This paper introduces a parsimonious and yet flexible semiparametric model to forecast financial volatility. The new model extends a related linear nonnegative autoregressive model previously used in the volatility literature by way of a power transformation. It is semiparametric in the sense that the distributional and functional form of its error component is partially unspecified. The statistical properties of the model are discussed and a novel estimation method is proposed. Simulation studies validate the new method and suggest that it works reasonably well in finite samples. The out-of-sample forecasting performance of the proposed model is evaluated against a number of standard models, using data on S&P 500 monthly realized volatilities. Some commonly used loss functions are employed to evaluate the predictive accuracy of the alternative models. It is found that the new model generally generates highly competitive forecasts.

Here we derive the Lévy characteristic triplet for the GNIG probability law. This characterizes the corresponding Lévy process. In addition we derive equivalent martingale measures with which to price simple put and call options. This is done under two different equivalent martingale measures. We also present a multivariate Lévy process where the marginal probability distribution follows a GNIG Lévy process. The main contribution is, however, a stochastic process which is characterized by autocorrelation in moments equal and higher than two, here a multivariate specification is provided as well. The main tool for achieving this is to add an integrated Feller square root process to the dynamics of the second moment in a time-deformed Browninan motion. Applications to option pricing are also considered, and a brief discussion is held on the topic of estimation of the suggested process.

The authors propose the class of normal inverse gaussian (NIG) distributions to approximate an unknown risk-neutral density The appeal of the NIG class of distributions is that it is characterized by the first four moments: mean, variance, skewness, mid kurtosis. These are the moments that are important to many risk management applications. One strength of this approach is that the authors link the pricing of individual derivatives to the moments of the risk-neutral distribution, which has an intuitive appeal in terms of how volatility, skewness, and kurtosis of the risk-neutral distribution can explain the behavior of derivative prices. The authors provide numerical and empirical evidence showing appealing features of their approach, notably its superior performance compared to the existing methods.

Here we investigate the statistical evidence for a novel class of constrained bivariate AR(1) processes to capture the temporal dynamics of the maximum and minimum for the hourly/daily inflow into a stochastic reservoir. The estimation is done using a perturbed maximum likelihood technique and is illustrated by an empirical example.

Here we present an extension to the generalized hyperbolic (GH) familiy of probability laws. In particular we add one more scaling parameter. We define the corresponding probability law and derive the Laplace transform. Further we develop a multivarite probability law which has one dimensional marginal laws that are distributed according to our extension of the GH familiy. This is also done for the special case when the marginal law is distributed according to the generalized normal inverse Gaussian (GING), a special case in our extension. Further we discuss the possibilities to use this univariate and multivariate probability law in financial modelling

This work consists of four articles concerning Gaussian probability laws with stochastic means and variances. The first paper introduces a new way of approximating the probability distribution of a ...

This work consists of four articles concerning Gaussian probability laws with stochastic means and variances. The first paper introduces a new way of approximating the probability distribution of a function of random variables. This is done with a Gaussian probability law with stochastic mean and variance. In the second paper an extension of the Generalized Hyperbolic class of probability distributions is presented. The third paper introduces, using a Gaussian probability law with stochastic mean and variance, a GARCH type stochastic process with skewed innovations. In the fourth paper a Levy process with second order stochastic volatility is presented, option pricing under such a process is also considered.

Here we present a general framework for a (1,1) type of process with innovations with a probability law of the mean- variance mixing type, therefore we call the process in question the mean variance mixing (1,1) or MVM GARCH\(1,1). One implication is a GARCH model with skewed innovations and constant mean dynamics. This is achieved without using a location parameter to compensate for time dependence that affects the mean dynamics. From a probabilistic viewpoint the idea is straightforward. We just construct our stochastic process from the desired behavior of the cumulants. Further we provide explicit expressions for the unconditional second to fourth cumulants for the process in question. In the paper we present a specification of the MVM-GARCH process where the mixing variable is of the inverse Gaussian type. On the basis on this assumption we can formulate a maximum likelihood based approach for estimating the process closely related to the approach used to estimate an ordinary (1,1). Under the distributional assumption that the mixing random process is an inverse Gaussian i.i.d process the MVM-GARCH process is then estimated on log return data from the Standard and Poor 500 index. An analysis for the conditional skewness and kurtosis implied by the process is also presented in the paper

Here we present a general framework for a GARCH (1,1) type of process with innovations with a probability law of the mean- variance mixing type, therefore we call the process in question the mean variance mixing GARCH (1,1) or MVM GARCH (1,1). One implication is a GARCH model with skewed innovations and constant mean dynamics. This is achieved without using a location parameter to compensate for time dependence that afiects the mean dynamics. From a probabilistic viewpoint the idea is straightforward. We just construct our stochastic process from the desired behavior of the cumulants. Further we provide explicit expressions for the unconditional second to fourth cumulants for the process in question. In the paper we present a speciflcation of the MVM-GARCH process where the mixing variable is of the inverse Gaussian type. On the basis on this assumption we can formulate a maximum likelihood based approach for estimating the process closely related to the approach used to estimate an ordinary GARCH (1,1). Under the distributional assumption that the mixing random process is an inverse Gaussian i.i.d process the MVM-GARCH process is then estimated on log return data from the Standard and Poor 500 index. An analysis for the conditional skewness and kurtosis implied by the process is also presented in the paper.

Here we present a general framework for a GARCH (1,1) process with innovations with a probability law of the mean- variance mixing type, therefore we call th ep rocess in question the mean variance mixing GARCH (1,1) or MVM GARCH (1,1). One implication is a GARCH model with skewed innovations and constant mean dynamics. This is achieved without using a location parameter to compensate for time dependence that affects the mean dynamics. From a probabilistic viewpoint the idea is straight forward we just construct our process from the desired behavior of the moments. Further we provide explicit expressions for the unconditional first to fourth cumulants for the process in question. In the paper we present a specification of the MVM- GARCH process where the mixing variable is of the inverse Gaussian type On basis on this assumption we can formulate a maximum likelihood based approach for estimating the process closely related to the approach used to estimate an ordinary GARCH (1,1).

We introduce a new approximation method for the distribution of functions of random variables that are real-valued. The approximation involves moment matching and exploits properties of the class of normal inverse Gaussian distributions. We show that the new method provides better approximations than Gram- Charlier and Edgeworth expansions. Further we provide an example on the usefulness of the approximation in relation to the temporal aggregation and unconditional distribution of the GARCH (1,1) process.