Fast Greeks by Algorithmic Differentiation

Abstract

We show how Algorithmic Differentiation can be used to implement efficiently the Pathwise Derivative method for the calculation of option sensitivities with Monte Carlo. The main practical difficulty of the Pathwise Derivative method is that it requires the differentiation of the payout function. For the type of structured options for which Monte Carlo simulations are usually employed, these derivatives are typically cumbersome to calculate analytically, and too time consuming to evaluate with standard finite-differences approaches. In this paper we address this problem and show how Algorithmic Differentiation can be employed to calculate very efficiently and with machine precision accuracy these derivatives. We illustrate the basic workings of this computational technique by means of simple examples, and we demonstrate with several numerical tests how the Pathwise Derivative method combined with Algorithmic Differentiation – especially in the adjoint mode – can provide speed-ups of several orders of magnitude with respect to standard methods.