Large-scale Application of Automatic Differentiation in Computational Finance (and beyond)

提案摘要：自动微分在计算金融(及以后)中的大规模应用*自动微分(AD)领域近年来取得了长足的进步。然而，由于效率方面的考虑，AD在处理大规模问题时并未大量使用。这在包括经济和金融在内的许多应用领域都是如此。例如，用标准方法评估可变年金的大型投资组合可能需要几个小时；对冲通常需要确定衍生品，需要的时间要长得多。显然，任何基于衍生品的此类投资组合的快速对冲方法都是有问题的。这对有效的风险管理提出了严峻的挑战。其他投资组合也存在类似的挑战(例如，信用价值调整(CVA))。*我们建议进行研究，以提高AD的效率和对大规模结构化问题的适用性。虽然我们提出的进展适用于计算科学和工程的许多分支，但我们强调计算金融和风险管理应用。我们一直在参与AD方法对结构问题的适应，在大多数实际的大规模问题都是结构的假设下工作。结构化问题的一般和实用定义见([1]，4.16)。其思想是，该函数可以被写成一个(偏序的)步骤序列--每个步骤本身就是一个非线性映射，是先前定义的(中间)变量的(子)函数。此结构定义捕获复合函数计算(即，一系列链式计算)、广义部分可分函数和(嵌套)蒙特卡罗函数(以及许多其他常见结构)。*如先前的工作所指出的，此结构涵盖许多应用，通常是表示求值代码的最自然的方式。给定一个暴露这种结构的代码，AD可以以一片接一片的方式应用，以获得显著的效率(在空间和时间上)。一般来说，这是因为分量函数的雅可比不是稀疏的(即隐藏稀疏的)就是紧致的(行很少)，而整体(原始目标)函数的雅可比通常是稠密的。我们在这里提出的想法可以对这些领域产生非常重要的影响-当然对于应用优化问题，以及我们的目标类：计算金融/风险管理中的敏感性问题。*托马斯·F·科尔曼和魏旭，SIAM，2016。**

Summary of Proposal: Large-scale Application of Automatic Differentiation in Computational Finance (and beyond)******The field of Automatic Differentiation (AD) has taken great strides in recent years. Nevertheless, AD is not heavily used for large-scale problems often due to efficiency concerns. This is true in many application areas, including economics and finance. For example, large portfolios of variable annuities can take many hours to evaluate by standard methods; hedging, typically requiring the determination of derivatives, would require significantly more time. Clearly, any rapid hedging methods based on derivatives for such portfolios is problematic. This poses a serious challenge for effective risk management. Similar challenges exist for other portfolios (e.g., Credit Value Adjustment (CVA's)).*******We propose research to increase the efficiency and applicability of AD to large-scale structured problems. While our proposed advances are applicable to many branches of computational science and engineering, we emphasize computational finance and risk management applications.*******AD technology has become more efficient in recent years. We have been involved in the adaption of AD methodology to problems with structure, working under the assumption that most practical large-scale problems are structured. A general and practical definition of a structured problem is given in ([1], 4.16) . The idea is that the function can be written as a (partially ordered) sequence of steps -- each step is a nonlinear mapping in itself and is a (sub-) function of previously defined (intermediate) variables. This structure definition captures composite function computations (i.e., a sequence of chained computations), generalized partially separable functions, and (nested) Monte-Carlo functions (and many other common structures).*******As indicated in previous work this structure covers many applications and is often the most natural way to express the evaluation code . Given a code that exposes this structure AD can be applied in a “slice-by-slice” manner to gain significant efficiency (both in space and time). Generally this gain is because the Jacobians of the component functions are either sparse (i.e., hidden sparsity) or compact (with few rows), whereas the Jacobian of the overall (original objective) function is often dense.*******In conclusion, the determination of derivatives is a fundamental computational task in much of computational science and engineering. The ideas we propose here can have a very significant impact across these areas - certainly with respect to applied optimization problems, as well as our target class: sensitivity problems in computational finance/risk management.*******[1] Automatic Differentiation in MATLAB Using ADMAT with Applications. Thomas F. Coleman and Wei Xu, SIAM, 2016. **