Projection of Functionals and Fast Pricing of Exotic Options

We investigate the approximation of path functionals. In particular, we advocate the use of the Karhunen-Loève 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-Loève 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.

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