英文标题：
《Functional Ito Calculus, Path-dependence and the Computation of Greeks》
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作者：
Samy Jazaerli and Yuri F. Saporito
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最新提交年份：
2018
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英文摘要：
Dupire\'s functional It\\^o calculus provides an alternative approach to the classical Malliavin calculus for the computation of sensitivities, also called Greeks, of path-dependent derivatives prices. In this paper, we introduce a measure of path-dependence of functionals within the functional It\\^o calculus framework. Namely, we consider the Lie bracket of the space and time functional derivatives, which we use to classify functionals accordingly to their degree of path-dependence. We then revisit the problem of efficient numerical computation of Greeks for path-dependent derivatives using integration by parts techniques. Special attention is paid to path-dependent functionals with zero Lie bracket, called locally weakly path-dependent functionals in our classification. Hence, we derive the weighted-expectation formulas for their Greeks. In the more general case of fully path-dependent functionals, we show that, equipped with the functional It\\^o calculus, we are able to analyze the effect of the Lie bracket on the computation of Greeks. Moreover, we are also able to consider the more general dynamics of path-dependent volatility. These were not achieved using Malliavin calculus.
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中文摘要：
Dupire的函数It^o演算为计算路径相关衍生品价格的敏感性（也称为希腊人）提供了经典Malliavin演算的替代方法。本文在泛函It^o演算框架下，引入了泛函的路径依赖性测度。也就是说，我们考虑空间和时间泛函导数的李括号，我们用它来根据路径依赖程度对泛函进行分类。然后，我们重新讨论了使用分部积分技术对路径相关导数进行有效数值计算的问题。我们特别关注具有零Lie括号的路径依赖泛函，在我们的分类中称为局部弱路径依赖泛函。因此，我们推导了加权期望公式。在更一般的完全路径依赖泛函的情况下，我们证明，通过泛函It^o演算，我们能够分析李括号对计算的影响。此外，我们还可以考虑路径依赖波动性的更一般动力学。这些都不是通过Malliavin演算实现的。
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分类信息：

一级分类：Quantitative Finance 数量金融学
二级分类：Computational Finance 计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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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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