In this paper, a rapid and high accurate numerical method for pricing
discrete single and double barrier knock-out call options is presented.
With regard to the well-known Black-Scholes model, the price of an
option in each monitoring date could be calculated by computing a
recursive integral formula that is based on the heat equation solution.
We have approximated these recursive solutions with the aid of Lagrange
interpolation on Jacobi polynomial nodes. After that, an operational
matrix, that makes our computation significantly fast, has been derived.
In some theorems, the convergence of the presented method has been shown
and the rate of convergence has been derived. The most important benefit
of this method is that its complexity is very low and does not depend on
the number of monitoring dates. The numerical results confirm the
accuracy and efficiency of the presented numerical algorithm.