Efficient Computation of Option Prices and Greeks by Quasi--Monte Carlo Method with Smoothing and Dimension Reduction

Abstract

Discontinuities and high dimensionality are common in the problems of pricing and hedging of derivative securities. Both factors have a tremendous impact on the accuracy of the quasi--Monte Carlo (QMC) method. An ideal approach to improve the QMC method is to transform the functions to make them smoother and having smaller effective dimension. This paper develops a two-step procedure to tackle the challenging problems of both the discontinuity and the high dimensionality concurrently. In the first step, we adopt the smoothing method to remove the discontinuities of the payoff function, improving the smoothness. In the second step, we propose a general dimension reduction method (called the CQR method) to reduce the effective dimension such that the better quality of QMC points in their initial dimensions can be fully exploited. The CQR method is based on the combination of the $k$-means clustering algorithm and the QR decomposition. The $k$-means clustering algorithm, a classical algorithm of machine lear...