(a) A new beneficiary under a "Cash Refund-Contributions" option (Option One-half), or "a Cash Refund-Initial Value" option (Option One) may be designated at any time during the member's life.

(b) A new beneficiary under any other option may be designated at any time within the period provided for the making of an election pursuant to this section. d. In the event of the death of a retired member, the installment of his retirement allowance, which would have become due and payable next following his death, shall be pro-rated as of the date of his death. The amount of such installment, as so pro-rated, shall be paid as follows: 1. If the member shall have:

(a) Elected to receive an optional benefit pursuant to this section, and

(b) Designated a beneficiary pursuant to this section, such amount shall be paid to such beneficiary, if such beneficiary survives him. In any other case such amount shall be paid to the retired member's estate or pursuant to section one thousand three hundred ten of the surrogate's court procedure act. 2. If the member shall not have elected to receive an optional benefit, such amount shall be paid to the beneficiary designated by him pursuant to subdivision d of section fifty-one of this article. In the event the appropriately designated beneficiary does not survive such member, or if he shall not have so designated a beneficiary, such amount shall be payable to the retired member's estate or pursuant to section one thousand three hundred ten of the surrogate's court procedure act. e. Notwithstanding any other provision of this article, an option selection previously filed by a member or retired member subject to the provisions of this section may be changed no later than thirty days following the date of payability of his or her retirement allowance. A retired member who has been retired for disability may change an option selection previously filed no later than (1) thirty days following the date on which such member's application for disability retirement was approved by the retirement board or (2) thirty days following the date on which such retiree was retired for disability, whichever is later.