摘要翻译：
金融资产价格的数学模型，例如包括随机波动或跳跃，是不完整的，因为衍生证券通常不能通过基础交易复制。在早期的工作（2004）中，第一作者提供了一个几何条件，在这个条件下，在基础和有限数量的香草期权的交易完成市场。我们用几种方法来补充这个结果。首先，我们证明了几何条件不是必要条件，并给出了一个较弱的充要条件。当定价函数为实解析函数时，这个条件一般不能直接证明，但我们证明了它在单点上是矩阵非退化的。特别是，任何随机波动率模型都是用一个任意的欧式期权来完成的。此外，我们还表明，将路径依赖的期权（如方差互换）添加到初级资产集合中，而不是普通的普通期权，也会使市场变得完整。
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英文标题：
《Market completion using options》
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作者：
Mark Davis and Jan Obloj
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最新提交年份：
2008
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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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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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英文摘要：
  Mathematical models for financial asset prices which include, for example, stochastic volatility or jumps are incomplete in that derivative securities are generally not replicable by trading in the underlying. In earlier work (2004) the first author provided a geometric condition under which trading in the underlying and a finite number of vanilla options completes the market. We complement this result in several ways. First, we show that the geometric condition is not necessary and a weaker, necessary and sufficient, condition is presented. While this condition is generally not directly verifiable, we show that it simplifies to matrix non-degeneracy in a single point when the pricing functions are real analytic functions. In particular, any stochastic volatility model is then completed with an arbitrary European type option. Further, we show that adding path-dependent options such as a variance swap to the set of primary assets, instead of plain vanilla options, also completes the market.
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PDF链接：
https://arxiv.org/pdf/0710.2792