Let a market consist of a stock \(S_t\) and a bond \(B_t\) governed by the equations \[ dS_t= a(t,S_t)S_tdt+ \sigma_tS_tdw_t \] and \[ dB_t=r_t B_tdt,\;B_0=1, \] where \(r_t\) is a bounded, nonnegative, progressively measurable interest rate process. The volatility \(\sigma_t\) is supposed to be random and satisfying on another stochastic differential equation. The market \((S_t,B_t)\) is shown to be incomplete, and a PDE for the risk-minimizing price of any contingent claim is derived, using its minimal equivalent martingale measure. A similar result for an another model with stochastic volatility was obtained by \textit{G. B. Di Masi}, \textit{Yu. M. Kabanov} and \textit{W. J. Runggaldier} [Theory Probab. Appl. 39, 172-182 (1994; Zbl 0836.60075)]. Next, it is assumed that the \(\sigma_t\) is observed through a process \(Y_t\) subject to random error. A price formula and a PDE are derived regarding the \(S_t\) and the \(Y_t\) as parameters. Finally, \(S_t\) is supposed to be observed. In this case the market is complete, and any contingent claim is priced by an arbitrage argument instead of risk-minimizing.