Convex ordering for stochastic control: the swing contracts case

Abstract

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.