次分数Black-Scholes模型的套利机会  

Arbitrage opportunities in sub-fractional Black-Scholes model

作　　者：肖炜麟[1] 周清[2] 吴卫星[3] Weilin Xiao;Qing Zhou;Weixing Wu

机构地区：[1]浙江大学管理学院,杭州310058 [2]北京邮电大学理学院,北京100876 [3]对外经济贸易大学金融学院,北京100029

出　　处：《中国科学：数学》2021年第11期1877-1894,共18页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:71871202,11871010,11971040和71733004)资助项目。

摘　　要：为了刻画金融时间序列的长记忆性和非平稳性,众多学者采用次分数Brown运动来描述金融资产价格变化的行为模式.然而,次分数Brown运动不是半鞅,能否直接将其应用于金融市场一直是金融数学领域的热点问题.基于Hurst指数H>1/2情形下次分数Brown运动Donsker逼近定理,本文研究几何次分数Brown运动框架下金融市场的套利问题.首先,采用Skorokhod拓扑下的随机游走理论,构建一个弱收敛于次分数Brown运动驱动的Black-Scholes市场模型.其次,利用积分不等式和次分数二元市场理论证明次分数Brown运动驱动的Black-Scholes金融模型存在套利机会.最后,采用Monte Carlo模拟说明套利发生的可能性,并展示套利发生过程.In order to capture the long range dependence and the nonstationary of financial time series,many scholars have used the sub-fractional Brownian motion to describe the price fluctuations of the underlying asset.Since the sub-fractional Brownian motion fails to be a semimartingale,its application in financial markets is in doubt and becomes a hot topic in mathematical finance.Using the Donsker type approximation for the subfractional Brownian motion with Hurst parameter H>1/2,this paper considers the arbitrage opportunities in the geometric sub-fractional Brownian motion.First,using the random walk theory in the Skorokhod topology with H>1/2,we construct a model,which converges weakly to the sub-fractional Black-Scholes model.Then,using the integral inequality and the theory of the sub-fractional binary market,we prove that there exist arbitrage opportunities in the sub-fractional Black-Scholes model.Finally,the Monte Carlo simulation illustrates the possibility of arbitrage opportunities and shows the generation of arbitrage opportunities.

关 键 词：次分数Brown运动 弱收敛 随机游走 套利机会 

分 类 号：F224[经济管理—国民经济] F830.9