Office of Scientific and International Cooperation

Abstract: Several path-dependent options with discrete monitoring like barrier, lookback and alpha-quantile can be priced exploiting fluctuation identities, which provide the z-transform of the characteristic function of the extrema of a L’evy stochastic process, possibly conditional on touching certain levels. A numerical inverse z-transform (IZT) is necessary to retrieve the characteristic function for pricing. Implementing the IZT with an Euler summation, the CPU time of Fourier-z methods is independent of the number of monitoring dates, whereas an alternative approach that iterates over all monitoring dates has a CPU time proportional to the latter. Both pricing methods share as a key component a fast Hilbert transform with a sinc functions expansion a> nd achieve exponential convergence of the error on the grid size. So far there was a tradeoff between the higher efficiency of the Fourier-z method and the higher accuracy of the iterative Hilbert transform method. We overcome this limitation developing an improved IZT algorithm that better exploits the machine accuracy. We validate our new IZT on an analytic z-transform paier and on the pricing of a double-barrier option with normal inverse Gaussian, Kou and variance gamma processes. Thanks to the higher accuracy, we are able to show for the first time that pricing \alpha-quantile options with the Spitzer identities has exponential error convergence on the grid size. Beyond our actual motivation and test cases, the IZT has many applications other than with fluctuation identities, and the latter have many applications in applied probability outside finance; the problem is connected with the accurate numerical computation of Fourier integrals and thus is even more general.