A bivariate infinitely divisible law for modeling the magnitude and duration of monotone periods of log‐returns

In this paper, we show that spectral functions relevant for commonly used models of the non-Debye relaxation are related to the Stieltjes functions supported on the positive semi-axis. Using only this property, it can be shown that the response and relaxation functions are non-negative. They are connected to each other and obey the time evolution provided by integral equations involving the memory function M(t), which is the Stieltjes function as well. This fact is also due to the Stieltjes character of the spectral function. Stochastic processes-based approach to the relaxation phenomena gives the possibility to identify the memory function M(t) with the Laplace (Lévy) exponent of some infinitely divisible stochastic processes and to introduce its partner memory k(t). Both memories are related by the Sonine equation and lead to equivalent evolution equations which may be freely interchanged in dependence of our knowledge on memories governing the process.

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Learning for infinitely divisible GARCH models in option pricing

Studies in Nonlinear Dynamics & Econometrics ◽

10.1515/snde-2019-0088 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Fumin Zhu ◽

Michele Leonardo Bianchi ◽

Young Shin Kim ◽

Frank J. Fabozzi ◽

Hengyu Wu

Keyword(s):

Option Pricing ◽

Stock Returns ◽

Bayesian Learning ◽

Space Structure ◽

Estimation Methods ◽

Garch Models ◽

Learning Approaches ◽

Infinitely Divisible ◽

Asymmetric Garch ◽

Non Gaussian

AbstractThis paper studies the option valuation problem of non-Gaussian and asymmetric GARCH models from a state-space structure perspective. Assuming innovations following an infinitely divisible distribution, we apply different estimation methods including filtering and learning approaches. We then investigate the performance in pricing S&P 500 index short-term options after obtaining a proper change of measure. We find that the sequential Bayesian learning approach (SBLA) significantly and robustly decreases the option pricing errors. Our theoretical and empirical findings also suggest that, when stock returns are non-Gaussian distributed, their innovations under the risk-neutral measure may present more non-normality, exhibit higher volatility, and have a stronger leverage effect than under the physical measure.

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Tail asymptotics of an infinitely divisible space-time model with convolution equivalent Lévy measure

Journal of Applied Probability ◽

10.1017/jpr.2020.73 ◽

2021 ◽

Vol 58 (1) ◽

pp. 42-67 ◽

Cited By ~ 1

Author(s):

Mads Stehr ◽

Anders Rønn-Nielsen

Keyword(s):

Space Time ◽

Time Interval ◽

Lévy Measure ◽

Infinitely Divisible ◽

Time Model ◽

Spatial Object ◽

Tail Behaviour ◽

Fixed Radius ◽

The Right ◽

Space Time Model

AbstractWe consider a space-time random field on ${{\mathbb{R}^d} \times {\mathbb{R}}}$ given as an integral of a kernel function with respect to a Lévy basis with a convolution equivalent Lévy measure. The field obeys causality in time and is thereby not continuous along the time axis. For a large class of such random fields we study the tail behaviour of certain functionals of the field. It turns out that the tail is asymptotically equivalent to the right tail of the underlying Lévy measure. Particular examples are the asymptotic probability that there is a time point and a rotation of a spatial object with fixed radius, in which the field exceeds the level x, and that there is a time interval and a rotation of a spatial object with fixed radius, in which the average of the field exceeds the level x.

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A general approach to sample path generation of infinitely divisible processes via shot noise representation

Statistics & Probability Letters ◽

10.1016/j.spl.2021.109091 ◽

2021 ◽

pp. 109091

Author(s):

Reiichiro Kawai

Keyword(s):

Shot Noise ◽

Sample Path ◽

Path Generation ◽

Infinitely Divisible ◽

Infinitely Divisible Processes

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An analysis of the Pólya point process

Advances in Applied Probability ◽

10.1017/s000186780002098x ◽

1983 ◽

Vol 15 (01) ◽

pp. 39-53 ◽

Cited By ~ 1

Author(s):

Ed Waymire ◽

Vijay K. Gupta

Keyword(s):

Point Process ◽

Point Processes ◽

Critical Value ◽

General Representation ◽

Infinitely Divisible ◽

Generating Functional ◽

Representation Problem ◽

Polya Process ◽

Pólya Process ◽

Stochastic Point Processes

The Pólya process is employed to illustrate certain features of the structure of infinitely divisible stochastic point processes in connection with the representation for the probability generating functional introduced by Milne and Westcott in 1972. The Pólya process is used to provide a counterexample to the result of Ammann and Thall which states that the class of stochastic point processes with the Milne and Westcott representation is the class of regular infinitely divisble point processes. So the general representation problem is still unsolved. By carrying the analysis of the Pólya process further it is possible to see the extent to which the general representation is valid. In fact it is shown in the case of the Pólya process that there is a critical value of a parameter above which the representation breaks down. This leads to a proper version of the representation in the case of regular infinitely divisible point processes.

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