@article{discovery10063605,
            note = {Copyright {\copyright} 2018 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY license. (http://creativecommons.org/licenses/by/4.0/).},
          volume = {271},
           pages = {210--223},
           month = {November},
          number = {1},
         journal = {European Journal of Operational Research},
            year = {2018},
           title = {Fluctuation identities with continuous monitoring and their application to the pricing of barrier options},
       publisher = {ELSEVIER SCIENCE BV},
             url = {https://doi.org/10.1016/j.ejor.2018.04.016},
            issn = {0377-2217},
        abstract = {We present a numerical scheme to calculate fluctuation identities for exponential L{\'e}vy 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{\'e}vy 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.},
          author = {Phelan, CE and Marazzina, D and Fusai, G and Germano, G},
        keywords = {Option pricing, Finance, Wiener-Hopf factorisation, Hilbert transform, Laplace transform, Spectral filter}
}