I have committed Zina in the month of Ramadhaan and broke my fast intentionally without any reason. I came to understanding that the ruling is: 1) to free a slave, 2) to fast 2 months consecutively, 3) to feed 60 indigents. But since there are no slaves to free, my second option is to fast 2 months and I am able to do that, but the problem is that I live with my parents and don't want to let them know that I have committed this grave major sin. I don't want to lose the respect of my parents. I am confused and don't know what to do, so please brother can you advice me.

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 is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

The fornication that you have committed in Ramadan is a great major sin which necessitates a true repentance to Allaah. Allaah says (which means): {And come not near the unlawful sexual intercourse. Verily it is a fahishah (i.e. anything that transgresses its limits; a great sin) and an evil way (that leads one to Hell) unless Allaah forgives him/her.}[17:32]. This sin is even more abominable as it happened during the day in Ramadan, and this is transgressing the honourable rank of this month.

Therefore, you have to repent sincerely to Allaah fulfilling the conditions of a true repentance; for details refer to Fatwa: 87903. Additionally, you have to expiate for intentionally breaking the fast, which is to free a slave, and if you could not find one, then you have to fast for two consecutive months, and if again you are unable to fast then you have to feed sixty poor persons. However, since you could not find a slave to free and you are able to fast, as you have stated, then it is obligatory on you to fast for two consecutive months. What you have mentioned, which is that you are living with your parents, is not an excuse for not fasting. It was more appropriate in the first place to be ashamed of Allaah when you committed that great and abominable sin. So you have to start fasting before the angel of death comes to you. Being ashamed that people would discover your sin does not mean that you can abandon the expiation that is required for that sin. Of course, a Muslim is required to conceal his sins, as the Prophet said: "All my nation is pardoned except those who openly expose their sins." [Muslim]

However, you can fast and use Tawriyah (saying something which has more than one meaning and intending a meaning different from what the listener is likely to understand). A  Muslim could fast because of a vow that he made to Allaah, to educate himself or to diminish his desires and so forth. So you can say something and make them understand that you are fasting to fulfil a vow. 

Allaah knows best.