Root's and Rost's embeddings : construction, optimality and applications to variance options
Root's and Rost's embeddings : construction, optimality and applications to variance options
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发表时间：
2011
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通讯作者：
Jiajie Wang
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作者：
Jiajie Wang
Root的Skorokhod嵌入问题的解(Root[1969])可以被描述为时空过程(Xt，t)在所谓的屏障上的第一次命中时间，其特征是使得停止的底层过程X具有给定的分布。Dupire[2005]和Carr and Lee[2010]最近的工作强调了理解独立于模型的方差期权套期保值的Root解决方案的重要性。当基本过程是一维具有给定初始分布的时齐扩散时，我们考虑了求解Root解的问题。我们对用偏微分方程组构造根解很感兴趣。我们首先证明，在一些较温和的条件下，当基础过程是从0开始的布朗运动时，构造根解等价于求解一个特殊的抛物型自由边界问题。然后将这一结果推广到时间均匀扩散。代替了构造自由边界所需的一些条件，我们还考虑了用变分不等式构造根解。最后，我们考虑了Root解的最优性及其应用。与现有的基于势论的最优性证明(Rost[1976])不同，本文给出了另一种证明方法，即找到一个路径不等式，它对于构造金融环境中的子套期保值策略具有重要的应用。此外，我们还考虑了Rost解的构造性和最优性等问题，Rost解也被称为Root解的逆。
Root’s solution (Root [1969]) to the Skorokhod embedding problem can be described as the first hitting time of a space-time process (Xt, t) on a so-called barrier, charac- terised by certain properties, such that the stopped underlying process X has a given distribution. Recent work of Dupire [2005] and Carr and Lee [2010] has highlighted the importance of understanding the Root’s solution for the model-independent hedging of variance options. We consider the problem of finding Root’s solutions when the underlying process is a time-homogeneous diffusion with a given initial distribution in one dimension. We are interested in constructing Root’s solution by partial differential equations. We begin by showing that, under some mild conditions, constructing Root’s solution is equiv- alent to solving a specialized parabolic free boundary problem in the case where the underlying process is a Brownian motion starting at 0. This result is then extended to time-homogeneous diffusions. Replacing some conditions needed in the free boundary construction, we then also consider the construction of Root’s solutions by variational inequalities. Finally we consider the optimality and applications of Root’s solutions. Unlike the existing proof of optimality (Rost [1976]), which relies on potential theory, an alternative proof is given by finding a path-wise inequality which has an impor- tant application for the construction of subhedging strategies in the financial context. In addition, we also consider these questions, construction and optimality, for Rost’s solution, which is also known as the reverse of the Root’s solution.