Abu Hurairah (May Allah be pleased with him) said: The Messenger of Allah (PBUH) said, "It is not lawful for a woman to observe an optional Saum (fast) without the permission of her husband when he is at home. Nor should she allow anyone to enter his house without his permission."[Al-Bukhari and Muslim].
Fasting here signifies voluntary fasting. Similar is the case of other voluntary prayers, i.e., voluntary Salat, recitation, etc. It is not permissible to do such acts without the permission of the husband if he is present. It should be borne in mind that a women is not permitted to let even her natural relations (who are Mahram) come in her home in the absence of her husband. She can, however, allow such persons to come about whom her husband has given express permission or in respect of whom he keeps silent.