Paper: arXiv 2503.13328

Authors: David Hobson, Dominykas Norgilas

Abstract

Suppose $μ$ and $ν$ are probability measures on $\mathbb R$ satisfying $μ\leq_{cx} ν$. Let $a$ and $b$ be convex functions on $\mathbb R$ with $a \geq b \geq 0$. We are interested in finding [ \sup_{\mathcal M} \sup_τ \mathbb{E}^{\mathcal M} \left[ a(X) I_{ { τ= 1 } } + b(Y) I_{ { τ= 2 } } \right] ] where the first supremum is taken over consistent models $\mathcal M$ (i.e., filtered probability spaces $(Ω, \mathcal F, \mathbb F, \mathbb P)$) such that $Z=(z,Z_1,Z_2)=(\int_{\mathbb R} x μ(dx) = \int_{\mathbb R} y ν(dy), X, Y)$ is a $(\mathbb F,\mathbb P)$ martingale, where $X$ has law $μ$ and $Y$ has law $ν$ under $\mathbb P$) and $τ$ in the second supremum is a $(\mathbb F,\mathbb P)$-stopping time taking values in ${1,2}$. Our contributions are first to characterise and simplify the dual problem, and second to completely solve the problem in the symmetric case under the dispersion assumption. A key finding is that the canonical set-up in which the filtration is that generated by $Z$ is not rich enough to define an optimal model and additional randomisation is required. This holds even though the marginal laws $μ$ and $ν$ are atom-free. The problem has an interpretation of finding the robust, or model-free, no-arbitrage bound on the price of a Bermudan option with two possible exercise dates, given the prices of co-maturing European options.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a rigorous theoretical contribution to martingale optimal transport and model-independent pricing of Bermudan options. It relies on deep mathematical proofs and duality arguments, though it contains no empirical market data or backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Find model-free upper bounds for 2-period Bermudan options with convex payoffs"] --> B["Inputs: Marginal measures (μ ≤cx ν) & Convex payoff functions (a ≥ b ≥ 0)"]
    B --> C["Formulate Dual Problem via Martingale Optimal Transport (MOT)"]
    C --> D["Characterize & Simplify Dual Optimization"]
    D --> E["Solve Symmetric Case under Dispersion Assumption"]
    E --> F["Outcome 1: Model-independent upper price bound established"]
    E --> G["Outcome 2: Optimal model requires external randomization beyond standard filtration"]