Stable Monte-Carlo Sensitivities of Bermudan Callable Products

Algorithmic Hessians and the fast computation of cross-gamma risk

“…This extends the "sausage Monte Carlo" notion of Piterbarg (2003) and ensures that all our functions have Lipschitz continuous first derivatives. It also bears some similarity to the smoothing function of Fries (2009). While we have a slightly suboptimal exercise strategy, we shall see that this has only a small effect on the gammas.…”

“…For Bermudan-type derivatives, the discontinuity in the first derivative arises in this case from the exercise decision and, indeed, if care is not taken, one can obtain pathwise values that are not even continuous functions of the evolving rates; a small change in level can lead to a change in exercise time and thus greatly different cash flows. Our approach is an adaptation of the sausage Monte Carlo in Piterbarg (2003) with an additional smoothing and is also influenced by the work of Fries (2009). For European derivatives, our approach is very similar to that of Liu and Hong (2010), and we refer the reader to that paper for analysis of the method.…”