Paper: arXiv 2406.07464

Authors: Gilles Pagès, Christian Yeo

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides rigorous theoretical proofs regarding convexity propagation and convex ordering for discrete-time stochastic optimal control applied to swing contracts. While mathematically dense and novel in extending convex ordering to optimal control problems, empirical validation is minimal and limited to illustrative numerical examples.

Research Flowchart

  flowchart TD
    A[Research Goal: Prove convexity propagation & convex ordering for swing options] --> B[Input: Euler scheme of Brownian diffusion with affine drift & convex volatility]
    B --> C[Methodology: Stein's formula, regularization, & stochastic optimal control]
    C --> D[Analysis: Relax 1D volatility to semi-convexity & establish domination criterion]
    D --> E[Numerical Illustrations & Validations]
    E --> F[Outcomes: Value function is convex in asset price & monotone w.r.t. parameters]

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1 of 46 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.