Adjoint Algorithmic Differentiation and the Derivative of the Cholesky Decomposition

Abstract

Adjoint Algorithmic Differentiation is an efficient method to calculate the sensitivities of financial instruments to the input parameters. It does require an implementation of the derivative of every elementary step in the pricing algorithm. In particular, for calculating correlation sensitivities in a Monte Carlo simulation it has been proposed to use a step-by-step adjoint derivative of the Cholesky factorization. In this paper we introduce a one-line matrix-level algorithm which uses only matrix inversion and multiplication and can thus be performed by a standard library for linear algebra. Moreover, it allows for a more time-efficient calculation of the standard errors on these sensitivities.