A Robust Numerical Simulation of a Fractional Black–Scholes Equation for Pricing American Options

IF 1.4 4区 物理与天体物理 Q2 MATHEMATICS, APPLIED
S. M. Nuugulu, F. Gideon, K. C. Patidar
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引用次数: 0

Abstract

After the discovery of fractal structures of financial markets, fractional partial differential equations (fPDEs) became very popular in studying dynamics of financial markets. Available research results involves two key modelling aspects; firstly, derivation of tractable asset pricing models, those that closely reflects the actual dynamics of financial markets. Secondly, the development of robust numerical solution methods. Often times, most the effective models are of a nonlinear nature, and as such, reliable analytical solution methods are seldomly available. On the other hand, the accurate value of American options strongly lies on the unknown free boundaries associated with these types of derivative contracts. The free boundaries emanates from the flexibility of the early exercise features with American options. To the best of our knowledge, the approach of pricing American options under the fractional calculus framework has not been extensively explored in literature, and an obvious wider research gap still exist on the design of robust solution methods for pricing American option problems formulated under the fractional calculus framework. Therefore, this paper serve to propose a robust numerical scheme for solving time-fractional Black–Scholes PDEs for pricing American put option problems. The proposed scheme is based on the front-fixing algorithm, under which the early exercise boundaries are transformed into fixed boundaries, allowing for a simultaneous computation of optimal exercise boundaries and their corresponding fair premiums. Results herein indicate that, the proposed numerical scheme is consistent, stable, convergent with order \({\mathcal {O}}(h^2,k)\), and also does guarantee positivity of solutions under all possible market conditions.

Abstract Image

为美式期权定价的分式布莱克-斯科尔斯方程的稳健数值模拟
在发现金融市场的分形结构之后,分形偏微分方程(fPDEs)在研究金融市场动态方面变得非常流行。现有的研究成果涉及两个关键的建模方面:第一,推导出可操作性强的资产定价模型,这些模型能密切反映金融市场的实际动态。第二,开发稳健的数值求解方法。很多时候,大多数有效模型都是非线性的,因此很少有可靠的分析求解方法。另一方面,美式期权的准确价值很大程度上取决于与这类衍生品合约相关的未知自由边界。自由边界源于美式期权提前行使特征的灵活性。据我们所知,美式期权在分式微积分框架下的定价方法还没有在文献中得到广泛的探讨,而且在为分式微积分框架下的美式期权定价问题设计稳健的求解方法方面还存在明显的研究空白。因此,本文提出了一种用于求解美式看跌期权定价问题的时间分式 Black-Scholes PDEs 的稳健数值方案。该方案基于前固定算法,将早期行使边界转化为固定边界,从而可以同时计算最优行使边界及其相应的公平权利金。本文的结果表明,所提出的数值方案是一致的、稳定的、收敛阶数为({\mathcal {O}}(h^2,k)\) 的,并且在所有可能的市场条件下都能保证解的正向性。
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来源期刊
Journal of Nonlinear Mathematical Physics
Journal of Nonlinear Mathematical Physics PHYSICS, MATHEMATICAL-PHYSICS, MATHEMATICAL
CiteScore
1.60
自引率
0.00%
发文量
67
审稿时长
3 months
期刊介绍: Journal of Nonlinear Mathematical Physics (JNMP) publishes research papers on fundamental mathematical and computational methods in mathematical physics in the form of Letters, Articles, and Review Articles. Journal of Nonlinear Mathematical Physics is a mathematical journal devoted to the publication of research papers concerned with the description, solution, and applications of nonlinear problems in physics and mathematics. The main subjects are: -Nonlinear Equations of Mathematical Physics- Quantum Algebras and Integrability- Discrete Integrable Systems and Discrete Geometry- Applications of Lie Group Theory and Lie Algebras- Non-Commutative Geometry- Super Geometry and Super Integrable System- Integrability and Nonintegrability, Painleve Analysis- Inverse Scattering Method- Geometry of Soliton Equations and Applications of Twistor Theory- Classical and Quantum Many Body Problems- Deformation and Geometric Quantization- Instanton, Monopoles and Gauge Theory- Differential Geometry and Mathematical Physics
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