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非流动性市场对冲模型的对称性约化与求解

文章 arXiv papers · 作者: Ljudmila A. Bordag et al.
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总结

这项数学研究分析了一个由Schoenbucher和Wilmott最初提出的非流动性市场自融资对冲一般模型。在该模型中，对冲策略满足一个非线性偏微分方程，其系数函数与投资者效用函数相关。作者使用李对称性分析刻画该方程的对称性，并将其约化为常微分方程。

论文对允许额外对称性的系数函数进行分类，将由此得到的三个特例分别与早期模型或一种新情形联系起来，并为一般模型和新特例模型推导最优子代数系统及约化形式。论文还给出了新特例的显式解，包括幂次型衍生品的解。这些是关于模型结构和可解性的分析结果；所提供的描述未报告实证检验、市场数据，也未提供这些解能改善实际对冲结果的证据。因此，该文主要有助于理解理论框架及其约化形式，实际适用性尚未评估。

核心观点

  • 模型中的自融资对冲策略满足非线性PDE，其系数与效用函数相关。
  • 研究使用李对称性分析描述对称性，并将PDE约化为常微分方程。
  • 论文对允许扩展对称性的系数函数进行分类，并将特例与既有模型联系起来。
  • 研究为一种新的特例模型推导了显式解，包括幂次型衍生品的解。
  • 所述贡献属于数学研究；文中未报告实证评估。

标签

期权衍生品定价风险管理统计学
全文
# Models of self-financing hedging strategies in illiquid markets: symmetry reductions and exact solutions


# Models of self-financing hedging strategies in illiquid markets: symmetry reductions and exact solutions









We study the general model of self-financing trading strategies in illiquid markets introduced by Schoenbucher and Wilmott, 2000. A hedging strategy in the framework of this model satisfies a nonlinear partial differential equation (PDE) which contains some function g(alpha). This function is deep connected to an utility function. We describe the Lie symmetry algebra of this PDE and provide a complete set of reductions of the PDE to ordinary differential equations (ODEs). In addition we are able to describe all types of functions g(alpha) for which the PDE admits an extended Lie group. Two of three special type functions lead to models introduced before by different authors, one is new. We clarify the connection between these three special models and the general model for trading strategies in illiquid markets. We study with the Lie group analysis the new special case of the PDE describing the self-financing strategies. In both, the general model and the new special model, we provide the optimal systems of subalgebras and study the complete set of reductions of the PDEs to different ODEs. In all cases we are able to provide explicit solutions to the new special model. In one of the cases the solutions describe power derivative products.

在遵守原作品许可的前提下，附作者信息全文展示。 许可协议: abstract CC0

此摘要由 Stratmill 研究智能体根据原文撰写，并非原文副本。