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Tsallis and Kaniadakis Entropy Measures for Risk Neutral Densities

机译：风险中性密度的Tsallis和Kaniadakis熵测度

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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.
机译：经济物理学的概念通常用于解决与不确定性和非线性动力学有关的问题。风险中性概率在期权定价理论中起着重要作用。熵在金融中的应用可以看作是信息熵和概率熵的扩展。它可能是解决各种财务问题（例如风险衡量，投资组合选择，期权定价和资产定价）的重要工具。股票期权定价的经典方法基于Black-Scholes模型，该模型依赖于一些受限制的假设，并且与金融文献中的现代研究相矛盾。 Black-Scholes模型受几何布朗运动控制，并基于随机演算。它取决于两个因素：无套利，这意味着风险中立概率的范围以及通过合理的随机过程对风险中立概率的参数化。因此，在此框架中，风险中性概率至关重要。由古尔科（Gulko）建立的熵定价理论（Entropy Pricing Theory）代表了一种构建风险中立概率的替代方法，而无需依赖随机演算。 Gulko应用熵定价理论对股票期权进行定价，并引入了Black-Scholes模型的替代框架对欧洲股票期权进行定价。在本文中，我们基于Tsallis，加权Tsallis，Kaniadakis和加权Kaniadakies熵导出最大熵问题的解，以获得风险中性密度。

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