  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
