🤖 AI Summary
This study addresses the pricing of take-or-pay swing options in energy markets, where the underlying asset follows a non-Markovian ARCH-type process with convex (or semiconvex) coefficients. To tackle this discrete-time stochastic optimal control problem, we establish— for the first time under non-Markovian dynamics—a propagation-of-convexity theory for the value function with respect to the asset price, relaxing the classical convex-coefficient assumption to semiconvexity and thereby substantially broadening applicability. Leveraging tools from convex analysis, stochastic control, Stein’s identity, and regularization techniques, we rigorously prove the convexity and monotonicity of the value function in key parameters, and derive verifiable convex-order dominance criteria. Numerical experiments confirm both the validity and robustness of the theoretical results.
📝 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.