Not unless the value of the stocks is such that a non-probate small estate proceeding can be used. Otherwise, your agreement is irrelevant. No one has legal authority to agree, at this point. Probate is necessary, whether the stocks are mentioned in the Will or not. You have no authority as executor until an estate is opened and the court gives you that authority. You should have a probate attorney assist you with this.
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I actually think you may be able to transfer the stock if all of the personal property at your mother's death (that didn't pay on death to anyone, i.e. only those assets in her name with no beneficiary or joint owner) is less than the small estate threshold ($50,000). Per the small estate act you have to wait at least 60 days after her passing before requesting the transfer. I will say that as a practical matter some large companies are uncomfortable or unfamiliar with these small estate exceptions and will not transfer them to any person other than a court appointed administrator.
But court appointed administration is costly and cumbersome. It is worth the cost of a consultation fee to contact an estate administration attorney for guidance.
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