Model-independent upper bounds for the prices of Bermudan options with convex payoffs

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 $$\sup_{\mathbf{M}} \sup_τ \mathbb{E}^{\mathbf{M}} \left[ a(X) I_{ \{ τ= 1 \} } + b(Y) I_{ \{ τ= 2 \} } \right] $$ where the first supremum is taken over consistent models $\mathbf{M}$ (i.e., filtered probability spaces $(Ω, \mathbf{F}, \mathbb{F}, \mathbb{P})$ 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 under some structural assumptions on the measures $μ$ and $ν$ (namely that $μ$ and $ν$ are absolutely continuous probability measures that satisfy 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.