Vibrato Monte Carlo Sensitivities

Abstract

We show how the benefits of the pathwise sensitivity approach to computing Monte Carlo Greeks can be extended to discontinuous payoff functions through a combination of the pathwise approach and the Likelihood Ratio Method. With a variance reduction modification, this results in an estimator which for timestep h has a variance which is O(h −1/2 ) for discontinuous payoffs and O(1) for continuous payoffs. Numerical results confirm the variance is much lower than the O(h −1 ) variance of the Likelihood Ratio Method, and the approach is also compatible with the use of adjoints to obtain multiple first order sensitivities at a fixed cost.

On the data-driven COS method

“…Existing approaches are based on Monte Carlobased techniques, like on finite-differences (bump and revalue), pathwise or likelihood ratio techniques, for which details can be found in [14] , chapter 7. Several extensions and improvements of these approaches have appeared, for example, based on adjoint formulations [15] , the ChF [16,17] , Malliavin calculus [18,19] , algorithmic differentiation [20,21] or combinations of these [22][23][24] . Intuitively, the ddCOS method follows a similar approach as the likelihood ratio method, i.e.…”

Computing Greeks Using Multilevel Path Simulation

“…This leads to new challenges: the use of naive algorithms is often impossible because of the inapplicability of pathwise sensitivities to discontinuous payoffs. These challenges can be addressed in three different ways: payoff smoothing using conditional expectations of the payoff before maturity [5]; an approximation of the above technique using path splitting for the final timestep [1]; the use of a hybrid combination of pathwise sensitivity and the Likelihood Ratio Method [4]. We discuss the strengths and weaknesses of these alternatives in different multilevel Monte Carlo settings.…”