Convex order for path-dependent derivatives: a dynamic programming approach

TL;DR

This paper develops a systematic dynamic programming approach to analyze convex order relations for path-dependent derivatives in continuous martingale models, providing bounds and extensions for European and American options.

Contribution

It introduces a novel methodology for propagating convexity in discrete models and extends convex order results to path-dependent derivatives and optimal stopping problems.

Findings

Derived bounds for European option prices in local volatility models.

Extended convex order results to American options with path-dependent payoffs.

Provided a systematic numerical scheme for propagating convexity in discretized models.

Abstract

We investigate the (functional) convex order of for various continuous martingale processes, either with respect to their diffusions coefficients for L\'evy-driven SDEs or their integrands for stochastic integrals. Main results are bordered by counterexamples. Various upper and lower bounds can be derived for path wise European option prices in local volatility models. In view of numerical applications, we adopt a systematic (and symmetric) methodology: (a) propagate the convexity in a {\em simulatable} dominating/dominated discrete time model through a backward induction (or linear dynamical principle); (b) Apply functional weak convergence results to numerical schemes/time discretizations of the continuous time martingale satisfying (a) in order to transfer the convex order properties. Various bounds are derived for European options written on convex pathwise dependent payoffs. We retrieve and extend former results obtains by several authors since the seminal 1985 paper by Hajek . In a second part, we extend this approach to Optimal Stopping problems using a that the Snell envelope satisfies (a') a Backward Dynamical Programming Principle to propagate convexity in discrete time; (b') satisfies abstract convergence results under non-degeneracy assumption on filtrations. Applications to the comparison of American option prices on convex pathwise payoff processes are given obtained by a purely probabilistic arguments.