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

TL;DR

The paper develops a model-free framework for pricing Bermudan options with convex payoffs in a two-period setting, given marginals $\mu\leq_{cx}\nu$ from co-maturing European prices. It derives a dual formulation via martingale optimal transport and shows a sharp simplification: the cheapest superhedge is generated by convex $\psi\ge b$, reducing the dual to a convex-envelope problem and establishing no duality gap under the dispersion and symmetry assumptions. In the symmetric, dispersion setting, the authors construct explicit optimal models using left-curtain, right-curtain, and HK couplings, and provide exact stopping rules and hedges for three cases, proving primal-dual optimality. They also discuss extensions to cases without $a\ge b$, and the necessity of randomization beyond canonical filtration, including stopping at time 0, to capture enhancements in model-based prices. Overall, the work characterizes robust price bounds for Bermudan-style payoffs and clarifies structural requirements for achieving model-free optimality in this class of problems.

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.

Figures (4)

• Figure 1: Sketch of symmetric densities $\rho$ and $\eta$ (under the dispersion assumption, note that $\rho>\eta$ on $(-e,e)$, and $\eta>\rho$ on $[-e,e]^c$), and the locations of $x_0$, $e$, $g^R(x_0)=-f^L (-x_0)$ and $g^L(-x_0)=-f^R (x_0)=0$. Mass in $(f^L(-x_0),-x_0)$ according to the initial law is mapped to the interval $(f^L(-x_0),0)$ according to the target law. Similarly, mass in $(x_0,g^R(x_0))$ is mapped to $(0,g^R(x_0))$. On the other hand, the mass in $(-\alpha,f^L(-x_0))\cup(g^R(x_0),\alpha)$ according to the initial law stays put, while the mass in $(-x_0,x_0)$ according to the initial law is mapped to the tails $(-\beta,f^L(-x_0))\cup(g^R(x_0),\beta)$.
• Figure 2: The Case (C1). The top part of the figure represents the stylized plots of functions $f^L$ and $g^L$ (on $[-e,-x_0]$) (resp. $f^R$ and $g^R$ (on $[e,x_0]$)) that support the left-curtain (resp. right-curtain) martingale coupling. Note that $g^L$ (resp. $f^R$) is non-decreasing, while $f^L$ (resp. $g^R$) is non-increasing on $[-e,-x_0]$ (resp. $[e,x_0]$). Furthermore, the shaded areas correspond to the sets (and associated exercise rules) on which all the optimal models $\mathcal{M}^*$ concentrate: the Bermudan option is exercised at time-1 if $Z_1\notin(-x_0,x_0)$, and then the mass in $(-\alpha,f^L(-x_0))\cup(g^R(x_0),\alpha)$ stays put (i.e., remains on the diagonal) while the mass in $[f^L(-x_0),-x_0]$ is mapped to $[f^L(-x_0), g^L(x_0)=0]$ and the mass in $[x_0,g^R(x_0)]$ is mapped to $[0=f^R(x_0), g^R(x_0)]$. On the other hand, if $Z_1\in(-x_0,x_0)$, then the option is not exercised at time-1 (it will be exercised at time-2) and the mass in $(-x_0,x_0)$ is mapped to the tails $(-\beta,f^L(-x_0))\cup(g^R(x_0),\beta)$ (recall Figure \ref{['fig:densities']}). The bottom part of the figure depicts the payoff functions $a$ and $b$ (with $a>b$), and the candidate convex function $\psi^{*,1}$ in the case (C1). In particular, we have that $\psi^{*,1}=l^L$ on $[f^L(-x_0),g^L(-x_0)=0]$ and $\psi^{*,1}=l^R$ on $[f^R(x_0)=0,g^R(x_0)]$, while $\psi^{*,1}=b$ on $(-\beta,f^L(-x_0))\cup(g^R(x_0),\beta)$.
