Several authors have shown the ability of the variance gamma model to correct some biases of the Black-Scholes model. The variance gamma distribution has two additional parameters that allow to capture the skewness and kurtosis observed in financial data. However its density has not got a simple form formula and this implies numerical issues for historical estimation and option pricing. This paper investigates the possibility of approximating the variance gamma distribution to a finite mixture of normals. Therefore, we apply this result to derive a simple historical estimation procedure by means of the Expectation Maximization algorithm and we obtain a simple formula to price a European call option.

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

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 ∆τ (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. We also discuss a theoretical comparison with implicit VaR and CVaR introduced in Barone Adesi (2016).

In this paper we show how to simulate and estimate a COGARCH(p, q) model in the R package yuima. Several routines for simulation and estimation are introduced. In particular, for the generation of a COGARCH(p, q) trajectory, the user can choose between two alternative schemes. The first is based on the Euler discretization of the stochastic differential equations that identify a COGARCH(p, q) model while the second considers the explicit solution of the equations defining the variance process.Estimation is based on the matching of the empirical with the theoretical autocorrelation function. Three different approaches are implemented: minimization of the mean squared error, minimization of the absolute mean error and the generalized method of moments where the weighting matrix is continuously updated. Numerical examples are given in order to explain methods and classes used in the yuima package.

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 leaves place 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 modeling univariate returns. Our results suggest that it is enough flexible to accomodate 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.