《Vibrato and automatic differentiation for high order derivatives and
sensitivities of financial options》
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作者:
Gilles Pag\\`es (UPMC), Olivier Pironneau (LJLL), Guillaume Sall (LJLL)
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最新提交年份:
2016
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英文摘要:
This paper deals with the computation of second or higher order greeks of financial securities. It combines two methods, Vibrato and automatic differentiation and compares with other methods. We show that this combined technique is faster than standard finite difference, more stable than automatic differentiation of second order derivatives and more general than Malliavin Calculus. We present a generic framework to compute any greeks and present several applications on different types of financial contracts: European and American options, multidimensional Basket Call and stochastic volatility models such as Heston\'s model. We give also an algorithm to compute derivatives for the Longstaff-Schwartz Monte Carlo method for American options. We also extend automatic differentiation for second order derivatives of options with non-twice differentiable payoff. 1. Introduction. Due to BASEL III regulations, banks are requested to evaluate the sensitivities of their portfolios every day (risk assessment). Some of these portfolios are huge and sensitivities are time consuming to compute accurately. Faced with the problem of building a software for this task and distrusting automatic differentiation for non-differentiable functions, we turned to an idea developed by Mike Giles called Vibrato. Vibrato at core is a differentiation of a combination of likelihood ratio method and pathwise evaluation. In Giles [12], [13], it is shown that the computing time, stability and precision are enhanced compared with numerical differentiation of the full Monte Carlo path. In many cases, double sensitivities, i.e. second derivatives with respect to parameters, are needed (e.g. gamma hedging). Finite difference approximation of sensitivities is a very simple method but its precision is hard to control because it relies on the appropriate choice of the increment. Automatic differentiation of computer programs bypass the difficulty and its computing cost is similar to finite difference, if not cheaper. But in finance the payoff is never twice differentiable and so generalized derivatives have to be used requiring approximations of Dirac functions of which the precision is also doubtful. The purpose of this paper is to investigate the feasibility of Vibrato for second and higher derivatives. We will first compare Vibrato applied twice with the analytic differentiation of Vibrato and show that it is equivalent, as the second is easier we propose the best compromise for second derivatives: Automatic Differentiation of Vibrato. In [8], Capriotti has recently investigated the coupling of different mathematical methods -- namely pathwise and likelihood ratio methods -- with an Automatic differ
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中文摘要:
本文讨论金融证券的二阶或更高阶希腊人的计算。它结合了颤音和自动微分两种方法,并与其他方法进行了比较。我们表明,这种组合方法比标准有限差分法更快,比二阶导数的自动微分法更稳定,比Malliavin演算更通用。我们提出了一个通用框架来计算任何希腊人,并给出了几种不同类型金融合同的应用:欧洲和美国期权、多维篮子看涨期权和随机波动率模型,如赫斯顿模型。我们还给出了计算美式期权Longstaff-Schwartz蒙特卡罗方法导数的算法。我们还推广了收益不可二次微分的期权二阶导数的自动微分。1、简介。根据巴塞尔协议III的规定,要求银行每天评估其投资组合的敏感性(风险评估)。其中一些投资组合规模巨大,准确计算敏感度非常耗时。面对为这项任务构建软件以及不信任不可微函数的自动微分的问题,我们转向了MikeGiles提出的一个想法,称为颤音。核心振动是似然比方法和路径评估相结合的一种区别。Giles【12】、【13】表明,与全蒙特卡罗路径的数值微分相比,计算时间、稳定性和精度都有所提高。在许多情况下,需要双重敏感性,即参数的二阶导数(例如伽马对冲)。灵敏度的有限差分近似是一种非常简单的方法,但其精度很难控制,因为它依赖于增量的适当选择。计算机程序的自动微分绕过了这一困难,其计算成本与有限差分法相似,甚至更便宜。但在金融学中,收益从来都不是二次可微的,因此必须使用广义导数,需要近似狄拉克函数,其精度也值得怀疑。本文旨在研究二阶导数和高阶导数振动的可行性。我们将首先比较两次应用的可控震源与可控震源的解析微分,并证明它是等效的,因为第二次更容易,我们提出了二阶导数的最佳折衷:可控震源的自动微分。在[8]中,卡普里奥蒂最近研究了不同数学方法(即路径法和似然比法)与自动差分法的耦合
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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:Computer Science 计算机科学
二级分类:Computational Engineering, Finance, and Science 计算工程、金融和科学
分类描述:Covers applications of computer science to the mathematical modeling of complex systems in the fields of science, engineering, and finance. Papers here are interdisciplinary and applications-oriented, focusing on techniques and tools that enable challenging computational simulations to be performed, for which the use of supercomputers or distributed computing platforms is often required. Includes material in ACM Subject Classes J.2, J.3, and J.4 (economics).
涵盖了计算机科学在科学、工程和金融领域复杂系统的数学建模中的应用。这里的论文是跨学科和面向应用的,集中在技术和工具,使挑战性的计算模拟能够执行,其中往往需要使用超级计算机或分布式计算平台。包括ACM学科课程J.2、J.3和J.4(经济学)中的材料。
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