Fluctuation identities with continuous monitoring and their application to the pricing of barrier options

Abstract

• We calculate fluctuation identities for the continuous-monitoring case. • To do so, we obtain the Wiener–Hopf factors in the Fourier-Laplace domain. • We apply this to price continuously monitored single- and double-barrier options. • We provide analytical error bounds which match well the numerical results. • This gives further insight into the convergence of the discretely monitored case. We present a numerical scheme to calculate fluctuation identities for exponential Lévy processes in the continuous monitoring case. This includes the Spitzer identities for touching a single upper or lower barrier, and the more difficult case of the two-barriers exit problem. These identities are given in the Fourier-Laplace domain and require numerical inverse transforms. Thus we cover a gap in the literature that has mainly studied the discrete monitoring case; indeed, there are no existing numerical methods that deal with the continuous case. As a motivating application we price continuously monitored barrier options with the underlying asset modelled by an exponential Lévy process. We perform a detailed error analysis of the method and develop error bounds to show how the performance is limited by the truncation error of the sinc-based fast Hilbert transform used for the Wiener–Hopf factorisation. By comparing the results for our new technique with those for the discretely monitored case (which is in the Fourier- z domain) as the monitoring time step approaches zero, we show that the error convergence with continuous monitoring represents a limit for the discretely monitored scheme.