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

David Hobson and Dominykas Norgilas

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Abstract: Suppose $\mu$ and $\nu$ are probability measures on $\mathbb R$ satisfying $\mu \leq_{cx} \nu$. Let $a$ and $b$ be convex functions on $\mathbb R$ with $a \geq b \geq 0$. We are interested in finding \[ \sup_{\mathcal M} \sup_{\tau} \mathbb{E}^{\mathcal M} \left[ a(X) I_{ \{ \tau = 1 \} } + b(Y) I_{ \{ \tau = 2 \} } \right] \] where the first supremum is taken over consistent models $\mathcal M$ (i.e., filtered probability spaces $(\Omega, \mathcal F, \mathbb F, \mathbb P)$) 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 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 $\mu$ 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.

Date: 2025-03, Revised 2025-03
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