【摘要】：傳統(tǒng)二叉樹模型的收斂階數(shù)最高是O(1/n),并且是非光滑的。我們推廣了Chang-Palmer (2007)單參數(shù)的方法,通過適當(dāng)選取兩個(gè)參數(shù)值,使得二叉樹模型能夠以O(shè)(1/n)的階光滑收斂到對應(yīng)的歐式期權(quán)價(jià)格或者數(shù)值期權(quán)價(jià)格,這不僅拓寬了這兩個(gè)參數(shù)的取值空間,還可以使收斂階數(shù)提高到0(1／n)。另外,我們在Joshi(2010)單步二叉樹的啟發(fā)下提出了對應(yīng)的雙步二叉樹,并證明適當(dāng)選取上移概率展開式的系數(shù)可以使它以任意有限正數(shù)的階數(shù)光滑收斂。 離散時(shí)間對沖策略和理想的連續(xù)對沖策略之間誤差的研究也是一個(gè)熱門話題。我們在某些技巧性條件下研究了關(guān)于一般Levy-Ito過程的對沖誤差的L2收斂性。并且在某些附加條件下,利用Tankov和Voltchkova (2009)等時(shí)間間隔對沖誤差的研究思路證明了不均等時(shí)間間隔總對沖誤差的穩(wěn)定弱收斂性。 另外,根據(jù)Hayashi和Mykland (2005)關(guān)于連續(xù)擴(kuò)散過程的結(jié)果,我們證明了關(guān)于更一般的Levy-Ito過程離散數(shù)據(jù)驅(qū)動(dòng)策略相對對沖誤差和總誤差的穩(wěn)定弱收斂性。注意到其極限不是鞅,但是通過對正則化過程設(shè)立門限(threshold)的方法,可以使其穩(wěn)定弱收斂到鞅。 在金融非參數(shù)檢驗(yàn)中,以往關(guān)于Lévy過程積分波動(dòng)率(Ⅳ)的門限估計(jì)量的收斂性研究都是在有限活性跳的前提下進(jìn)行的。我們把Mancini (2011)允許無限活性跳的研究方法應(yīng)用到門限估計(jì)版本(version)的Bipower variation收斂速度的研究上面,并分析了在不同情況下的收斂速度。我們發(fā)現(xiàn)該方法同樣可以應(yīng)用于Integrated quarticity (IQ)的門限估計(jì)量的收斂速度的研究上面,并得到了在不同情況下的收斂性。
[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，

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