We derive probabilistic representations for the probability density function
of the arbitrage price of a financial asset and the price of European call and
put options in a linear stochastic volatility model with correlated Brownian
noises. In such models the asset price satisfies a linear SDE with coefficient
of linearity being the volatility process. 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.