Fluctuation identities with continuous monitoring and their application to price barrier options

TL;DR

This paper develops a numerical method to compute fluctuation identities for exponential Lévy processes under continuous monitoring, enabling accurate pricing of barrier options and filling a gap in existing discrete-only approaches.

Contribution

The authors introduce a novel numerical scheme for continuous monitoring fluctuation identities, including the two-barrier exit problem, with detailed error analysis and application to barrier option pricing.

Findings

The method accurately computes fluctuation identities in continuous monitoring.

Error bounds are established based on sinc-based Hilbert transform truncation.

Continuous monitoring results serve as a limit for discretely monitored schemes as time steps approach zero.

Abstract

We present a numerical scheme to calculate fluctuation identities for exponential L\'evy 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 L\'evy 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.