Tsallis and Kaniadakis Entropy Measures for Risk Neutral Densities

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

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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 