Algorithmic Hessians and the Fast Computation of Cross-Gamma Risk

Abstract

We introduce a new methodology for computing Hessians from algorithms for function evaluation, using backwards methods. We show that the complexity of the Hessian calculation is a linear function of the number of state variables times the complexity of the original algorithm. We apply our results to computing the Gamma matrix of multi-dimensional financial derivatives including Asian Baskets and cancellable swaps. In particular, our algorithm for computing Gammas of Bermudan cancellable swaps is order O(n^2) per step in the number of rates. We present numerical results demonstrating that the computing all n(n 1)/2 Gammas in the LMM takes roughly n/3 times as long as computing the price.

Keywords: automatic differentiation, Monte Carlo simulation, Greeks, Gamma, LIBOR market model, cancellable

JEL Classification: G13