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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高阶衍生品的颤音和自动微分与金融期权的敏感性

本文涉及金融证券的二阶或更高阶希腊字母的计算。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。我们表明这种组合技术比标准有限差分更快，比二阶导数的自动微分更稳定，比 Malliavin 微积分更通用。我们提出了一个通用框架来计算任何希腊语，并在不同类型的金融合约上提出了几种应用：欧式和美式期权、多维篮子看涨期权和随机波动率模型，例如赫斯顿模型。我们还给出了计算美式期权的 Longstaff-Schwartz Monte Carlo 方法的导数的算法。我们还扩展了具有非两倍可微收益的期权的二阶导数的自动微分。一、介绍。由于 BASEL III 规定，银行需要每天评估其投资组合的敏感性（风险评估）。其中一些投资组合非常庞大，准确计算敏感度非常耗时。面对为此任务构建软件以及不信任不可微函数的自动微分的问题，我们转向了由 Mike Giles 开发的名为 Vibrato 的想法。核心的颤音是似然比法和路径评估相结合的差异化。在 Giles [12]、[13] 中，表明与完整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。在很多情况下，需要双重敏感性，即关于参数的二阶导数（例如伽马对冲）。灵敏度的有限差异近似是一种非常简单的方法，但它的精确性是难以控制的，因为它依赖于适当的增量选择。计算机程序的自动微分绕过了难度，其计算成本与有限差分相似，甚至更便宜。但在金融中，收益永远不会是两次可微的，因此必须使用广义衍生品，需要狄拉克函数的近似值，其精度也值得怀疑。本文的目的是研究颤音用于二阶和更高阶导数的可行性。我们将首先将两次应用的颤音与颤音的解析微分进行比较，并证明它是等效的，由于第二个更容易，我们提出了二阶导数的最佳折衷方案：颤音的自动微分。在 [8] 中，Capriotti 最近研究了不同数学方法（即路径和似然比方法）与自动差异的耦合。