Spectral binomial tree: New algorithms for pricing barrier options

• Binomial tree is popular but slow to converge to calculate barrier option prices.
• We incorporate the spectral expansion method into it for pricing barrier options.
• The original idea comes from the eigenexpansion approach in PDEs.
• It can compute double barrier options with one billion steps within 0.07 s.
• The prices are always the same as those by conventional binomial trees.

This paper introduces new and significantly fast algorithms to evaluate the price of double barrier options using binomial trees. To compute the price of double barrier options accurately, trees with large numbers of steps must be used, which is time consuming. In order to overcome this weakness, we develop new computational algorithms based on the spectral expansion method. The original idea of this method is coming from the eigenexpansion approach in PDEs. We show that this method enables us to compute double barrier options within 0.07 s, even if we use binomial trees with one billion steps. Moreover, this algorithm is easy to implement. In addition, the prices obtained by the proposed approach are always the same as those obtained by conventional binomial trees and show a good approximation to those by earlier studies.