It is very common for homes to sell at appraised value. Now, that being said I am talking about appraised value from an appraiser not from the city tax assessment. Those can unfortunately be all over the board these days. If it was an independent appraisal which you or your grandmother had done, then look at the homes which the appraiser used as comps to determine your value. Are those homes actually similar and close to the subject property? If you are getting a loan on the property, you will also have to get yet another appraisal done in order to get your loan anyway. If you end up with two similar appraisals, then I would feel comfortable moving forward. Even if you did end up paying a little more for the home than you would from a seller you did not know, at least you are keeping the money in the family!