On the effects of dimension reduction techniques on some high-dimensional problems in finance

Many problems in finance can be formulated as high-dimensional integrals, which are often attacked by quasi-Monte Carlo (QMC) algorithms. To enhance QMC algorithms, dimension reduction techniques, such as the Brownian bridge (BB) and principal component analysis (PCA), are used to reduce the effective dimension. This paper explores in depth the effects of these techniques on the dimension structure of some typical high-dimensional problems from finance: the pricing of path-dependent options and bond valuation according to term structure models. By deriving explicit expressions for the underlying integrands and the associated weights that control the relative importance of different variables, and by investigating the variance ratios, the effective dimensions, the mean dimension, and their limiting behavior as the nominal dimension tends to infinity, we show theoretically and empirically how and to what extent the BB and PCA algorithms change the dimension structure (including the degree of additivity) of the underlying functions. They change the functions to be strongly weighted and substantially reduce the effective dimensions and the mean dimension; and they enhance the degree of additivity, which is particularly important for QMC. Moreover, the resulting functions are of low effective dimension. not only in the superposition sense, but also in the truncation sense. The variance ratios, the effective dimensions, and the mean dimension associated with these techniques are very insensitive to the nominal dimension (they are essentially constant), which highlights the possibility of removing the curse of dimensionality when dimension reduction techniques are used in combination with QMC. A counterexample is also shown for which the BB and PCA may increase the effective dimension. The investigation provides further insight into the effects of dimension reduction techniques.