Bards are designed as a light-armor wearing class that uses spells to complement his abilities. Since the assassin is designed as a light-armor wearing that uses spells to complement his abilities, I believe that the intent of the rules is for assassins to cast spells while wearing light armor.
Going by RAW, how does a bard cast spells? He casts arcane spells which suffer no ASF% while wearing light armor, so an assassin that casts spells "as a bard" would be the same. If it were intended that assassins suffer ASF%, then I believe they would be labeled as casting spells "as a sorcerer," who casts spells spontaneously and whose spells do suffer the arcane spell failure chance. The only differences between ASN and BRD spellcasting that are listed is that assassins use Int instead of Cha, and that their spell lists are different.