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Tsallis and Kaniadakis Entropy Measures for Risk Neutral Densities

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Computer Aided Systems Theory – EUROCAST 2017 (EUROCAST 2017)

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Abstract

Concepts of Econophysics are usually used to solve problems related to uncertainty and nonlinear dynamics. The risk neutral probabilities play an important role in the theory of option pricing. The application of entropy in finance can be regarded as the extension of both information entropy and probability entropy. It can be an important tool in various financial issues such as risk measures, portfolio selection, option pricing and asset pricing. The classical approach of stock option pricing is based on Black-Scholes model, which relies on some restricted assumptions and contradicts with modern research in financial literature. The Black-Scholes model is governed by Geometric Brownian Motion and is based on stochastic calculus. It depends on two factors: no arbitrage, which implies the universe of risk-neutral probabilities and parameterization of risk-neutral probability by a reasonable stochastic process. Therefore, risk-neutral probabilities are vital in this framework. The Entropy Pricing Theory founded by Gulko represents an alternative approach of constructing risk-neutral probabilities without depending on stochastic calculus. Gulko applied Entropy Pricing Theory for pricing stock options and introduced an alternative framework of Black-Scholes model for pricing European stock options. In this paper we derive solutions of maximum entropy problems based on Tsallis, Weighted-Tsallis, Kaniadakis and Weighted-Kaniadakies entropies, in order to obtain risk-neutral densities.

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References

  1. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–659 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  2. Borwein, J., Choksi, R., Maréchal, P.: Probability distributions of assets inferred from option prices via the principle of maximum entropy. J. Soc. Ind. Appl. Math. 14, 464–478 (2003)

    MathSciNet  MATH  Google Scholar 

  3. Belis, M., Guiasu, S.: A quantitative-qualitative measure of information in cybernetic systems. IEEE Trans. Inf. Theory 14(4), 593–594 (1968)

    Article  Google Scholar 

  4. Cressie, N., Richardson, S., Jaussent, I.: Ecological bias: use of maximum entropy approximations. ANZ J. Stat. 46(2), 233–255 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Guiasu, S.: Weighted entropy. Rep. Math. Phys. 2(3), 165–179 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gulko, L.: Dart boards and asset prices: introducing the entropy pricing theory. In: Fomby, T.B., Hill, R.C. (eds.) Advances in Econometrics. JAI Press, Greenwich (1997)

    Google Scholar 

  7. Gulko, L.: The Entropy Theory of Bond Option Pricing, Working Paper, Yale School of Management, New Haven, CT, October 1995

    Google Scholar 

  8. Guo, W.Y.: Maximum entropy in option pricing: a convex-spline smoothing method. J. Futures Markets 21, 819–832 (2001)

    Article  Google Scholar 

  9. Kaniadakis, G.: Non-linear kinetics underlying generalized statistics. Phys. A 296, 405–425 (2001)

    Article  MATH  Google Scholar 

  10. Preda, V., Sheraz, M.: Risk-neutral densities in entropy theory of stock options using lambert function and a new approach. Proc. Rom. Acad. 16(1), 20–27 (2015)

    MathSciNet  Google Scholar 

  11. Preda, V., Dedu, S., Sheraz, M.: New measure selection for Hunt-Devolder semi-Markov regime switching interest rate models. Phys. A 407, 350–359 (2014)

    Article  MathSciNet  Google Scholar 

  12. Rényi, A.: On measures of entropy and information. In: Proceedings of the 4th Berkely Sympodium on Mathematics of Statistics and Probability, vol. 1, pp. 547–561. Berkeley University Press, Berkeley (1961)

    Google Scholar 

  13. Rompolis, L.S.: Retrieving risk neutral densities from European option prices based on the principle of maximum entropy. J. Empir. Finan. 17, 918–937 (2010)

    Article  Google Scholar 

  14. Shanon, C.E., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press, Urbana (1963)

    Google Scholar 

  15. Shafee, F.: Lambert function and a new non-extensive form of entropy. J. Appl. Math. 72, 785–800 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Sheraz, M., Dedu, S., Preda, V.: Entropy measures for assessing volatile markets. Procedia Econ. Finan. 22, 655–662 (2015)

    Article  Google Scholar 

  17. Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  18. Trivellato, B.: Deformed exponentials and applications to finance. Entropy 15(9), 3471–3489 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ubriaco, M.R.: Entropies based on fractional calculus. Phys. Lett. A 373, 2516–2519 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

“This work was partially supported by Ningbo Natural Science Foundation (No. 2016A610077) and K.C. Wong Magna Fund in Ningbo University.”

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Authors and Affiliations

  1. Department of Mathematical Sciences, Department of Economics and Finance, Institute of Business Administration Karachi, KU Circular Road, Karachi, 75270, Pakistan

    Muhammad Sheraz

  2. National Institute for Economic Research “Costin C. Kiritescu”, Romanian Academy, 13 Septembrie 13, 050711, Bucharest, Romania

    Vasile Preda

  3. “Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of Romanian Academy, 13 Septembrie 13, 050711, Bucharest, Romania

    Vasile Preda

  4. Department of Applied Mathematics, Bucharest University of Economic Studies, Calea Dorobanţi 15-17, 010552, Bucharest, Romania

    Silvia Dedu

  5. School of Advanced Studies of the Romanian Academy, Calea Victoriei 125, 010071, Bucharest, Romania

    Silvia Dedu

Authors
  1. Muhammad Sheraz
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  2. Vasile Preda
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  3. Silvia Dedu
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Correspondence to Muhammad Sheraz .

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Editors and Affiliations

  1. University of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain

    Roberto Moreno-Díaz

  2. Johannes Kepler University Linz, Linz, Austria

    Franz Pichler

  3. University of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain

    Alexis Quesada-Arencibia

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Sheraz, M., Preda, V., Dedu, S. (2018). Tsallis and Kaniadakis Entropy Measures for Risk Neutral Densities. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds) Computer Aided Systems Theory – EUROCAST 2017. EUROCAST 2017. Lecture Notes in Computer Science(), vol 10672. Springer, Cham. https://doi.org/10.1007/978-3-319-74727-9_7

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  • DOI: https://doi.org/10.1007/978-3-319-74727-9_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-74726-2

  • Online ISBN: 978-3-319-74727-9

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