金融数学中的若干极限定理
发布时间：2018-10-09 12:38
【摘要】：传统二叉树模型的收敛阶数最高是O(1/n),并且是非光滑的。我们推广了Chang-Palmer (2007)单参数的方法,通过适当选取两个参数值,使得二叉树模型能够以O(1/n)的阶光滑收敛到对应的欧式期权价格或者数值期权价格,这不仅拓宽了这两个参数的取值空间,还可以使收敛阶数提高到0(1／n)。另外,我们在Joshi(2010)单步二叉树的启发下提出了对应的双步二叉树,并证明适当选取上移概率展开式的系数可以使它以任意有限正数的阶数光滑收敛。 离散时间对冲策略和理想的连续对冲策略之间误差的研究也是一个热门话题。我们在某些技巧性条件下研究了关于一般Levy-Ito过程的对冲误差的L2收敛性。并且在某些附加条件下,利用Tankov和Voltchkova (2009)等时间间隔对冲误差的研究思路证明了不均等时间间隔总对冲误差的稳定弱收敛性。 另外,根据Hayashi和Mykland (2005)关于连续扩散过程的结果,我们证明了关于更一般的Levy-Ito过程离散数据驱动策略相对对冲误差和总误差的稳定弱收敛性。注意到其极限不是鞅,但是通过对正则化过程设立门限(threshold)的方法,可以使其稳定弱收敛到鞅。 在金融非参数检验中,以往关于Lévy过程积分波动率(Ⅳ)的门限估计量的收敛性研究都是在有限活性跳的前提下进行的。我们把Mancini (2011)允许无限活性跳的研究方法应用到门限估计版本(version)的Bipower variation收敛速度的研究上面,并分析了在不同情况下的收敛速度。我们发现该方法同样可以应用于Integrated quarticity (IQ)的门限估计量的收敛速度的研究上面,并得到了在不同情况下的收敛性。
[Abstract]:The convergence order of the traditional binary tree model is the highest O (1 / n), and it is not smooth. In this paper, we generalize the method of Chang-Palmer (2007) single parameter. By properly selecting two parameter values, the binary tree model can converge to the corresponding European option price or numerical option price with the order smooth of O (1 / n). This not only broadens the value space of these two parameters, but also increases the convergence order to 0 (1 / n). In addition, we propose the corresponding two-step binary tree inspired by Joshi (2010) single-step binary tree, and prove that it can converge smoothly with the order of any finite positive number by properly selecting the coefficients of the upshift probability expansion. The error between discrete time hedging strategy and ideal continuous hedging strategy is also a hot topic. We study the L2 convergence of hedging errors for general Levy-Ito processes under some technical conditions. Under some additional conditions, the stable weak convergence of the total hedge error of unequal time interval is proved by using the research ideas of Tankov and Voltchkova (2009). In addition, according to the results of Hayashi and Mykland (2005) on the continuous diffusion process, we prove the stable weak convergence of the more general discrete data-driven strategy for Levy-Ito processes relative to the hedging error and the total error. It is noted that the limit is not a martingale, but by setting a threshold (threshold) for the regularization process, it can be stabilized and weakly converged to the martingale. In the financial nonparametric test, the convergence of the threshold estimator for the integral volatility (鈪，
本文编号：2259364
[Abstract]:The convergence order of the traditional binary tree model is the highest O (1 / n), and it is not smooth. In this paper, we generalize the method of Chang-Palmer (2007) single parameter. By properly selecting two parameter values, the binary tree model can converge to the corresponding European option price or numerical option price with the order smooth of O (1 / n). This not only broadens the value space of these two parameters, but also increases the convergence order to 0 (1 / n). In addition, we propose the corresponding two-step binary tree inspired by Joshi (2010) single-step binary tree, and prove that it can converge smoothly with the order of any finite positive number by properly selecting the coefficients of the upshift probability expansion. The error between discrete time hedging strategy and ideal continuous hedging strategy is also a hot topic. We study the L2 convergence of hedging errors for general Levy-Ito processes under some technical conditions. Under some additional conditions, the stable weak convergence of the total hedge error of unequal time interval is proved by using the research ideas of Tankov and Voltchkova (2009). In addition, according to the results of Hayashi and Mykland (2005) on the continuous diffusion process, we prove the stable weak convergence of the more general discrete data-driven strategy for Levy-Ito processes relative to the hedging error and the total error. It is noted that the limit is not a martingale, but by setting a threshold (threshold) for the regularization process, it can be stabilized and weakly converged to the martingale. In the financial nonparametric test, the convergence of the threshold estimator for the integral volatility (鈪，
本文编号：2259364
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