﻿ 多资产期权确定最佳实施边界问题的研究 Research on the Implementation of the Optimal Implementation of the Multi-Asset Option

Pure Mathematics
Vol.06 No.06(2016), Article ID:19111,31 pages
10.12677/PM.2016.66068

Research on the Implementation of the Optimal Implementation of the Multi-Asset Option

Xiaoqing Wu

College of Science, Southwest Petroleum University, Chengdu Sichuan

Received: Nov. 10th, 2016; accepted: Nov. 24th, 2016; published: Nov. 30th, 2016

ABSTRACT

In this paper, we study the problem of determining the optimal implementation boundary of multi- asset option, and establish a mathematical model of multidimensional Black-Scholes equation with singular inner boundary function vector In multi-dimension region, the option price function is an unknown function. The exact solution of the mathematical model is obtained by using the matrix theory and the generalized characteristic function method. And the exponential function vector expression of the singular inner boundary is obtained. It is demonstrated that: when any, the maximum value of the solution of the region is obtained on the singular boundary, namely. The free boundary problem A and free boundary problem B of Black-Scholes equation are solved. The free boundary of problem A and B is expressed by the function vector. The free boundary of the problem A and problem B coincides with the singular inner boundary. So the vector expression of the exponential function is the best implementation of the boundary. The exponential function vector satisfies the condition; and is calculated by; the formula shows that is only determined by all the parameters appearing in the multidimensional Black-Scholes equation.

Keywords:Multi-Asset Option, Best Implementation Boundary, Free Boundary Problem, Multi-Dimension Black-Scholes Equation

1. 引言

(01)

(02)

(03)

2. 主要结果

2.1. 多资产期权的数学模型I的研究

(4)

(5)

(6)

2.1.1. Black-Scholes方程数学模型I的求解

(7)

1) 正线下三角矩阵的行列式

2) 由唯一确定；由唯一确定

3) 记，则为正线上三角矩阵，

；由唯一确定

4) 记；则当时，有。从而当，有，有

(10)

(11)

(12)

(13)

(14)

(15)

，作的线性变换

(16)

(17)

(18)

(19)

(20)

(21)

m维Euler方程在半无界区域的特征值问题II

(25)

(26)

，由引理1.1矩阵为正线上三角矩阵。

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

(58)式即(54)式。引理证毕。

(59)

(60)

(61)

(62)

(63)

(64)

(65)

(66)

(67)

(68)

(69)

(70)

(71)

(72)

(73)

(74)

(75)

(76)

(77)

(78)

1)为正定对称矩阵，

2)为充分光滑的单调函数；

3)

(79)

(80)

(81)

2.1.2. 多维Black-Scholes方程奇异内边界的确定

，使其满足

1)为正定对称矩阵；

2)为充分光滑的单调函数；

3)

(89)

(90)

(91)

(92)

(93)

(94)

(95)

(96)

(97)

(98)

(99)

(100)

1) 当

(101)

2) 当

(102)

(103)

(104)

(105)

(106)

(107)

(108)

2) 充分性，若(104)式成立，即有

(111)

(112)

(113)

(114)

(115)

(116)

(117)

(118)

(119)

(120)

(121)

1) 当；数学模型I.1的解

(122)

(123)

(124)

(125)

2) 当，则数学模型I.1的解

(126)

(127)

3) 当，数学模型I.1的解

(128)

(129)

(130)

(131)

(132)

(133)

(134)

(135)

(136)

(137)

。由(132)对关于求偏导得到

(138)

(139)

(140)

(141)

；积分变量，有

(142)

(143)

；由(142)，有，引理1.1的结论4)即有

(144)

(145)

，从而有成立。

(146)

，有,

2.2. 关于多维Black-Scholes方程的自由边界问题的研究

1)为正定矩阵；

2)

(152)

(153)

(156)

(159)

(160)

1)为正定矩阵；

2)

(166)

(167)

(170)

(171)

(172)

(173)

2.3. 数学模型III与自由边界问题A和问题B的关系

，使其满足

1)为正定矩阵；

2)

3)

4)

5)

(181)

(182)

(183)

1)为正定矩阵；

2)

3)

4)

5)

(186)

(187)

1)为正定矩阵；

2)

3)

4)

5)

(190)

(191)

(192)

(193)

(194)

(195)

3. 结论

Research on the Implementation of the Optimal Implementation of the Multi-Asset Option[J]. 理论数学, 2016, 06(06): 496-526. http://dx.doi.org/10.12677/PM.2016.66068

1. 1. 姜礼尚. 期权定价的数学模型和方法[M]. 第2版. 北京: 高等教育出版社, 2008.

