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

Projection of Functionals and Fast Pricing of Exotic Options