What is the legal valuation date of the property I want to buy out?

Let's keep this simple. You clearly have an investment worthy of paying a lawyer to review/prepare appropriate paperwork. You can obtain free counseling and assistance in self-representation from the Family Law Facilitator at the Courthouse.

So long as you each own half the property, the valuation would be the day the contract is drawn up and executed by both parties, unless you both want to agree to a certain amount or an earlier value and put that in writing.

Not being able to read the controlling documents re exit time and valuation of property, as to when the valuation occurs, generally, valuation occurs when property can be sold or otherwise disposed of to a willing buyer by a willing seller.

Why should he sell to you for the 2013 value if the value is more today? If you have no written agreement or court order, the value is what you agree on.

There are some facts missing. I assume you are speaking as to your ex-husband. How did he get the interest in your mother's property. If it was from your mother transferring the interest to you and you then treated that interest as being community property so he was entitled to half, you may be overly generous to him. A gift or sale of the property by your mother to you specifically and not the two of you, whether you were married at the time or not, is your private property and is not community property unless you commingled it with your community property. If it is community property, if you used non-community assets to take care of it pay taxes, make improvements] you would first be entitled to be refunded for that. The date you determine the value of the property is the date you agreed to with him. If you did not agree to any date and the only agreement was that when you had the funds you would buy him out, I guess the date of valuation would be the date you actually offer him the funds that are his [which has not yet happened]. The sale price of the home back in 2013 would not be the date of the valuation nor the determination of its value. At the time of the sale, the value would have been what the house sold for less any mortgages, costs involved in paying off the mortgage, costs of selling the house, and taxes that had to be paid. If the two of you were in a different tax bracket so the taxes you owed were different, that would add another complication. So there is not a simple answer to your question.

A community property pierce of real estate will be valued at the date of the buyout. There can be exceptions, but they are rare.