This paper discusses two classes of distributions, and stochastic processes derived from them: modified stable (MS) laws and normal modified stable (NMS) laws. This extends corresponding results for the generalised inverse Gaussian (GIG) and generalised hyperbolic (GH) or normal generalised inverse Gaussian (NGIG) laws. The wider framework thus established provides, in particular, for added flexibility in the modelling of the dynamics of financial time series, of importance especially as regards OU based stochastic volatility models for equities. In the special case of the tempered stable OU process an exact option pricing formula can be found, extending previous results based on the inverse Gaussian and gamma distributions.

In this paper we study the detailed distributional properties of integrated non-Gaussian OU (intOU) processes. Both exact and approximate results are given. We emphasise the study of the tail behaviour of the intOU process. Our results have many potential applications in financial economics, for OU processes are used as models of instantaneous volatility in stochastic volatility (SV) models. In this case an intOU process can be regarded as a model of integrated volatility. Hence the tail behaviour of the intOU process will determine the tail behaviour of returns generated by SV models.

Risk management approaches that do not incorporate randomly changing volatility tend to under- or overestimate the risk depending on current market conditions. We show how some popular stochastic volatility models in combination with the hyperbolic model introduced in Eberlein and Keller (1995) can be applied quite easily for risk management purposes. Moreover, we compare their relative performance on the basis of German stock index data.

In this paper we overview the pricing of several so-called exotic options in the nowdays quite popular exponential Lévy models.

Abstract. The question of the measurement of strategic long-term financial risks is of considerable importance. Existing modelling instruments allow for a good measurement of market risks of trading books over relatively small time intervals. However, these approaches may have severe deficiencies if they are routinely applied to longer time periods. In this paper we give an overview on methodologies that can be used to model the evolution of risk factors over a one-year horizon. Different models are tested on financial time series data by performing backtesting on their expected shortfall predictions. 1

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We study some properties of a new continuous-time model for financial time series which is driven by a class of Lévy processes instead of a Brownian motion. This model emerged from extensive empirical investigations. We discuss path properties of the driving process and aspects of the valuation of derivatives.

The aim of this work is to examine an existing relation between prices of put and call options, of both the European and the American type. This relation, based on a change of numeraire corresponding to a change of the probability measure through Girsanov’s Theorem, is called put–call duality. It includes as a particular case, the relation known as put–call symmetry. Necessary and sufficient conditions for put–call symmetry to hold are obtained, in terms of the triplet of predictable characteristic of the Lévy process, answering in this way a question raised by Carr and Chesney (1996).

The aim of this work is to use a duality approach to study the pricing of derivatives depending on two stocks driven by a bidimensional Levy process. The main idea is to apply Girsanov's Theorem for Levy processes, in order to reduce the posed problem to the pricing of a one Levy driven stock in an auxiliary market, baptized as \dual market". In this way, we extend the results obtained by Gerber and Shiu (1996) for two dimensional Brownian motion. Also we examine an existing relation between prices of put and call options, of both the European and the American type. This relation, based on a change of numeraire corresponding to a change of the probability measure through Girsanov's Theorem, is called put{call duality. It includes as a particular case, the relation known as put{call symmetry. Necessary and sucient conditions for put{call symmetry to hold are obtained, in terms of the triplet of predictable characteristic of the Levy process.

Abstract. The development of a methodology that could be used for the measurement of strategic long-term financial risks is becoming an important task. Existing modelling instruments allow for a good measurement of market risks of trading books over relatively small time intervals. However, these approaches might have some severe deficiencies if they are applied to longer time periods. In this paper we give an overview on methodologies that are proposed to model the evolution of risk factors over a long horizon. We investigate in detail the statistical properties and the behaviour of financial time series at different frequencies. Then, we test the different models on these data by backtesting expected shortfall predictions.