Convex ordering for stochastic control: the (path dependent) swing contracts case

TL;DR

This paper studies how convexity properties propagate in stochastic control problems, specifically in swing option pricing, demonstrating convexity of the value function and introducing criteria for monotonicity with numerical validation.

Contribution

It establishes convexity of the value function in stochastic control for swing options and introduces a domination criterion for parameter monotonicity, relaxing volatility assumptions.

Findings

Value function is convex in the underlying asset price.

Convexity assumption on volatility can be relaxed to semi-convexity.

Numerical illustrations validate theoretical results.

Abstract

We investigate propagation of convexity and convex ordering on a typical discrete-time stochastic optimal control problem, namely the pricing of swing option. The dynamics of the underlying asset is modelled by the Euler scheme of a Brownian diffusion with affine drift, and convex volatility. 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 functional monotonicity of the value function with respect to parameters of the underlying dynamics. 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 volatility dynamics can be relaxed with a semi-convexity assumption. Finally, to validate our results, we also conduct numerical illustrations.