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 近年来，自动微分（AD）领域取得了长足的进步。 然而，AD通常由于效率问题而不被大量用于大规模问题。 这在许多应用领域都是如此，包括经济和金融。例如，用标准方法评估大型可变年金投资组合可能需要很多时间;对冲通常需要确定衍生工具，需要的时间要多得多。显然，任何基于衍生品的快速对冲方法都存在问题。这对有效的风险管理提出了严峻挑战。其他投资组合也存在类似的挑战（例如，信用价值调整（CVA））。 我们提出的研究，以提高AD的效率和适用性的大型结构化问题。虽然我们提出的进步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技术变得更加高效。我们已经参与了适应AD方法的结构问题，工作的假设下，大多数实际的大型问题的结构。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。该结构定义捕获复合函数计算（即，链式计算序列）、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）。 正如在以前的工作中指出，这种结构涵盖了许多应用程序，往往是最自然的方式来表达评估代码。给定暴露此结构的代码，AD可以以“逐片”的方式应用以获得显著的效率（在空间和时间上）。通常，该增益是因为分量函数的雅可比矩阵是稀疏的（即，隐藏的稀疏性）或紧凑（具有很少的行），而整体（原始目标）函数的雅可比矩阵通常是稠密的。 总之，导数的确定是许多计算科学和工程中的基本计算任务。 我们在这里提出的想法可以在这些领域产生非常重大的影响-当然是在应用优化问题方面，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进行自动微分及其应用托马斯F.科尔曼和Wei Xu，SIAM，2016年。

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.