Revisiting linear and lognormal stochastic volatility models

In this paper we revisit linear stochastic volatility models with correlated Brownian noises. In such models the asset price satisfies a linear SDE with coefficient of linearity being the volatility process, and a volatility equation with time-dependent coefficients. This class contains among others the Black–Scholes model, the Heston model and the log-normal stochastic volatility model. We present a representation theorem for the density of price, conditions ensuring smoothness of density and some other properties. As an application of using of our general framework we can refine the results for the log-normal stochastic volatility model with correlated noises.