论文标题
广义分数布朗运动及其混合物与资产定价的混合物的半明星性能
Semimartingale properties of a generalized fractional Brownian motion and its mixtures with applications in asset pricing
论文作者
论文摘要
我们研究了Pang and Taqqu（2019）引入的广义分数布朗运动（GFBM）的半明星特性，并讨论了GFBM及其在金融资产价格上的混合物的应用。 GFBM是自相似的，并且具有非平稳增量，其Hurst索引$ h \ in（0,1）$由两个参数确定。我们确定了这两个参数值的区域，其中GFBM是半木马。 下一步，我们研究由独立的BM和GFBM组成的混合过程，并确定其为半明星的参数范围，该参数导致GFBM的$ h \ in（1/2,1）$。我们还得出了相关的等效布朗措施。与Cheridito（2001）证明的$ h \ in \ {1/2 \} \ cup（3/4,1] $的混合FBM相比，该结果与$ h \ in \ {1/2 \} \ cup（2001）相反，并显示了GFBM中引入的附加参数的重要性。 然后，我们在远距离依赖性的情况下使用混合GFBM研究了半明天的资产定价理论，以及在期权定价和投资组合优化中的应用。最后，我们讨论了使用GFBM对套利理论的含义，特别是提供了一个远程依赖性的Semimartingale资产定价模型，而无需套利。
We study the semimartingale properties for the generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019) and discuss the applications of the GFBM and its mixtures to financial asset pricing. The GFBM is self-similar and has non-stationary increments, whose Hurst index $H \in (0,1)$ is determined by two parameters. We identify the regions of these two parameter values where the GFBM is a semimartingale. We next study the mixed process made up of an independent BM and a GFBM and identify the range of parameters for it to be a semimartingale, which leads to $H \in (1/2,1)$ for the GFBM. We also derive the associated equivalent Brownian measure. This result is in great contrast with the mixed FBM with $H \in \{1/2\}\cup(3/4,1]$ proved by Cheridito (2001) and shows the significance of the additional parameter introduced in the GFBM. We then study the semimartingale asset pricing theory with the mixed GFBM, in presence of long range dependence, and applications in option pricing and portfolio optimization. Finally we discuss the implications of using GFBM on arbitrage theory, in particular, providing an example of semimartingale asset pricing model of long range dependence without arbitrage.
