A Functional Analysis Approach to Static Replication of European Options

The replication of any European contingent claim by a static portfolio of calls and puts with strikes forming a continuum, formally proven by Carr and Madan (1998), is part of the more general theory of integral equations. We apply spectral decomposition techniques to show that replication may also be achieved with a discrete portfolio of special options. We propose a numerical application for fast pricing of vanilla options that may be suitable for large option books or high frequency option trading, and we use a reflected Brownian motion model to show how pricing formulas for the special options may be obtained