Efficient simulation of Greeks of multiasset European and Asian style options by Malliavin calculus and quasi-Monte Carlo methods

Abstract

This paper discusses simulation of sensitivities or Greeks of multiasset European and Asian style options by Malliavin calculus combined with Monte Carlo and quasi-Monte Carlo methods. By using the Malliavin calculus, we are able to express Greeks of both European and Asian style multiasset options explicitly in terms of the expectations of the option payoff functions multiplied by the Malliavin weights. For European multiasset options, these expectations are further converted to integrals over hypercubes in order to make use the better uniformity of low discrepancy sequences. Numerical results show the advantages of Malliavin calculus method to the finite difference method for options with nonsmooth payoffs. The superiority of the first method over the second one is even more significant when these methods are combined with quasi-Monte Carlo methods. For example, when simulating Δ1 of a basket binary option or a basket “double” binary option, the efficiencies reach up to tens of thousands with good lattice points sequences; and the efficiencies are up to tens of millions or higher with the same type of sequences when simulating Γ11 or Γ12 for the same type of options.