Convex ordering for stochastic control: the swing contracts case

We investigate propagation of convexity and convex ordering on a typical stochastic optimal control problem, namely the pricing of {Take-or-Pay} swing option, a financial derivative product commonly traded on energy markets. The dynamics of the underlying asset is modelled by an 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 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 ARCH coefficients can be relaxed with a semi-convexity assumption. To validate the results presented in this paper, we also conduct numerical illustrations.