• Figure 3: The Case (C2). The shaded areas in the top part of the figure represent the sets (and associated exercise rules) on which all the optimal models $\mathcal{M}^*$ concentrate: the Bermudan option is exercised at time-1 if $Z_1\notin(-x_1,x_1)$, and the mass in $(-\alpha,-h(x_1))\cup(h(x_1),\alpha)$ stays put, while the mass in $[-h(x_1),-x_1]\cup[x_1,h(x_1)]$ is mapped to $[-h(x_1),h(x_1)]$. On the other hand, if $Z_1\in(-x_1,x_1)$, then the option will be exercised at time-2 and the mass in $(-x_1,x_1)$ is mapped to the tails $(-\beta,-h(x_1))\cup(h(x_1),\beta)$. The bottom part of the figure shows how a candidate convex function $\psi^{*,1}$ (from Case (C1)) needs to be modified in order to obtain the cheapest superhedging strategy in the Case (C2). Under the assumptions of case (C2), $\psi^{*,1}\geq b$ but it is not convex (see the dash-dotted piece-wise linear curve on $[f^L(-x_0),g^R(x_0)]$ (linear sections correspond to $l^L$ and $l^R$) that has a strictly positive (resp. negative) slope to the left (resp. right) of $0$). However, we can find a pair $(x_1,h(x_1))$, with $x_1\in(0,x_0)$ and $h(x_1)\in(g^R(x_0),\beta)$, and such that the line $l_1$, that goes through $(x_1,a(x_1))$ and $(h(x_1),b(h(x_1)))$, has zero slope. By symmetry, $l_1$ also goes through $(-x_1,a(-x_1)=a(x_1))$ and $(-h(x_1),b(-h(x_1))=b(h(x_1)))$. Then $\psi^{*,2}=\max\{b,l_1\}$ is convex and thus generates a candidate (and in fact optimal) superhedging strategy.
• Figure 4: The Case (C3) with $a(e)>b(e)$. The shaded areas in the top part of the figure represent the sets (and associated exercise rules) on which all the optimal models $\mathcal{M}^*$ concentrate: the Bermudan option is exercised at time-1 if $Z_1\notin(-x_3,x_3)$ (and then the mass in $(-\alpha,f^L(-x_3))\cup(g^R(x_3),\alpha)$ stays put, while the mass in $[f^L(-x_3),-x_3]$ is mapped to $[f^L(-x_3),g^L(-x_3)]$ and the mass in $[x_3,g^R(x_3)]$ is mapped to $[f^R(x_3),g^R(x_3)]$ and if $Z_1\in(g^L(-x_3),f^R(x_3))$ and $U\leq(\eta(Z_1)/\rho(Z_1))$ (note that only a portion of the mass in $(g^L(-x_3),f^R(x_3))$ stays put). On the other hand, the option will be exercised at time-2 if either $Z_1\in(-x_3,g^L(-x_3))\cup(f^R(x_3),x_3)$ (and then the mass in $(-x_3,g^L(-x_3))\cup(f^R(x_3),x_3)$ is mapped to the tails $(-\beta,f^L(-x_3))\cup(g^R(x_3),\beta)$), or $Z_1\in(g^L(-x_3),f^R(x_3))$ and $U>(\eta(Z_1)/\rho(Z_1))$ (and then this portion of mass in $(g^L(-x_3),f^R(x_3))$ is (again) mapped to the tails $(-\beta,f^L(-x_3))\cup(g^R(x_3),\beta)$). In the bottom part of the figure we observe that, in the setting of Case (C3), $\psi^{*,1}$ is convex (see the dash-dotted piece-wise linear curve on $[f^L(-x_0),g^R(x_0)]$), but since $\psi^{*,1}(0)<a(0)$, it is not optimal (even if $\psi^{*,1}\geq b$). However, we can find $x_3\in(x_0,e)$ such that the line $l^R_3$, that goes through $(x_3,a(x_3))$ and $(g^R(x_3),b(g^R(x_3)))$, is such that $l^R_3(f^R(x_3))=a(f^R(x_3))$. By taking $l^L_3(\cdot)=l^R_3(-\cdot)$, and setting $\psi^{*,3}=\max\{b,l^L_3,l^R_3\}$ on $(-\beta,g^L(-x_3))\cup(f^R(x_3),\beta)$, and $\psi^{*,3}=a$ otherwise, we have that $\psi^{*,3}$ is in fact optimal.

Theorems & Definitions (43)

• Definition 1: Superhedge, Hobson and Neuberger HobsonNeuberger:17, Hobson and Norgilas HobsonNorgilas:19
• Definition 2: Hedging cost
• Remark 1
• Lemma 1: Hobson and Norgilas HobsonNorgilas:19
• proof
• Definition 3: Superhedge generated by $\psi$
• Theorem 1
• Lemma 2: BeiglbockHobsonNorgilas:22
• Lemma 3: BeiglbockHobsonNorgilas:22
• Lemma 4