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Tsallis and Kaniadakis Entropy Measures for Risk Neutral Densities

  • Muhammad SherazEmail author
  • Vasile Preda
  • Silvia Dedu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10672)

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.

Keywords

Entropy measures Risk neutral densities Entropy pricing theory Tsallis entropy Kaniadakis entropy 

Notes

Acknowledgements

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

References

  1. 1.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–659 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Belis, M., Guiasu, S.: A quantitative-qualitative measure of information in cybernetic systems. IEEE Trans. Inf. Theory 14(4), 593–594 (1968)CrossRefGoogle Scholar
  4. 4.
    Cressie, N., Richardson, S., Jaussent, I.: Ecological bias: use of maximum entropy approximations. ANZ J. Stat. 46(2), 233–255 (2004)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Guiasu, S.: Weighted entropy. Rep. Math. Phys. 2(3), 165–179 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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. 7.
    Gulko, L.: The Entropy Theory of Bond Option Pricing, Working Paper, Yale School of Management, New Haven, CT, October 1995Google Scholar
  8. 8.
    Guo, W.Y.: Maximum entropy in option pricing: a convex-spline smoothing method. J. Futures Markets 21, 819–832 (2001)CrossRefGoogle Scholar
  9. 9.
    Kaniadakis, G.: Non-linear kinetics underlying generalized statistics. Phys. A 296, 405–425 (2001)CrossRefzbMATHGoogle Scholar
  10. 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)MathSciNetGoogle Scholar
  11. 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)MathSciNetCrossRefGoogle Scholar
  12. 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. 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)CrossRefGoogle Scholar
  14. 14.
    Shanon, C.E., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press, Urbana (1963)Google Scholar
  15. 15.
    Shafee, F.: Lambert function and a new non-extensive form of entropy. J. Appl. Math. 72, 785–800 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Sheraz, M., Dedu, S., Preda, V.: Entropy measures for assessing volatile markets. Procedia Econ. Finan. 22, 655–662 (2015)CrossRefGoogle Scholar
  17. 17.
    Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Trivellato, B.: Deformed exponentials and applications to finance. Entropy 15(9), 3471–3489 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ubriaco, M.R.: Entropies based on fractional calculus. Phys. Lett. A 373, 2516–2519 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, Department of Economics and FinanceInstitute of Business Administration KarachiKarachiPakistan
  2. 2.National Institute for Economic Research “Costin C. Kiritescu”, Romanian AcademyBucharestRomania
  3. 3.“Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of Romanian AcademyBucharestRomania
  4. 4.Department of Applied MathematicsBucharest University of Economic StudiesBucharestRomania
  5. 5.School of Advanced Studies of the Romanian AcademyBucharestRomania

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