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arXiv · 1712.00077

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

Carolyn E. Phelan·Daniele Marazzina·Gianluca Fusai·Guido Germano
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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.

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Carolyn E. Phelan, Daniele Marazzina, Gianluca Fusai, Guido Germano. 2017-11-30. Fluctuation identities with continuous monitoring and their application to price barrier options. https://arxiv.org/abs/1712.00077

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