In this paper we study the asset pricing problem when the volatility is random. First, we derive a PDE for the risk-minimizing price of any contingent claim. Secondly, we assume that the volatility process sigma (t) is observed through an observation process Y-t subject to random error. A price formula and a PDE are then derived regarding the stock price S-t and the observation process Y-t as parameters. Finally, we assume that S-t is observed. In this case we have a complete market and any contingent claim is then priced by an arbitrage argument instead of by risk-minimizing.