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Game theoretic analysis of incomplete markets: emergence of probabilities, nonlinear and fractional Black-Scholes equations

机译:不完全市场的博弈论分析:概率的出现,非线性和分数布莱克-舒尔斯方程

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摘要

Expanding the ideas of the author's paper "Nonexpansive maps and option pricing theory" [Kibernetica 34(6) (1998), 713-724] we develop a pure game-theoretic approach to option pricing, by-passing stochastic modeling. Risk neutral probabilities emerge automatically from the robust control evaluation. This approach seems to be especially appealing for incomplete markets encompassing extensive, so to say untamed, randomness, when the coexistence of infinite number of risk neutral measures precludes one from unified pricing of derivative securities. Our method is robust enough to be able to accommodate various markets rules and settings including path dependent payoffs, American options and transaction costs. On the other hand, it leads to rather simple numerical algorithms. Continuous time limit is described by nonlinear and/or fractional Black-Scholes type equations.
机译:扩展作者论文“非扩张性地图和期权定价理论”的思想[Kibernetica 34(6)(1998),713-724],我们开发了一种纯博弈论的期权定价方法,绕过了随机建模。稳健的控制评估会自动得出风险中性概率。当无限数量的风险中性措施的共存使衍生证券的统一定价无法实现时,这种方法似乎尤其适合那些包含广泛的,即说是不受限制的随机性的不完整市场。我们的方法足够强大,能够适应各种市场规则和设置,包括依赖于路径的收益,美国期权和交易成本。另一方面,它导致相当简单的数值算法。连续时间限制由非线性和/或分数Black-Scholes型方程式描述。

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