Analytic smoothing properties of a general, strongly parabolic linear Cauchy problem of second order in RN×(0,T) with analytic coefficients (in space and time variables) are investigated. They are expressed in terms of holomorphic continuation of global (weak) L2-type solutions to the system. Given 0<T′<T⩽∞, it is proved that any L2-type solution u:RN×(0,T)→RM possesses a bounded holomorphic continuation u(x+iy,σ+iτ) into a complex domain in CN×C defined by (x,σ)∈RN×(T′,T), |y|<A′ and |τ|<B′, where A′,B′>0 are constants depending upon T′. The proof uses the extension of a solution to an L2-type solution in a domain in CN×C, such that this extension satisfies the Cauchy–Riemann equations. The holomorphic extension is thus obtained in a Hardy space H2. Applications include market completion by European options in Finance.