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Tham khảo tài liệu 'an introduction to financial option valuation: mathematics, stochastics and computation_14', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 24.5 Notes and references 263 L 0 Time Fig. 24.1. An example of a finite difference grid jh ik N 0 N0. Crosses mark points used by the binomial method 24.13 to obtain a single time-zero option value. finite difference schemes for such problems see Wilmott et al. 1995 for example. A promising but often overlooked alternative is to use a penalty method. Indeed the basic binomial method of Chapter 18 is an example of a simple explicit penalty method. More accurate versions are developed and analysed in Forsyth and Vetzal 2002 . Our illustration in Section 24.4 of the connection between binomial and finite difference methods was based on Appendix C of Forsyth and Vetzal 2002 . A fuller treatment of this topic can be found in Kwok 1998 . It is worth making the point that the development and implementation of numerical methods for PDEs is an area where a beginner is generally best advised to make use of existing technology off the shelf is preferable to roll your own . However a basic understanding of the nature of simple numerical methods at the level of these last two chapters gives a good feel for what to expect from PDE solvers. MATLAB comes with a fairly simple built-in PDE solver pdepe and may be augmented with a PDE toolbox. Generally there is an abundance of numerical PDE software available both commercially and in the public domain. Good places to start are the Netlib Repository www.netlib.org liblist.html and the Differential Equations and Related Topics page http www.maths.dundee. ac.uk software index.html DEs maintained by David Griffiths at the University of Dundee. 264 Finite difference methods for the Black-Scholes PDE CH24 Program for Chapter 24 Crank-Nicolson for a European put clf Problem and method parameters E 4 sigma 0.3 r 0.03 T 1 L 10 Nx 50 Nt 50 k T Nt h L Nx T1 diag ones Nx-2 1 1 - diag ones Nx-2 1 -1 T2 -2 eye Nx-1 Nx-1 diag ones Nx-2 1 1 diag ones Nx-2 1 -1 mvec 1 Nx-1 D1 diag mvec D2 diag mvec. 2 F 1-r k eye Nx-1 Nx-1 0.5 k sigma 2 D2 T2 0.5 k
