What is the virtue of fasting the first day of Muharram?

All perfect praise be to Allaah, the Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

We do not know of any authentic text pinpointing the virtue of fasting on the first day of Muharram in particular. However, texts generally prove that it is permissible to fast on it. The Prophet, sallallaahu ‘alayhi wa sallam, was authentically reported to have encouraged to observe much fasting in the month of Muharram. He said: "The best fasting second to the obligatory one is in the month of Allaah that is called Muharram." [Muslim]

It was also narrated that the Prophet, sallallaahu ‘alayhi wa sallam, used to fast three days at the beginning of every month, such as Saturday, Sunday and Monday of one month or Tuesday, Wednesday and Thursday from another. [At-Tirmithi: Hasan]

So, if a Muslim observes fasting on the first day of Muharram in that way and based on these general texts, it would be a good act. As to believing that this has a special virtue, however, we do not know any text to that effect.

Allaah Knows best.