﻿ 分数布朗运动环境下的资产配置策略多期收益保证价值的测算 Pricing Multi-Period Return Guarantees Combined with Asset Allocation Strategy under Fractional Brownian Motion

Finance
Vol.06 No.02(2016), Article ID:17377,10 pages
10.12677/FIN.2016.62007

Pricing Multi-Period Return Guarantees Combined with Asset Allocation Strategy under Fractional Brownian Motion

Yanlian Deng, Yunsheng Lun

Department of Mathematics, Donghua University, Shanghai

Received: Apr. 3rd, 2016; accepted: Apr. 14th, 2016; published: Apr. 21st, 2016

ABSTRACT

In this paper, we consider that the price processes of risky assets are driven by fractional Brownian motion (). With the Wick-Itô integral and the quasi-conditional expectation, we compute the value of multi-period return guarantees under CM strategy and under CPPI strategy. Through the numerical simulation, the influence on the value of multi-period return guarantees under the two strategies is compared and analyzed, which is made by the periods of multi-period return guarantees and the important parameters of the financial market and asset allocation strategy.

Keywords:Fractional Brownian Motion, Quasi-Conditional Expectation, CM Strategy, CPPI Strategy

1. 引言

2. 准备知识

2.1. 分数布朗运动

(1)为分数Hida检验泛函空间，若对任意的

(2)为分数Hida分布空间，若对任意的都具有如下展开形式：

，满足对任意的，有，则存在唯一的使得对任意的

(1)

，定义Wick积如下：

，若可表示为

，定义关于自然域流的拟条件期望为：

2.2. 金融市场

.

[6] 中指出，在分数Black-Scholes市场中虽然不存在等价鞅测度，但存在唯一的等价测度，即风险中性测度。上式中的市场瞬时无风险利率由Vasicek模型刻画，即满足：

.

2.3. 多期收益保证水平的设定

， (2)

3. 资产配置策略

3.1. CM策略

。 (3)

.

(4)

，则根据Wick-Itô积分易知，

3.2. CPPI策略

1987年，Black和Jones最先提出固定比例投资组合保险策略(constant proportion portfolio insurance)，简称CPPI策略。目前，CPPI策略以模型简单，参数的设定能充分反映不同投资人的风险偏好，且易于操作与实施等特点备受大型收益保证性金融产品投资机构的青睐。当资产配置策略为CPPI策略时，投资于风险性资产的比例(策略函数)与t时刻资产组合的市场价值相关，即，则

4. 数值分析

Table 1. Simulating parameters

Figure 1. The influence of volatility

Figure 2. The influence of return guarantees

Figure 3. The influence of parameter in CM strategy

Figure 4. The influence of parameters in CPPI strategy

Pricing Multi-Period Return Guarantees Combined with Asset Allocation Strategy under Fractional Brownian Motion[J]. 金融, 2016, 06(02): 64-73. http://dx.doi.org/10.12677/FIN.2016.62007

1. 1. Brennan, J.M. and Schwartz, E.S. (1976) The Pricing of Equity-Linked Life Insurance Policies with an Asset Value Guarantee. Journal of Financial Economics, 3, 195-213. http://dx.doi.org/10.1016/0304-405X(76)90003-9

2. 2. 张飞, 刘海龙. 价格跳跃风险下CPPI策略多期收益保证价值的测算[J]. 系统工程理论与实践, 2014, 34(8): 1944- 1951.

3. 3. 王亦奇, 刘海龙. 结合资产配置策略测算多期收益保证价值[J]. 管理科学学报, 2011, 14(11): 42-51.

4. 4. Black, F. and Scholes, M. (1973) The Pricing of Options and Gorporate Liabilities. Journal of Political Economy, 81, 637-654. http://dx.doi.org/10.1086/260062

5. 5. Shiryaev, A.N. (1999) Essentials of Stochastic Finance. World Scientific, Singapore.

6. 6. Rogers, L. (1997) Arbitrage with Fractional Brownian Motion. Mathematical Finance, 7, 95-105. http://dx.doi.org/10.1111/1467-9965.00025

7. 7. Bjork, T. and Hult, H. (2005) A Note on Wick Products and the Fractional Black-Scholes Model. Finance and Stochastics, 9, 197-209. http://dx.doi.org/10.1007/s00780-004-0144-5

8. 8. Sottinen, T. (2001) Fractional Brownian Motion, Random Walks and Binary Market Models. Finance and Stochastics, 5, 343-355. http://dx.doi.org/10.1007/PL00013536

9. 9. Necula, C. (2002) Option Pricing in a Fractional Brownian Motion Environment. Work Papers. http://dx.doi.org/10.2139/ssrn.1286833

10. 10. Hu, Y. and Oksendal, B. (2003) Fractional White Noise Calculus and Applications to Finance. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 6, 1-32. http://dx.doi.org/10.1142/S0219025703001110

11. 11. Biagini, F. and Hu, Y., Øksendal, B. and Zhang, T. (2008) Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, Berlin.

12. 12. Rostek, S. and Schobel, R. (2006) Risk Preference Based Option Pricing in a Fractional Brownian Market. Tubinger Diskussinsbeitrag, Tu-ebingen.