If the spouses owned the stock as joint tenants with right of survivorship, then the surviving spouse became the sole owner after the first spouse died. A call to the brokerage firm will usually tell you what documentation you need to provide.
As far as the taxes, when you sell the shares, the way in which you’re taxed depends on where you live. Most states say that half of the investment’s tax basis was stepped up when the first spouse died. This means that when you sell, the capital gains or losses on your half of the investment will be based on the stock’s value when you originally purchased it. However, the basis of the spouse’s share will be based on its value at the time of your spouse’s death.
Let’s say that you and your wife purchased shares of stock for $20,000. The stock was worth $70,000 when she died, and you sold the shares for $80,000 sometime later. Each spouse started out with a basis of $10,000 (one-half of the original $20,000 investment). Because the stock was worth $70,000 when the spouse died, the basis of that half got bumped up to $35,000. When you eventually sell all the shares, the basis will be $45,000 (your original $10,000 and the stepped-up $35,000). The surviving spouse will be taxed on a capital gain of $35,000 ($80,000 minus $45,000). Even if the shares are sold less than a year after she died, you’ll still pay long-term capital-gains taxes on the profit.
However, if you live in a community-property state like Arizona, California, Idaho, Louisiana, Nevada, New Mexico, Texas, Washington, and Wisconsin, the calculations are different. In those nine community-property states, the entire investment would be stepped up to its value at the date of the first spouse’s death. If the shares were worth $70,000 when she died, and you sold them for $80,000, you’d be taxed only on the $10,000 increase.
Keep records of the value of the investment when you originally purchased it, as well as its value when the joint owner died. If you don’t have those types of records, your financial institution should be able to help you find the information.
Reference: Kiplinger (April 7, 2017) “Selling Jointly Owned Stock After the Death of a Spouse”