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

提案摘要：自动微分在计算金融（及其他领域）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并没有大量用于大规模问题。在包括经济和金融在内的许多应用领域都是如此。例如，用标准方法评估大型可变年金投资组合可能需要很长时间；套期保值通常需要确定衍生品，这将需要更多的时间。显然，针对此类投资组合的任何基于衍生品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））。*******我们建议研究如何提高AD对大规模结构化问题的效率和适用性。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。*******近年来，AD技术的效率越来越高。我们一直致力于将AD方法应用于结构问题，并假设大多数实际的大规模问题都是结构化的。结构化问题的一般和实用定义在（[1],4.16）中给出。其思想是，函数可以写成一个（部分有序的）步骤序列——每个步骤本身都是一个非线性映射，并且是先前定义的（中间）变量的（子）函数。这个结构定义捕获了复合函数计算（即一系列链式计算）、广义部分可分函数和（嵌套的）蒙特卡罗函数（以及许多其他常见结构）。*******如前所述，此结构涵盖了许多应用程序，并且通常是表示求值代码的最自然的方式。给定一个公开此结构的代码，可以以“逐片”的方式应用AD，以获得显著的效率（在空间和时间上）。通常，这种增益是因为组成函数的雅可比矩阵要么稀疏（即隐藏稀疏性），要么紧凑（只有几行），而总体（原始目标）函数的雅可比矩阵通常是密集的。*******总之，在计算科学和工程中，导数的确定是一个基本的计算任务。我们在这里提出的想法可以在这些领域产生非常重要的影响——当然是关于应用优化问题，以及我们的目标类别：计算金融/风险管理中的敏感性问题。*******[1]在MATLAB中使用ADMAT自动微分的应用。Thomas F. Coleman和Wei Xu， 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. **