时变混合分数布朗运动下带交易费用的亚式期权定价

数学物理学报 ›› 2021, Vol. 41 ›› Issue (4): 1135-1146.

时变混合分数布朗运动下带交易费用的亚式期权定价

丁毅(),郭精军*()

^{兰州财经大学统计学院兰州 730020}

收稿日期:2020-10-06 出版日期:2021-08-26 发布日期:2021-08-09
通讯作者: 郭精军 E-mail:2014295797@qq.com;guojj@lzufe.edu.cn
作者简介:丁毅, E-mail: 2014295797@qq.com
基金资助:
国家自然科学基金(71961013);国家自然科学基金(72061020);甘肃省飞天学者项目和兰州财经大学科研创新团队支持计划

Pricing Asian Options Under Time-Changed Mixed Fractional Brownian Motion with Transactions Costs

Yi Ding(),Jingjun Guo*()

^{School of Statistics, Lanzhou University of Finance and Economics, Lanzhou 730020}

Received:2020-10-06 Online:2021-08-26 Published:2021-08-09
Contact: Jingjun Guo E-mail:2014295797@qq.com;guojj@lzufe.edu.cn
Supported by:
the NSFC(71961013);the NSFC(72061020);the Feitian Scholars Project in Gansu Province and the Research Innovation Team Support Plan of Lanzhou University of Finance and Economics

摘要/Abstract

摘要：

考虑到经典的Black-Scholes（B-S）期权定价模型不能描述金融资产价格常值周期性、长相依性等特征，因此采用时变混合分数布朗运动描述金融资产价格的变动，并且对亚式期权定价时也考虑了交易费用.运用自融资Delta对冲策略得出了在离散情形下几何平均亚式看涨期权价格所满足的偏微分方程以及几何平均亚式看涨、看跌期权的定价公式，并分析了定价模型中的参数对期权价格的影响.最后，选取万科股票的日收盘价对建立的定价模型进行了实证分析，验证了定价模型的有效性.

关键词: 亚式期权, 时变过程, 混合分数布朗运动, 交易费用

Abstract:

Considering that the classical Black-Scholes(B-S) option pricing model can not describe the characteristics of constant value periodicity and long-term dependence of financial asset prices, the time-changed mixed fractional Brownian motion is used to describe the changes of financial asset prices. By using self-financing delta hedging strategy, the partial differential equation of geometric average Asian call option price in discrete case and the pricing formula of geometric average Asian call and put option are obtained, and the influence of parameters in pricing model on option price is analyzed. Based on the daily closing price of Vanke stock, this article makes an empirical analysis on the established pricing model, and verifies the effectiveness of the pricing model.

Key words: Asian option, Time-changed process, Mixed fractional Brownian motion, Transaction cost

中图分类号:

O211.6

丁毅,郭精军. 时变混合分数布朗运动下带交易费用的亚式期权定价[J]. 数学物理学报, 2021, 41(4): 1135-1146.

Yi Ding,Jingjun Guo. Pricing Asian Options Under Time-Changed Mixed Fractional Brownian Motion with Transactions Costs[J]. Acta mathematica scientia,Series A, 2021, 41(4): 1135-1146.

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$K$	$C_{GBM}$	$C_{FBM}$	$C_{MFBM}$	$C_{TC-MFBM}$
90	12.818953	12.689548	13.990609	13.192214
91	12.081619	11.928910	13.335705	12.412176
92	11.366589	11.190936	12.699837	11.636384
93	10.674784	10.476815	12.083259	10.864681
94	10.007019	9.787618	11.486168	10.096917
95	9.363990	9.124284	10.908701	9.332946
96	8.746270	8.487611	10.350937	8.572629
97	8.154301	7.878242	9.812900	7.815831
98	7.588394	7.296662	9.294556	7.062422
99	7.048728	6.743191	8.795820	6.312278
100	6.535349	6.217986	8.316557	5.565279

	$C_{GBM}$	$C_{FBM}$	$C_{MFBM}$	$C_{TC-MFBM}$
均值	9.489636	9.256528	11.00682	9.351251
最大值	12.81895	12.68955	13.99061	13.19221
最小值	6.535349	6.217986	8.316557	5.565279
中位数	9.36399	9.124284	10.9087	9.332946
标准差	2.089258	2.152659	1.884268	2.529078

	均值	最大值	最小值	中位数	标准差	极差
收盘价	17.41	27.57	8.78	14.67	5.44	18.79
模拟价	17.63	27.00	9.44	16.22	4.55	17.56
BSM	17.57	31.47	8.28	15.63	7.56	23.19

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