  We investigate propagation of convexity and convex ordering on a typical
stochastic optimal control problem, namely the pricing of
\q{\emph{Take-or-Pay}} swing option, a financial derivative product commonly
traded on energy markets. The dynamics of the underlying asset is modelled by
an \emph{ARCH} model with convex coefficients. We prove that the value function
associated to the stochastic optimal control problem is a convex function of
the underlying asset price. We also introduce a domination criterion offering
insights into the monotonicity of the value function with respect to parameters
of the underlying \emph{ARCH} coefficients. We particularly focus on the
one-dimensional setting where, by means of Stein's formula and regularization
techniques, we show that the convexity assumption for the \emph{ARCH}
coefficients can be relaxed with a semi-convexity assumption. To validate the
results presented in this paper, we also conduct numerical illustrations.
