SWR算法在期权定价中的应用
|
摘要
本文首先借鉴Schwarz波形松弛算法求解热传导方程的思路分析了SWR算法在欧式期权定价问题中的可行性,然后将欧式看涨期权定价问题转换为一类热传导方程的初-边值问题,通过Schwarz迭代方法获得了相应的误差方程,进而给出了算法误差的收敛性结果及算法的流程.数值实验表明,与经典的欧式期权定价公式相比,本算法具有更好的估计效果.
In this paper,inspired by the idea of solving heat conduction equation by means of Schwarz waveform relaxation algorithm, we analyze the feasibility of applying SWR algorithm in European option pricing. After transforming European call option pricing into a class of initial boundary value problems for heat conduction equation, error functions are obtained with Schwarz iterative method. Then the convergence result of algorithm error and the flow diagrams of algorithm are given. It is showed by a numerical experiment that this algorithm has better estimation effect compared with the classical European option pricing formula.
In this paper,inspired by the idea of solving heat conduction equation by means of Schwarz waveform relaxation algorithm, we analyze the feasibility of applying SWR algorithm in European option pricing. After transforming European call option pricing into a class of initial boundary value problems for heat conduction equation, error functions are obtained with Schwarz iterative method. Then the convergence result of algorithm error and the flow diagrams of algorithm are given. It is showed by a numerical experiment that this algorithm has better estimation effect compared with the classical European option pricing formula.
引文
[1] Gander M J,Zhao H, Overlapping Schwarz waveform relaxation for parabolic problems in higher dimension [J]. Proceedings of Algorithm, 1997, 14: 42.
[2] Burrage K, Parallel and sequential methods for ordinary differential equations [M]. New York: Clarendon Press, 1995.
[3] Jeltsch R, Pohl B,Waveform relaxation with overlapping splittings [J]. SIAM J Sci Comput, 1995, 16: 40.
[4] BellenA, Zennaro B, The use of Runge-Kutta formulae in waveform relaxation methods [J]. Appl Numer Math, 1993, 11: 95.
[5] Bjárhus M, A note on the convergence of discretized dynamic iteration [J]. BIT Numer Math, 1995, 35: 291.
[6] Gander M J, Stuart A M, Space-time continuous analysis of waveform relaxation for the heat equation [J]. SIAM J Sci Comput, 1998, 19: 2014.
[7] Gander M J, Halpern L, Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems [J]. SIAM J Numer Anal, 2007, 45: 666.
[8] Bennequin D, Gander M J, Halpern L, A homographic best approximation problem with application to optimized Schwarz waveform relaxation [J]. Math Comput, 2009, 78: 185.
[9] Gander M J, A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations [J]. Numer Linear Algebra Appl, 1999, 6: 125.
[10] MartinV, Schwarz waveform relaxation algorithms for the linear viscous equatorial shallow water equations [J]. SIAM J Sci Comput, 2009, 31: 3595.
[11] Bouajaji M E, Dolean V, Gander M J, et al. Optimized Schwarz methods for the time-harmonic Maxwell equations with damping [J]. SIAM J Sci Comput, 2012, 34: A2048.
[12] Stampfli J, Goodman V, The mathematics of finance: modeling and hedging [M]. Providence: American Mathematical Society, 2001.
[2] Burrage K, Parallel and sequential methods for ordinary differential equations [M]. New York: Clarendon Press, 1995.
[3] Jeltsch R, Pohl B,Waveform relaxation with overlapping splittings [J]. SIAM J Sci Comput, 1995, 16: 40.
[4] BellenA, Zennaro B, The use of Runge-Kutta formulae in waveform relaxation methods [J]. Appl Numer Math, 1993, 11: 95.
[5] Bjárhus M, A note on the convergence of discretized dynamic iteration [J]. BIT Numer Math, 1995, 35: 291.
[6] Gander M J, Stuart A M, Space-time continuous analysis of waveform relaxation for the heat equation [J]. SIAM J Sci Comput, 1998, 19: 2014.
[7] Gander M J, Halpern L, Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems [J]. SIAM J Numer Anal, 2007, 45: 666.
[8] Bennequin D, Gander M J, Halpern L, A homographic best approximation problem with application to optimized Schwarz waveform relaxation [J]. Math Comput, 2009, 78: 185.
[9] Gander M J, A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations [J]. Numer Linear Algebra Appl, 1999, 6: 125.
[10] MartinV, Schwarz waveform relaxation algorithms for the linear viscous equatorial shallow water equations [J]. SIAM J Sci Comput, 2009, 31: 3595.
[11] Bouajaji M E, Dolean V, Gander M J, et al. Optimized Schwarz methods for the time-harmonic Maxwell equations with damping [J]. SIAM J Sci Comput, 2012, 34: A2048.
[12] Stampfli J, Goodman V, The mathematics of finance: modeling and hedging [M]. Providence: American Mathematical Society, 2001.