The Damped Crank–Nicolson Time-Marching Scheme for the Adaptive Solution of the Black–Scholes Equation

IF 0.8 4区经济学 Q4 BUSINESS, FINANCE

Journal of Computational Finance Pub Date : 2015-04-30 DOI:10.21314/JCF.2015.301

C. Goll, R. Rannacher, W. Wollner

引用次数: 22

Abstract

This paper is concerned with the derivation of a residual-based a posteriori error estimator and mesh-adaptation strategies for the space-time finite element approximation of parabolic problems with irregular data. Typical applications arise in the field of mathematical finance, where the Black–Scholes equation is used for modeling the pricing of European options. A conforming finite element discretization in space is combined with second-order time discretization by a damped Crank–Nicolson scheme for coping with data irregularities in the model. The a posteriori error analysis is developed within the general framework of the dual weighted residual method for sensitivity-based, goal-oriented error estimation and mesh optimization. In particular, the correct form of the dual problem with damping is considered.

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Black-Scholes方程自适应解的阻尼Crank-Nicolson时间推进格式

本文研究了基于残差的后验误差估计器的推导和网格自适应策略，用于不规则数据抛物型问题的时空有限元逼近。典型的应用出现在数学金融领域，其中布莱克-斯科尔斯方程被用于建模欧式期权的定价。采用阻尼的Crank-Nicolson格式将空间上的一致性有限元离散与二阶时间离散相结合，以处理模型中的数据不规则性。后验误差分析是在双加权残差法的一般框架内发展起来的，用于基于灵敏度的、面向目标的误差估计和网格优化。特别地，考虑了带阻尼的对偶问题的正确形式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Computational Finance BUSINESS, FINANCE-

CiteScore

0.90

自引率

0.00%

发文量

期刊介绍： The Journal of Computational Finance is an international peer-reviewed journal dedicated to advancing knowledge in the area of financial mathematics. The journal is focused on the measurement, management and analysis of financial risk, and provides detailed insight into numerical and computational techniques in the pricing, hedging and risk management of financial instruments. The journal welcomes papers dealing with innovative computational techniques in the following areas: Numerical solutions of pricing equations: finite differences, finite elements, and spectral techniques in one and multiple dimensions. Simulation approaches in pricing and risk management: advances in Monte Carlo and quasi-Monte Carlo methodologies; new strategies for market factors simulation. Optimization techniques in hedging and risk management. Fundamental numerical analysis relevant to finance: effect of boundary treatments on accuracy; new discretization of time-series analysis. Developments in free-boundary problems in finance: alternative ways and numerical implications in American option pricing.