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Analytically pricing double barrier options based on a time-fractional Black-Scholes equation

机译:基于时间分数Black-Scholes方程对双障碍期权进行解析定价

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This paper investigates the pricing of double barrier options when the price change of the underlying is considered as a fractal transmission system. In this scenario, the option price is governed by a modified Black-Scholes equation with a time-fractional derivative. In comparison with standard derivatives of integer order, the fractional-order derivatives are characterized by their "globalness", i.e., the rate of change of a function near a point is affected by the property of the function defined in the entire domain of definition rather than just near the point itself. The existence of the time-fractional derivative, in conjunction with the presence of two barriers of the double-barrier options, has added an additional degree of difficulty not only when a purely numerical solution is sought but also when an analytical method is attempted. Albeit difficult, we have managed to find an explicit closed-form analytical solution for double-barrier options, which has been taken to price the single barrier options and European path-independent options under the same framework as a special case of the current solution. In addition, not only have we provided a theoretical proof for the convergence of the newly-found analytic solution in series form, which is a vital step to show the closeness of our solution, we have also proposed an efficient numerical evaluation technique to facilitate the implementation of our formula so that it can be easily used in trading practice. Crown Copyright (C) 2015 Published by Elsevier Ltd. All rights reserved.
机译:当标的物的价格变化被视为分形传递系统时,本文研究了双重障碍期权的定价。在这种情况下,期权价格由带有时间分数导数的修正Black-Scholes方程控制。与整数阶的标准导数相比,分数阶导数的特征在于它们的“全局性”,即,在一个点附近的函数的变化率受在整个定义域中定义的函数性质的影响,而不是仅仅靠近点本身。时间分数导数的存在,加上双障碍选择的两个障碍的存在,不仅在寻求纯数值解时而且在尝试分析方法时也增加了额外的难度。尽管很困难,但我们设法找到了一种针对双重障碍期权的显式封闭式分析解决方案,该解决方案已在与当前解决方案的特殊情况相同的框架下为单个障碍期权和与欧洲路径无关的期权定价。此外,我们不仅为串联形式新发现的解析解的收敛提供了理论证明,这是证明我们的解的逼近性的重要步骤,而且我们还提出了一种有效的数值评估技术,以方便进行求解。实施我们的公式,以便可以轻松地在交易实践中使用。 Crown版权所有(C)2015,由Elsevier Ltd.发行。保留所有权利。

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