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摘要翻译：
本综述章的目的是提出一种转换技术，可用于分析和数值计算美国式香草期权的早期行使边界，该期权可由一类广义Black-Scholes方程建模。定性和定量地分析了具有波动率系数的线性和非线性Black-Scholes方程的早期行权边界，波动率系数可以是期权价格本身的二阶导数的非线性函数。研究具有非线性波动性的非线性Black-Scholes方程的动机来自于考虑非平凡交易成本、投资者偏好、反馈和非流动性市场效应以及波动（无保护）投资组合风险的期权定价模型。本文提出了一种将早期运动边界位置的自由边界问题转化为一个定域上的非线性非局部抛物方程的解的方法。我们进一步提出了一个迭代数值格式，可以用来寻找自由边界的近似。对于线性Black-Scholes方程，我们可以导出自由边界位置的非线性积分方程。本文给出了各种线性和非线性Black-Scholes方程早期运动边界的数值逼近结果，并讨论了自由边界对模型参数的依赖关系。最后讨论了转换方法在美式亚式期权定价方程中的应用。
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英文标题：
《Transformation methods for evaluating approximations to the optimal
exercise boundary for linear and nonlinear Black-Scholes equations》
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作者：
Daniel Sevcovic
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最新提交年份：
2008
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分类信息：
一级分类：Quantitative Finance 数量金融学
二级分类：Computational Finance 计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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一级分类：Mathematics 数学
二级分类：Analysis of PDEs 偏微分方程分析
分类描述：Existence and uniqueness, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDE's, conservation laws, qualitative dynamics
存在唯一性，边界条件，线性和非线性算子，稳定性，孤子理论，可积偏微分方程，守恒律，定性动力学
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一级分类：Mathematics 数学
二级分类：Numerical Analysis 数值分析
分类描述：Numerical algorithms for problems in analysis and algebra, scientific computation
分析和代数问题的数值算法，科学计算
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英文摘要：
The purpose of this survey chapter is to present a transformation technique that can be used in analysis and numerical computation of the early exercise boundary for an American style of vanilla options that can be modelled by class of generalized Black-Scholes equations. We analyze qualitatively and quantitatively the early exercise boundary for a linear as well as a class of nonlinear Black-Scholes equations with a volatility coefficient which can be a nonlinear function of the second derivative of the option price itself. A motivation for studying the nonlinear Black-Scholes equation with a nonlinear volatility arises from option pricing models taking into account e.g. nontrivial transaction costs, investor's preferences, feedback and illiquid markets effects and risk from a volatile (unprotected) portfolio. We present a method how to transform the free boundary problem for the early exercise boundary position into a solution of a time depending nonlinear nonlocal parabolic equation defined on a fixed domain. We furthermore propose an iterative numerical scheme that can be used in order to find an approximation of the free boundary. In the case of a linear Black-Scholes equation we are able to derive a nonlinear integral equation for the position of the free boundary. We present results of numerical approximation of the early exercise boundary for various types of linear and nonlinear Black-Scholes equations and we discuss dependence of the free boundary on model parameters. Finally, we discuss an application of the transformation method for the pricing equation for American type of Asian options.
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PDF链接：
https://arxiv.org/pdf/0805.0611
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