  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 accordingly 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 locally weakly path-dependent functionals in our
classification. Hence, we derive the weighted-expectation formulas for their
Greeks. 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 the computation of Greeks. Moreover, we are also
able to consider the more general dynamics of path-dependent volatility. These
were not achieved using Malliavin calculus.
