Jan Korbel

Option pricing beyond Black–Scholes based on double-fractional diffusion

We show how the prices of options can be determined with the help of double-fractional differential equation in such a way that their inclusion in a portfolio of stocks provides a more reliable hedge against dramatic price drops than the use of options whose prices were fixed by the Black–Scholes formula.

Modeling of financial processes with a space-time fractional diffusion equation of varying order

Abstract In this paper, a new model for financial processes in form of a space-time fractional diffusion equation of varying order is introduced, analyzed, and applied for some financial data. While the orders of the spatial and temporal derivatives of this equation can vary on different time intervals, their ratio remains constant and thus the global scaling properties of its solutions are conserved. In this way, the model covers both a possible complex short-term behavior of the financial processes and their long-term dynamics determined by its characteristic time-independent scaling exponent. As an application, we consider the option pricing and describe how it can be modeled by the space-time fractional diffusion equation of varying order. In particular, the real option prices of index S&P 500 traded in November 2008 are analyzed in the framework of our model and the results are compared with the predictions made by other option pricing models.

On q-non-extensive statistics with non-Tsallisian entropy

We combine an axiomatics of Renyi with the q-deformed version of Khinchin axioms to obtain a measure of information (i.e., entropy) which accounts both for systems with embedded self-similarity and non-extensivity. We show that the entropy thus obtained is uniquely solved in terms of a one-parameter family of information measures. The ensuing maximal-entropy distribution is phrased in terms of a special function known as the Lambert W-function. We analyze the corresponding “high” and “low-temperature” asymptotics and reveal a non-trivial structure of the parameter space. Salient issues such as concavity and Schur concavity of the new entropy are also discussed.

Point Information Gain and Multidimensional Data Analysis

We generalize the point information gain (PIG) and derived quantities, i.e., point information gain entropy (PIE) and point information gain entropy density (PIED), for the case of the Renyi entropy and simulate the behavior of PIG for typical distributions. We also use these methods for the analysis of multidimensional datasets. We demonstrate the main properties of PIE/PIED spectra for the real data with the examples of several images and discuss further possible utilizations in other fields of data processing.

Remarks on “Comments on ‘On q-non-extensive statistics with non-Tsallisian entropy’ ” [Physica A 466 (2017) 160]

Recently in [Physica A 411 (2014) 138] Ilić and Stanković h ave suggested that there may be problem for the class of hybri d entropies introduced in [P. Jizba and T. Arimitsu, Physica A 340 (2004) 110]. In this Comment we point out that the problem can be traced down to the q-additive entropic chain rule and to a peculiar behavior of t he DeFinetti–Kolmogorov relation for escort distributions. However, despite this, one can still safely use the proposed hybrid entropies in most of the statistical -thermodynamics considerations.Abstract Recently in Ilic and Stankovic (2017) have suggested that there may be problem for the class of hybrid entropies introduced in Jizba and Arimitsu (2004). In this Comment we point out that the problem can be traced down to the q -additive entropic chain rule and to a peculiar behavior of the De Finetti–Kolmogorov relation for escort distributions. However, despite this, one can still safely use the proposed hybrid entropies in most of the statistical-thermodynamics considerations.

On statistical properties of Jizba–Arimitsu hybrid entropy

Jizba–Arimitsu entropy (also called hybrid entropy) combines axiomatics of Renyi and Tsallis entropy. It has many common properties with them, on the other hand, some aspects as e.g., MaxEnt distributions, are different. In this paper, we discuss statistical properties of hybrid entropy. We define hybrid entropy for continuous distributions and its relation to discrete entropy. Additionally, definition of hybrid divergence and its connection to Fisher metric is also presented. Interestingly, Fisher metric connected to hybrid entropy differs from corresponding Fisher metrics of Renyi and Tsallis entropy. This motivates us to introduce average hybrid entropy, which can be understood as an average between Tsallis and Renyi entropy.

Rescaling the nonadditivity parameter in Tsallis thermostatistics

Abstract The paper introduces nonadditivity parameter transformation group induced by Tsallis entropy. We discuss simple physical applications such as systems in the contact with finite heat bath or systems with temperature fluctuations. With help of the transformation, it is possible to introduce generalized distributive rule in q -deformed algebra. We focus on MaxEnt distributions of Tsallis entropy with rescaled nonadditivity parameter under escort energy constraints. We show that each group element corresponds to one class of q -deformed distributions. Finally, we briefly discuss the application of the transformation to Jizba–Arimitsu hybrid entropy and its connection to Average Hybrid entropy.

Classification of complex systems by their sample-space scaling exponents

The nature of statistics, statistical mechanics and consequently the thermodynamics of stochastic systems is largely determined by how the number of states

Applications of Multifractal Diffusion Entropy Analysis to Daily and Intraday Financial Time Series

W(N)

Modeling Financial Time Series: Multifractal Cascades and Rényi Entropy

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