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

Abstract

Suppose $\mu$ and $\nu$ are probability measures on $\mathbb{R}$ satisfying $\mu {\le }_{cx}\nu$. Let $a$ and $b$ be convex functions on $\mathbb{R}$ with $a\ge b\ge 0$. We are interested in finding $$\underset{\mathbf{M}}{\sup }\underset{\tau }{\sup }{\mathbb{E}}^{\mathbf{M}}\left[a(X)I_{\{\tau =1\}}+b(Y)I_{\{\tau =2\}}\right]$$ where the first supremum is taken over consistent models $\mathbf{M}$ (i.e., filtered probability spaces $(\Omega ,\mathbf{F},\mathbb{F},\mathbb{P})$ such that $Z=(z,Z_1,Z_2)=({\int }_{\mathbb{R}}x\mu (dx)={\int }_{\mathbb{R}}y\nu (dy),X,Y)$ is a $(\mathbb{F},\mathbb{P})$ martingale, where $X$ has law $\mu$ and $Y$ has law $\nu$ under $\mathbb{P}$) and $\tau$ 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 $\mu$ and $\nu$ (namely that $\mu$ and $\nu$ 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 $\mu$ and $\nu$ 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.

55 pages, 6 figures. In the new version we work with arbitrary convex payoffs and marginal distributions that satisfy the Dispersion Assumption