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American options and semilinear parabolic partial differential equations in weighted Sobolev spaces.

机译：加权Sobolev空间中的美式期权和半线性抛物型偏微分方程。

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摘要

To value an American option as a function of time t and price of the underlying asset S is currently a major research problem in both Financial Markets and for academic purposes. Options and more general financial derivatives (also known as contingent claims) are now an important tool in risk management. One of the earliest models used in pricing derivatives is the Black-Scholes model, for which the movement in the price of the underlying asset on which the claim is based is modeled by geometric Brownian motion. Other models used for theoretical and numerical analysis of American options include the free boundary problem method, linear complimentarity problem method, and variational inequality methods. And there are others based on the Cox, Ross, and Rubinstein binomial approach.;Despite the existence of these methods, there is a strong practical demand to create new methods which firstly are more computationally efficient and make explicit the mathematical framework involved. Partial differential equations (PDEs) of parabolic type have fundamental applications to modelling processes with diffusion and uncertainty. The pricing of European options can be reduced to the calculation of certain solutions of parabolic equations, often called backward Kolmogorov's equations and obtained through Ito's lemma. In this work we are dealing with American options and we transform the Black-Scholes equation into a nonlinear parabolic equation in the entire space variable. The initial condition might be unbounded and so we strive to show the existence of the solution in some weighted Sobolev space.
机译：目前，根据时间t和基础资产S的价格对美式期权进行估价是金融市场和学术目的的主要研究问题。期权和更一般的金融衍生产品（也称为或有债权）现在是风险管理中的重要工具。用于定价导数的最早模型之一是Black-Scholes模型，基于该模型，索赔所基于的基础资产的价格变动是通过几何布朗运动建模的。用于美式期权理论和数值分析的其他模型包括自由边界问题方法，线性互补问题方法和变分不等式方法。还有其他基于Cox，Ross和Rubinstein二项式方法的方法。尽管存在这些方法，但仍然存在创建新方法的强烈实践需求，这些方法首先在计算上更有效，并且明确了所涉及的数学框架。抛物线型偏微分方程（PDE）在建模具有扩散和不确定性的过程中具有基本应用。可以将欧洲期权的定价简化为某些抛物线方程解的计算，通常被称为后向Kolmogorov方程，并通过Ito引理获得。在这项工作中，我们正在处理美国选择权，并将整个空间变量中的Black-Scholes方程转换为非线性抛物线方程。初始条件可能是不受限制的，因此我们努力证明某些加权Sobolev空间中解的存在。

著录项

  • 作者

    Muthoka, Terrence K.;

  • 作者单位

    The University of Alabama at Birmingham.;

  • 授予单位 The University of Alabama at Birmingham.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2014
  • 页码 101 p.
  • 总页数 101
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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