上证指数收益率的长相依性分析及其欧式期权定价

华东师范大学学报(自然科学版) ›› 2014, Vol. 2014 ›› Issue (3): 14-22.

上证指数收益率的长相依性分析及其欧式期权定价

徐林¹，徐婷²

1. 安徽师范大学数学计算机科学学院，芜湖 241003； 2. 华东师范大学金融统计学院，上海，200246

收稿日期:2013-08-24 修回日期:2013-10-24 出版日期:2014-05-25 发布日期:2014-07-25

Long range dependence of Shanghai stock market and pricing of European option

XU Lin¹, XU Ting²

1. School of Mathematics and Computer Science, Wuhu, 241003, China; 2. School of Finance and Statistics, Shanghai,200246, China

Received:2013-08-24 Revised:2013-10-24 Online:2014-05-25 Published:2014-07-25

摘要/Abstract

摘要： 对上证指数对数收益率的长相依性进行了统计检验并完成了相应的统计建模以及参数估计, 就所得的模型完成了在此模型下的欧式期权定价设计, 比较了具有长相依性质的模型下期权定价与经典的Black-Scholes 模型下期权定价的不同点, 分析了长相依性质对于期权定价的影响. 统计检验采用的是经典的R/S分析法和修正R/S分析法, 通过对上证指数收益率的时间序列进行了实证分析, 发现上证指数体现出长相依性质. 数值分析的结果显示分数布朗运动模型下欧式期权定价与经典Black-Scholes期权定价有很大的不同点, 主要表现在分数布朗运动模型下的定价从时间的角度来看表现得更为平稳.

关键词: 收益率, 长相依性, R/S分析法, Hurst指数, 分数布朗运动

Abstract: This paper aims at the empirical statistical test of the long range dependence of Shanghai Composite Index and finishes statistical modeling for corresponding financial data and derives the valuation formula for European call option under the model we proposed. Theoretical analysis and numerical examples are given to illustrate the impact of the long range dependence on the option pricing by comparing the option valuation formula under fractional Brownian motion (fBm for short) model and the one under Black-Scholes model. The main statistics we adopted are R/S and modified R/S statistics. Numerical results show that the valuation formula under fBm are much more stable than the one under Black-Scholes model due to the former one has the long range dependence property.

Key words: Log-return rate, Long-range dependence, Classical Rescaled Range Analysis, Hurst Exponent, Fractional Brownian Motion

中图分类号:

O211.6

徐林，徐婷. 上证指数收益率的长相依性分析及其欧式期权定价[J]. 华东师范大学学报(自然科学版), 2014, 2014(3): 14-22.

XU Lin, XU Ting. Long range dependence of Shanghai stock market and pricing of European option[J]. Journal of East China Normal University(Natural Sc, 2014, 2014(3): 14-22.

参考文献

1] BAILLIC R. Long memory process and fractional integration in econometrics [J]. Journal of Econometrics, 1996, 1: 5-59.

[2] BERAN J. Statistical methods for data with long range dependences [J]. Statistics Science, 1992, 7: 404-427.

[3] BROCKWELL P, Davis R. Time Series: Theory and Methods [M]. Springer-Verlag, 1991

[4] CHERIDITO P. Arbitrage in fractional Brownian motion models, Finance and Stochastics, 2003, 7 (4): 533-553.

[5] DUNCAN T E, HU Y Z, PASIC-DUNCAN B. Stochastic calculus for fractional Brownain motion I: Theorey. SIAM: Journal of Control and Optimizaiton, 2000(38): 582-612.

[6] PETERS E. Fractal Market Analysis [M], New York: John &Wiley, 1994.

[7] GRANGER C, DING Z. Varieties of long memory models [J]. Journal of Econometrics, 1996, 73(1): 61-77.

[8] HURST H E. Long term storage capacity of reservoirs. Transaction of the American Society of Civil Engineers, 1951, 116: 770-799

[9] LU Z, GUEGAN D. Testing unit roots and long range dependence of foreign exchange [J]. Journal of Time Series Analysis, 2010, 7: 1-8.

[10] LO A W. Long-term memory in stock market prices [J]. Econometrica, 1991(59): 1279–1313.

[11] LO A W, MacKinlay C. Stock market prices do not follow random walks: Evidence from a simple specification test [J]. Review of Financial Study, 1988(1): 41–66.

[12] MANDELBROT B B, NESS J W V. Fractional Brownian Motions, Fractional Noises and Applications [J]. SIAM Review, 1968(10): 422-437.

[13] ROBINSON, P. Efficient tests for no-stationary hypothesis [J]. Journal of the American Statistical Association, 1994, 89: 1420-1437.

[14] WILLINGER W, TAQQU M, TEVEROVSKY V. Stock market prices and long-range dependence [J]. Finance and Stochastics, 1999: 1-13.

[15] SOTTINEN T, VALKEILA E. On arbitrage and replication in the fractional Black–Scholes pricing model [J]. Statistics & Decisions, 2003. 21: 137–151.

[16]赵巍,何建敏. 分数布朗运动驱动的期权定价模型及其风险特征 [J]. 数理统计与管理, 2011, 30(6): 1002-1008

[1]	陈海珍, 周圣武, 孙祥艳. 混合分数布朗运动下的回望期权定价[J]. 华东师范大学学报(自然科学版), 2018, 2018(4): 47-58.
[2]	杨朝强. 一类特殊混合跳-扩散模型的欧式回望期权定价[J]. 华东师范大学学报(自然科学版), 2017, (4): 1-17.
[3]	张蓉, 邹勇. 可视图复杂网络度分布拟合比较研究[J]. 华东师范大学学报(自然科学版), 2017, 2017(2): 75-80.