Computing exponential moments of the discrete maximum of a Lévy process and lookback options

We present a fast and accurate method to compute exponential moments of the discretely observed maximum of a Lévy process. The method involves a sequential evaluation of Hilbert transforms of expressions involving the characteristic function of the (Esscher-transformed) Lévy process. It can be discretized with exponentially decaying errors of the form O(exp (-aM^b)) for some a,b > 0, where M is the number of discrete points used to compute the Hilbert transform. The discrete approximation can be efficiently implemented using the Toeplitz matrix-vector multiplication algorithm based on the fast Fourier transform, with total computational cost of O(N M log (M)), where N is the number of observations of the maximum. The method is applied to the valuation of European-style discretely monitored floating strike, fixed strike, forward start and partial lookback options (both newly written and seasoned) in exponential Lévy models.