARCH model in capturing the time dependence observed in the data.

Implicit Expectiles and Measures of Implied Volatility

We show how to compute the expectiles of the risk neutral distribution from the prices of European call and put options. Empirical properties of these implicit expectiles are studied on a dataset of closing daily prices of FTSE MIB index options. We introduce the interexpectile difference Δτ (X) := eτ (X) - e1-τ (X), for τ ∈ (1/2,1], and suggest that it is a natural measure of the variability of the risk neutral distribution. We investigate its theoretical and empirical properties and compare it with the VIX index computed by CBOE and with implicit VaR and CVaR introduced in Barone Adesi (2016).

Sensitivity Analysis of the Mixed Tempered Stable Parameters with Implications in Portfolio Optimization

This paper investigates the use, in practical financial problems, of the Mixed Tempered Stable distribution both in its univariate and multivariate formulation. In the univariate context, we study the dependence of a given coherent risk measure on the distribution parameters. The latter allows to identify the parameters that seem to have a greater influence on the given measure of risk. The multivariate Mixed Tempered Stable distribution enters in a portfolio optimization problem built considering a real market dataset of seventeen hedge fund indexes. We combine the flexibility of the multivariate Mixed Tempered Stable distribution, in capturing different tail behaviors, with the ability of the ARMA-GARCH model in capturing the time dependence observed in the data.