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

自动微分在计算金融（及其他领域）中的大规模应用

• 批准号: RGPIN-2017-03860
• 负责人: Coleman, Thomas
• 金额: $ 2.48万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=685287
• 关键词: Large; scale; Application; Automatic; Differentiation

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. **

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，由于效率问题，AD 并未大量用于解决大规模问题。 这在许多应用领域都是如此，包括经济和金融。例如，通过标准方法评估大型可变年金投资组合可能需要花费数小时； 对冲通常需要确定衍生品，因此需要更多的时间。显然，任何基于此类投资组合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（例如信用价值调整（CVA））也存在类似的挑战。********我们建议进行研究以提高 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。********AD 技术近年来变得更加高效。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。 ([1], 4.16) 中给出了结构化问题的一般且实用的定义。这个想法是，该函数可以写成（部分有序）步骤序列——每个步骤本身就是一个非线性映射，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（即一系列链式计算）、广义部分可分离函数和（嵌套）蒙特卡罗函数（以及许多其他常见结构）。********如之前的工作所示，该结构涵盖了许多应用程序，并且通常是表达评估代码的最自然方式。给定公开此结构的代码，AD 可以以“逐片”的方式应用，以获得显着的效率（在空间和时间上）。一般来说，这种增益是因为分量函数的雅可比行列式要么是稀疏的（即隐藏稀疏性）要么是紧凑的（行数很少），而整体（原始目标）函数的雅可比行列式通常是稠密的。 ******总之，导数的确定是许多计算科学和工程中的一项基本计算任务。 我们在这里提出的想法可以对这些领域产生非常重大的影响 - 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 和应用程序进行自动微分。 Thomas F. Coleman 和 Wei Xu，SIAM，2016 年。**