Personally, I think it's silly to discuss the budget in closed session. However, per
VA § 55-510.1:
The board of directors or any subcommittee or other committee thereof may convene in executive session to consider personnel matters; consult with legal counsel; discuss and consider contracts, pending or probable litigation and matters involving violations of the declaration or rules and regulations adopted pursuant thereto for which a member, his family members, tenants, guests or other invitees are responsible; or discuss and consider the personal liability of members to the association, upon the affirmative vote in an open meeting to assemble in executive session.
Therefore, where the budget addresses those areas, those areas ONLY may be discussed in executive session. However, the remaining budget discussion should be conducted in open session.
My Association has the Treasurer prepare the draft budget for Board approval. The Board discusses it, makes changes and then approves the budget for presentation to the membership (it now becomes a proposed budget). Once the membership has an opportunity to make comments at the annual meeting, the new board would review the proposed budget, make any changes they feel are necessary and adopt the budget by majority vote.