You can ask to surrender the annuity. If you have owned the annuity for less than seven years or so, you may have to pay a surrender charge. That fee can start at around 7% if you pull out in the first year you own the annuity, and then it typically declines by one percentage point a year until it disappears after seven or eight years. You also will have to pay income tax on all the investment earnings in your annuity, and if you are younger than 59 ½ you typically will be hit with a 10% early withdrawal penalty courtesy of the IRS.
Alternatively, you can opt to transfer your money to another annuity in what is known as a 1035 exchange without incurring any taxes. But you should be very wary if a salesperson tells you a 1035 swap is a great deal, because it comes with a fat sales commission for the insurance agent.
What's more, by moving into a new annuity, you will start a new surrender period. For example, say you have owned an annuity for 10 years. You probably could close out your account without paying a surrender charge. But if you swap that annuity for a new one, you will be hit with a surrender charge of about 7% to close the account within the next seven years or so.