非线性Black-Scholes方程最优执行边界近似评估的迭代算法
An iterative algorithm for evaluating approximations to the optimal exercise boundary for a nonlinear Black-Scholes equation
AI总结：
针对非线性Black-Scholes方程，提出一种将自由边界问题转化为固定域上非线性抛物型方程的方法，并设计迭代数值方案近似求解最优执行边界。
摘要：
本文旨在分析和计算一类非线性Black-Scholes方程的最优执行边界，其中非线性波动率可以是期权价格二阶导数的函数。研究非线性Black-Scholes方程的背景源于考虑非平凡交易成本、投资者偏好、反馈效应、非流动性市场影响以及波动（未保护）组合风险的期权定价模型。我们提出了一种新方法，将最优执行边界的自由边界问题转化为定义在固定域上的时变非线性抛物型方程的解。此外，我们提出了一种迭代数值方案，可用于近似求解自由边界。我们给出了不同类型非线性Black-Scholes方程的最优执行边界的数值近似结果，并讨论了自由边界对各种模型参数的依赖性。
英文摘要：
The purpose of this paper is to analyze and compute the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility which can be a 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 new method how to transform the free boundary problem for the early exercise boundary position into a solution of a time depending nonlinear parabolic equation defined on a fixed domain. We furthermore propose an iterative numerical scheme that can be used to find an approximation of the free boundary. We present results of numerical approximation of the early exercise boundary for various types of nonlinear Black--Scholes equations and we discuss dependence of the free boundary on various model parameters.
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