Second-order Monte Carlo sensitivities in linear or constant time

Abstract

This paper considers the problem of efficiently computing the full matrix of second-order sensitivities of a Monte Carlo price when the number of inputs is large. Specifically, the paper analyzes and compares methods with run times of at most O(NT), where N is the dimension of the input and T is the time required to compute the price. Since none of the alternatives from previous literature appears to be satisfactory in all settings, we propose two original methods: the first method is based on differentiation in a distributional sense, while the second method leverages a functional relation between first- and second-order derivatives. The former shows excellent generality and computational times to achieve a given target accuracy. The latter is by far the most effective in at least one relevant example and has theoretical interest in that it is the first practical estimator of the full Hessian whose complexity, as a multiple of that of the only-price implementation, does not grow with the dimension of the problem.