Smoking Adjoints: fast evaluation of Greeks in Monte Carlo calculations

Abstract

This paper presents an adjoint method to accelerate the calculation of Greeks by Monte Carlo simulation. The method calculates price sensitivities along each path; but in contrast to a forward pathwise calculation, it works backward recursively using adjoint variables. Along each path, the forward and adjoint implementations produce the same values, but the adjoint method rearranges the calculations to generate potential computational savings. The adjoint method outperforms a forward implementation in calculating the sensitivities of a small number of outputs to a large number of inputs. This applies, for example, in estimating the sensitivities of an interest rate derivatives book to multiple points along an initial forward curve or the sensitivities of an equity derivatives book to multiple points on a volatility surface. We illustrate the application of the method in the setting of the LIBOR market model. Numerical results confirm that the computational advantage of the adjoint method grows in proportion to the number of initial forward rates