Financial market models with Lévy processes and time-varying volatility

Kim, Young S.; Rachev, Svetlozar T.; Bianchi, Michele Leonardo; Fabozzi, Frank J.

doi:10.1016/j.jbankfin.2007.11.004

Cited by 86 publications

(54 citation statements)

References 18 publications

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“…The spectral measure R of the second example has a non-trivial bounded support and the derived TID distribution has exponential moments of any order. In the last example, the MTS distribution is considered, see [8,6], the spectral measure is defined on an unbounded support and there exist exponential moments of some order.…”

Section: Examplesmentioning

confidence: 99%

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Tempered Infinitely Divisible Distributions and Processes

Bianchi

Rachev

Kim

et al. 2011

Theory Probab. Appl.

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Abstract. In this paper, we construct the new class of tempered infinitely divisible (TID) distributions. Taking into account the tempered stable distribution class, as introduced by in the seminal work of Rosińsky [10], a modification of the tempering function allows one to obtain suitable properties. In particular, TID distributions may have exponential moments of any order and conserve all proper properties of the Rosiński setting. Furthermore, we prove that the modified tempered stable distribution is TID and give some further parametric example.M.S.C. classification: 60E07, 60G52.

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Section: Examplesmentioning

confidence: 99%

“…A parametric example of TID distributions has been already considered in the literature, the MTS distribution, see [8,6]. The Lévy measure of a MTS distribution is defined as…”

Section: Example 2: Non Trivial Spectral Measurementioning

confidence: 99%

Tempered Infinitely Divisible Distributions and Processes

Bianchi

Rachev

Kim

et al. 2011

Theory Probab. Appl.

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“…In order to calibrate asset returns models through exponential Lévy process or tempered stable GARCH model [13,14], one needs a correct evaluation of both the pdf and cdf functions. With the pdf function it is possible to construct a maximum likelihood estimator (MLE), while the cdf function allows one to assess the goodness of fit.…”

Section: Evaluating the Density Functionmentioning

confidence: 99%

“…The paper is related to some previous works of the authors [13,14] where the exponential Lévy and the tempered stable GARCH models has been studied. The remainder of this paper is organized as follows.…”

Section: Introductionmentioning

confidence: 98%

Tempered stable distributions and processes in finance: numerical analysis

Bianchi

Rachev

Kim³

et al. 2010

Mathematical and Statistical Methods for Actuarial Sciences and Finance

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Abstract. Most of the important models in finance rest on the assumption that randomness is explained through a normal random variable. However there is ample empirical evidence against the normality assumption, since stock returns are heavy-tailed, leptokurtic and skewed. Partly in response to those empirical inconsistencies relative to the properties of the normal distribution, a suitable alternative distribution is the family of tempered stable distributions. In general, the use of infinitely divisible distributions is obstructed by the difficulty to calibrate and simulate them. In this paper, we address some numerical issues resulting from tempered stable modelling, with a view toward the density approximation and simulation.M.S.C. classification: 60E07.

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“…This result is not in conflict with our stable mixture GARCH model because our goal is to (i) devise a model endowed with some plausible statistical and economic motivation, (ii) which yields relatively superior density and risk forecasts for daily (and possibly higher frequency) data, and (iii) can be used in a multivariate context via an ICA decomposition, but without concern for the stability (or lack thereof) aspect of returns. A possible way of incorporating all such features would be to use the tempered stable distribution, which also has a tractable characteristic function, mimics the shape of the stable distribution, but is such that, when iid copies are summed, the tail index increases; see Kim et al (2008) and Kim et al (2010). e.g., Samanidou et al (2007) for an overview of such models. This is in line with recent research with experimental data by Kirchler and Huber (2007), who show that heterogeneous fundamental information can be a major source for the emergence of fat tails and volatility clustering.…”

mentioning

confidence: 99%

Stable Mixture GARCH Models

et al. 2011

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A new model class for univariate asset returns is proposed which involves the use of mixtures of stable Paretian distributions, and readily lends itself to use in a multivariate context for portfolio selection. The model nests numerous ones currently in use, and is shown to outperform all its special cases. In particular, an extensive out-of-sample risk forecasting exercise for seven major FX and equity indices confirms the superiority of the general model compared to its special cases and other competitors. Estimation issues related to problems associated with mixture models are discussed, and a new, general, method is proposed to successfully circumvent these. The model is straightforwardly extended to the multivariate setting by using an independent component analysis framework. The tractability of the relevant characteristic function then facilitates portfolio optimization using expected shortfall as the downside risk measure. AbstractA new model class for univariate asset returns is proposed which involves the use of mixtures of stable Paretian distributions, and readily lends itself to use in a multivariate context for portfolio selection. The model nests numerous ones currently in use, and is shown to outperform all its special cases. In particular, an extensive out-of-sample risk forecasting exercise for seven major FX and equity indices confirms the superiority of the general model compared to its special cases and other competitors. Estimation issues related to problems associated with mixture models are discussed, and a new, general, method is proposed to successfully circumvent these. The model is straightforwardly extended to the multivariate setting by using an independent component analysis framework. The tractability of the relevant characteristic function then facilitates portfolio optimization using expected shortfall as the downside risk measure.

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Financial market models with Lévy processes and time-varying volatility

Cited by 86 publications

References 18 publications

Tempered Infinitely Divisible Distributions and Processes

Tempered Infinitely Divisible Distributions and Processes

Tempered stable distributions and processes in finance: numerical analysis

Stable Mixture GARCH Models

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