Tsallis and Kaniadakis Entropy Measures for Risk Neutral Densities
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.
KeywordsEntropy measures Risk neutral densities Entropy pricing theory Tsallis entropy Kaniadakis entropy
“This work was partially supported by Ningbo Natural Science Foundation (No. 2016A610077) and K.C. Wong Magna Fund in Ningbo University.”
- 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 1995Google 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
- 14.Shanon, C.E., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press, Urbana (1963)Google Scholar