Alternating Direction Implicit (ADI) Method
Alternating Direction Implicit (ADI) Method
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交替方向隐式 (ADI) 方法
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Stéphane Villeneuve
中科院分区：
文献类型：
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作者：
Stéphane Villeneuve
在Black-Scholes模型中，期权价值被描述为欧式期权的某些偏微分方程的解或美式期权的偏微分不等式的解。一般来说，特别是美式期权，没有封闭式公式，期权价值必须用数值计算，例如使用有限差分法。不幸的是，当考虑依赖于多个资产的期权定价时，有限差分方法受到维数灾难的影响。
为了克服这个困难，人们已经知道了50多年的交替方向隐式（ADI）算法是一个有效的程序，用于解决一个大规模的线性方程组所产生的有限差分离散椭圆或抛物方程。我们在这里描述ADI方法，并讨论它们的一些特点。
保留字：
有限差分;
拆分方法;
美式期权;
多维Black-Scholes模型;
粘性解
In the Black–Scholes model option values are characterized as solutions of certain partial differential equations for European options or partial differential inequalities for American options. In general and in particular for American options, there is no closed-form formulae and the option values have to be evaluated numerically, for example using the finite difference method. Unfortunately, when considering the pricing of options depending on several assets, the finite difference method suffers from the curse of dimensionality.
To circumvent this difficulty, it has been known for more than 50 years that the alternating direction implicit (ADI) algorithm is an efficient procedure for solving a large-scale system of linear equations arising from the finite difference discretization of elliptic or parabolic equations. We describe here the ADI methods and discuss some of their features.
Keywords:
finite difference;
splitting methods;
American options;
multidimensional Black–Scholes model;
viscosity solutions