This paper studies the pricing of American carbon emission derivatives and its numerical method under the mixed fractional Brownian motion. To capture the long memory properties such as self-similarity and long-range dependence in the price process, we proposed a model based on a fractional Black–Scholes, which is more in line with the actual characteristics of the option market. We have outlined a power penalty approach using parabolic variation inequality and linear complementarity (LCP) which arises from mixed fractional Brownian motion. In addition, we introduced a nonuniform grid-based modification of the fitted finite volume method for numerical solution. Numerically, we show the impact of Hurst exponent on the pricing and analyze the convergence rates of the proposed penalty method. In conclusion, since mfBm is a well-developed mathematical model of strongly correlated stochastic processes, this model will be an efficient model for pricing carbon financial derivative.

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混合分数布朗运动下美国碳排放衍生物的定价和数值方法

本文研究了混合分数布朗运动下美国碳排放衍生产品的定价及其数值方法。为了捕获价格过程中的长期记忆特性（如自相似性和长期依赖关系），我们提出了一种基于分数Black-Scholes的模型，该模型更符合期权市场的实际特征。我们概述了使用抛物线变化不等式和线性互补性（LCP）的幂罚方法，该线性互补性是由混合分数布朗运动引起的。另外，我们为数值解引入了一种基于有限网格的非均匀网格修正方法。在数值上，我们显示了赫斯特指数对定价的影响，并分析了提出的惩罚方法的收敛速度。综上所述，