  We investigate the approximation of path functionals. In particular, we
advocate the use of the Karhunen-Lo\`eve expansion, the continuous analogue of
Principal Component Analysis, to extract relevant information from the image of
a functional. Having accurate estimate of functionals is of paramount
importance in the context of exotic derivatives pricing, as presented in the
practical applications. Specifically, we show how a simulation-based procedure,
which we call the Karhunen-Lo\`eve Monte Carlo (KLMC) algorithm, allows fast
and efficient computation of the price of path-dependent options. We also
explore the path signature as an alternative tool to project both paths and
functionals.
