Fast quadrature methods for options with discrete dividends

Discontinuities in the stock price at ex-dividend dates make it hard to derive mathematically elegant solutions for European-style options with discrete dividends under the piecewise lognormal model. Numerical schemes such as non-recombining trees are either computationally intensive, lack accuracy when computations are fast, or are unable of handling discrete barriers. Quadrature techniques circumvent the latter two problems but due to the occurrence of time-consuming multifold integrals for multi-dividend options, their implementations have been limited to three-dividend European options only. This research proposes a faster computational method where the integrand is approximated by a Chebyshev polynomial and the fast Fejér quadrature is used to evaluate integral representations of European, American, Bermudan, continuous and discrete barrier options. The method has the capability of accurately handling an arbitrary number of discrete dividend payments in an efficient manner. The computational superiority and higher accuracy of the proposed approach is demonstrated via an extensive set of numerical results.