2. 2. 姜礼尚, 徐承龙, 任学敏, 李少华. 金融数学丛书:金融衍生产品定价的数学模型与案例分析[M]. 第2版. 北京: 高等教育出版社, 2013.

3. 3. 任学敏, 魏嵬, 姜礼尚, 梁进著. 信用风险估值的数学模型与案例分析[M]. 北京: 高等教育出版社, 2014.

4. 4. 姜礼尚. 金融衍生产品定价的数学模型与案例分析[M]. 北京: 高等教育出版社, 2008.

5. 5. 黄文礼, 李胜宏. 分数布朗运动驱动下带比例交易成本的期权定价[J]. 高校应用数学学报, 2011, 26(2): 201-208.

6. 6. 冯骅. 金融产品及其衍生物价格的数学模型研究[D]: [硕士学位论文]. 北京: 中国石油大学(北京), 2011.

7. 7. 袁世冉. 对一篮子期权定价模型的研究[D]: [硕士学位论文]. 秦皇岛: 燕山大学, 2014.

8. 8. 胡宗义. 投资选择及资产定价数学模型研究[D]: [博士学位论文]. 长沙: 湖南大学, 2004.

9. 9. 周杲昕. 多资产期权定价模型的数值新方法研究[D]: [硕士学位论文]. 北京: 华北电力大学, 2013.

10. 10. 黎伟, 周圣武, Li Wei, Zhou Shengwu. 带交易费的多资产期权定价模型及数值解法[J]. 江苏师范大学学报(自然科学版), 2011, 29(2): 53-56.

11. 11. 刘莉. 互换与极值期权定价的树网格法[D]: [硕士学位论文]. 桂林: 广西师范大学, 2008.

12. 12. 许德志. 若干期权定价模型的数值新方法研究[D]: [硕士学位论文]. 北京: 华北电力大学(北京), 2010.

13. 13. 张帆. 两类期权定价模型有限差分的并行计算[D]: [硕士学位论文]. 北京: 华北电力大学, 2014.

14. 14. Broadie, M and Detemple, J. (1996) American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods. Review of Financial Studies, 9, 1211-1250. https://doi.org/10.1093/rfs/9.4.1211

15. 15. Figlewski, S. and Gao, B. (1999) The Adaptive Mesh Model: A New Approach to Efficient Option Pricing. Journal of Financial Economics, 53, 313-351. https://doi.org/10.1016/S0304-405X(99)00024-0

16. 16. Figlewski, S. and Gao, B. (1998) The Adaptive Mesh Model: A New Approach to Efficient Option Pricing. New York University, Leonard N. Stern School of Business, New York.

17. 17. Whaley, R.E. (1981) On the Valuation of American Call Options on Stocks with Known Dividends. Journal of Financial Economics, 9, 207-211. https://doi.org/10.1016/0304-405X(81)90013-1

18. 18. Ankudinova, J. and Ehrhardt, M. (2008) On the Numerical Solution of Nonlinear Black-Scholes Equations. Computers & Mathematics with Applications, 56, 799-812. https://doi.org/10.1016/j.camwa.2008.02.005

19. 19. Lötstedt, P., Persson, J. and Sydow, L.V. (2007) Space-Time Adaptive Finite Difference Method for European Multi-Asset Options. Computers & Mathematics with Applications, 53, 1159-1180. https://doi.org/10.1016/j.camwa.2006.09.014

20. 20. Goodman, V. and Stampfli, J. (2001) The Mathematics of Finance: Modeling and Hedging. Pure and Applied Undergraduate Texts, 7, 250.

21. 21. Broadie, M. and Detemple, J. (1995) American Capped Call Options on Dividend-Paying Assets. Review of Financial Studies, 8, 161-191. https://doi.org/10.1093/rfs/8.1.161

22. 22. 史荣昌. 矩阵分析[M]. 北京: 北京理工大学出版社, 1996.

23. 23. 吴小庆. 数学物理方程及其应用[M]. 北京: 科学出版社, 2008.

24. 24. 吴小庆. 偏微分方程理论与实践[M]. 北京: 科学出版社, 2009.