英文标题：
《The existence of optimal bang-bang controls for GMxB contracts》
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作者：
Parsiad Azimzadeh, Peter A. Forsyth
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最新提交年份：
2015
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英文摘要：
  A large collection of financial contracts offering guaranteed minimum benefits are often posed as control problems, in which at any point in the solution domain, a control is able to take any one of an uncountable number of values from the admissible set. Often, such contracts specify that the holder exert control at a finite number of deterministic times. The existence of an optimal bang-bang control, an optimal control taking on only a finite subset of values from the admissible set, is a common assumption in the literature. In this case, the numerical complexity of searching for an optimal control is considerably reduced. However, no rigorous treatment as to when an optimal bang-bang control exists is present in the literature. We provide the reader with a bang-bang principle from which the existence of such a control can be established for contracts satisfying some simple conditions. The bang-bang principle relies on the convexity and monotonicity of the solution and is developed using basic results in convex analysis and parabolic partial differential equations. We show that a guaranteed lifelong withdrawal benefit (GLWB) contract admits an optimal bang-bang control. In particular, we find that the holder of a GLWB can maximize a writer\'s losses by only ever performing nonwithdrawal, withdrawal at exactly the contract rate, or full surrender. We demonstrate that the related guaranteed minimum withdrawal benefit contract is not convexity preserving, and hence does not satisfy the bang-bang principle other than in certain degenerate cases.
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中文摘要：
提供有保证的最低收益的大量金融合同通常被视为控制问题，在该问题中，在解域的任何一点上，控制都能够从可容许集合中获取不可数个值中的任何一个。通常，此类合同规定持有人在有限的确定时间内行使控制权。最优bang-bang控制的存在性是文献中的一个常见假设，即最优控制只接受可容许集合中有限个值的子集。在这种情况下，搜索最优控制的数值复杂性大大降低。然而，文献中并没有对何时存在最佳砰砰控制进行严格处理。我们向读者提供了一个bang-bang原理，根据该原理，对于满足一些简单条件的合同，可以建立这种控制的存在性。bang-bang原理依赖于解的凸性和单调性，是利用凸分析和抛物偏微分方程的基本结果发展起来的。我们证明了保证终身退出福利（GLWB）合同允许最优的bang-bang控制。特别是，我们发现，GLWB的持有者可以通过不退稿、完全按照合同利率退稿或完全退稿来最大化作者的损失。我们证明了相关的保证最小取款收益契约不是保凸的，因此除了某些退化情形外，它不满足bang-bang原理。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Pricing of Securities        证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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