Adjoint methods for computing sensitivities in local volatility surfaces

Abstract

In this paper we present the adjoint method of computing sensitivities of option prices with respect to nodes in the local volatility surface. We first introduce the concept of algorithmic differentiation and how it relates to
path-wise sensitivity computations within a Monte Carlo framework. We explain the two approaches available: forward mode and adjoint mode. We illustrate these concepts on the simple example of a model with a geometric Brownian motion driving the underlying price process, for which
we compute the Delta and Vega in forward and adjoint mode. We then go on to explain in full detail how to apply these ideas to a model where the underlying has a volatility term defined by a local volatility surface. We provide source codes for both the simple and the more complex case and
analyze numerical results to show the strengths of the adjoint approach.