Functional It\^o Calculus, Path-dependence and the Computation of Greeks

Dupire's functional It\^o calculus provides an alternative approach to the classical Malliavin calculus for the computation of sensitivities, also called Greeks, of path-dependent derivatives prices. In this paper, we introduce a measure of path-dependence of functionals within the functional It\^o calculus framework. Namely, we consider the Lie bracket of the space and time functional derivatives, which we use to classify functionals according to their degree of path-dependence. We then revisit the problem of efficient numerical computation of Greeks for path-dependent derivatives using integration by parts techniques. Special attention is paid to path-dependent functionals with zero Lie bracket, called weakly path-dependent functionals in our classification. We then derive the weighted-expectation formulas for their Greeks, that was first derived using Malliavin calculus. In the more general case of fully path-dependent functionals, we show that, equipped with the functional It\^o calculus, we are able to analyze the effect of the Lie bracket on computation of Greeks. This was not achieved using Malliavin calculus. Numerical examples are also provided.