Vibrato and automatic differentiation for high-order derivatives and sensitivities of financial options

摘要

This paper deals with the computation of second or higher order greeks offinancial securities. It combines two methods, Vibrato and automaticdifferentiation and compares with other methods. We show that this combinedtechnique is faster than standard finite difference, more stable than automaticdifferentiation of second order derivatives and more general than MalliavinCalculus. We present a generic framework to compute any greeks and presentseveral applications on different types of financial contracts: European andAmerican options, multidimensional Basket Call and stochastic volatility modelssuch as Heston's model. We give also an algorithm to compute derivatives forthe Longstaff-Schwartz Monte Carlo method for American options. We also extendautomatic differentiation for second order derivatives of options withnon-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 timeconsuming to compute accurately. Faced with the problem of building a softwarefor this task and distrusting automatic differentiation for non-differentiablefunctions, we turned to an idea developed by Mike Giles called Vibrato. Vibratoat core is a differentiation of a combination of likelihood ratio method andpathwise evaluation. In Giles [12], [13], it is shown that the computing time,stability and precision are enhanced compared with numerical differentiation ofthe full Monte Carlo path. In many cases, double sensitivities, i.e. secondderivatives with respect to parameters, are needed (e.g. gamma hedging). Finitedifference approximation of sensitivities is a very simple method but itsprecision is hard to control because it relies on the appropriate choice of theincrement. Automatic differentiation of computer programs bypass the difficultyand its computing cost is similar to finite difference, if not cheaper. But infinance the payoff is never twice differentiable and so generalized derivativeshave to be used requiring approximations of Dirac functions of which theprecision is also doubtful. The purpose of this paper is to investigate thefeasibility of Vibrato for second and higher derivatives. We will first compareVibrato applied twice with the analytic differentiation of Vibrato and showthat it is equivalent, as the second is easier we propose the best compromisefor second derivatives: Automatic Differentiation of Vibrato. In [8], Capriottihas recently investigated the coupling of different mathematical methods --namely pathwise and likelihood ratio methods -- with an Automatic differ