Before revision:  Examples of such models are considered, including a log-normal stochastic volatility model . In all examples a closed formula for the density function is given. In the Appendix we present a conditional version of the Donati-Martin and Yor formula .
After revision:  Examples of such models are considered, including a log-normal stochastic volatility model and Heston stochastic volatility model. For a linear stochastic volatility model we derive representations for the probability density function of the arbitrage price of a financial asset and the prices of European call and put options. A closed-form formulae for the density function and the prices of European call and put options are given for log-normal stochastic volatility model. We also obtain present some new results for Heston and extended Heston stochastic volatility models .