This is can be a tricky one. These Zivot Notes discuss a slightly more advanced way to select lags for the ADF. That being said, it is good to remember that purpose of including lags is to control for serial correlation. Consequently, you'll want to examine your error to assure that no serial correlation is present. Even a good model fit (i.e. low IC) does not ensure the absence of serial correlation.
It is important to remember that it is essential to include the lowest number of lags possible. Including erroneous lags will greatly diminish the test's power. This is especially problematic because it is well known that ADF test have low power, especially for near unit root processes.
Additionally, you should consider other tests for stochastic trends like PP or KPSS.
Finally, never loss prospective of the bigger picture. Stock prices, especially at a daily frequency, almost always follow a stochastic trend. If they did not follow a random walk, then you could forecast the prices with a relatively high level of certainty (i.e. the prediction interval for stochastic trends explode). If this was the case then you could easily forecast stock prices and make billions of dollars. But stock markets are efficient, really efficient and there is not billions of dollars laying around to be picked up. At least that is what Eugene Fama says.