簡易檢索 / 詳目顯示

回結果列表
研究生： 彭宇晨
Peng, Yu-Chen
論文名稱： 布萊克-休斯美式期權方程的高階緊緻方法
High Order Compact Methods for Black-Scholes Equation of American Options
指導教授： 王偉成
Wang, Wei-Cheng
口試委員: 鄭志豪
Teh, Jyh-Haur
韓傳祥
Han, Chuan-Hsiang
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2018
畢業學年度： 106
語文別： 英文
論文頁數： 33
中文關鍵詞： 布萊克-休斯方程 、美式買權 、自由邊界問題 、高階緊緻方法 、漸近展開
外文關鍵詞： Black-Scholes equation, American call option, free boundary problem, high order compact method, asymptotic expansion
相關次數： 點閱：2  下載：0 
分享至:
查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報

    • 在這篇論文中，我們將介紹布萊克-休斯方程（美式買權）的高階緊緻方法。我們先計算自由邊界上的函數跳躍條件。藉由跳躍條件，我們可以更精準地找出自由邊界的位置。我們首先介紹二階收歛的方法，然後推廣至高階緊緻的方法。

    In this thesis, we introduce high order compact methods for the Black-Scholes equation of American call options. We compute some jump conditions depend on the free boundary. By applying the jump conditions, we can locate the free boundary more precisely. At first we would introduce a second order method, and then attempt to a higher order method with compact scheme.

    1 Introduction . . . . . . . . . . . . . . . . . . . . . 1 2 The Black-Scholes equation of the American call option 2 2.1 Problem state . . . . . . . . . . . . . . . . . . . 2 2.2 Some properties of the solution . . . . . . . . . . 3 3 Introduction of the method of Han & Wu . . . . . . . . 4 4 Second order modifi ed method . . . . . . . . . . . . . 7 4.1 Second order backward di fferentiation formula (BDF2) 7 4.2 Modi fied value of right boundary condition . . . . . 7 4.3 The case while Sb() crossing the space lattice . . . 10 4.4 Some problems near the expiry . . . . . . . . . . . 11 4.5 Algorithm . . . . . . . . . . . . . . . . . . . . . 12 4.6 Numerical results . . . . . . . . . . . . . . . . . 14 5 Remove singularity by asymptotic expansion . . . . . . 17 6 High order compact method (3rd order) . . . . . . . . 21 6.1 Fundamental scheme . . . . . . . . . . . . . . . . . 21 6.2 Jump values of the interface . . . . . . . . . . . . 22 6.3 Idea of first 3 time steps . . . . . . . . . . . . . 24 6.4 Numerical results for 3rd order . . . . . . . . . . 24 7 4th order scheme . . . . . . . . . . . . . . . . . . . 25 7.1 Fundamental scheme . . . . . . . . . . . . . . . . . 25 7.2 Jump value of the interface . . . . . . . . . . . . 25 7.3 Equation for variable d . . . . . . . . . . . . . . 26 7.4 Idea of fi rst 4 time steps . . . . . . . . . . . . . 27 7.5 Numerical result for 4th order . . . . . . . . . . . 28 8 Graphs of solutions . . . . . . . . . . . . . . . . . 29 9 Conclusion . . . . . . . . . . . . . . . . . . . . . . 32

    [1] H. Han and X. Wu, A fast numerical method for the Black-Scholes equation of American options, SIAM J. Numer. Anal., 41 (2003), pp. 2081-2095.
    [2] M. Ehrhardt and R. E. Mickens, A fast, stable and accurate numerical method for the Black-Scholes equation of American options, International J. of Theoretical and Applied Finance, 11 (2008), pp. 471-501.
    [3] P. A. Forsyth and K. R. Vetzal, Quadratic convergence for valuing American options using a penalty method, SIAM J. Scienti c Computing, 23 (2002), pp. 2095-2122.
    [4] P. Wilmott, J. Dewynne, and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, UK, 1993.
    [5] G. Alobaidi, American options and their strategies, The University of Western Ontario, London, 2000.
    [6] G. Alobaidi and R. Mallier, Asymptotic analysis of American call options, IJMMS 27 (2001), pp. 177-188.
    [7] J. D. Evans, R. Kuske, and Joseph B. Keller, American options on assets with dividends near expiry, Math. Finance, 12 (2002), pp. 219-237.
    [8] S. Kazemi, M. Dehghan, and A. F. Bastani, Asymptotic expansion of solutions to the Black-Scholes equation arising from American option pricing near the expiry, Journal of Computational and Applied Mathematics, 311 (2017), pp. 11-37.
    [9] J. Goodman and D. N. Ostrov, On the early exercise boundary of the American put option, SIAM J. Appl. Math. 62 (2002), pp. 1823-1835.
    [10] X. Chen and J. Chadam, Mathematical analysis of the optimal exercise boundary for American put options, SIAM J. Math. Anal. 38 (2007), pp. 1613-1641.
    [11] L. Wu, Y. K. Kwok, A front- xing nite di erence method for the valuation of American options, J. Financial Eng. 6 (2)(1997), pp. 83-97
    [12] D. Y. Tangman, A. Gopaul, and M. Bhuruth, Numerical pricing of options using high-order compact nite di erence schemes, Journal of Computational and Applied Mathematics, 218 (2008), pp. 270-280.
    [13] Merton, R. C. (1973): Theory of Rational Option Pricing, Bell J. Econ. Mgt. Sci., 4, pp.141-183
    [14] Jacka, S. D. (1991): Option Stopping and the American Put, Math Finance, 1, pp.1-14

    QR CODE