Absinthe Pernod, Kashrus, and Passover

I just called the Kashrus Department of The London Beis Din, KLBD, at +44 20 8343 6255. The KLBD site lists two Pernod absinthes, "Absinthe 40% Vol" and "Absinthe 68% Vol", as being kosher (and parve). I asked the gentleman who answered what alcohol they use. He asked me why I want to know, and I told him that I want to know if I can keep them at home during Pesach. He answered that he couldn't give me too much detail, since the information they get from the manufacturer is covered by a confidentiality agreement, but he does have a full list of ingredients. His answer to my question was quite clear, if a little cagy, as he warned me it would be: The Pernod absinthes listed definitely contain grain - I assume that this means that they are distilled from grain alcohol, which was my original question - and "they would have to be sold on Pesach".