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 (−aMb)) 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(NMlog (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.

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