Non-zero-sum optimal stopping game with continuous versus periodic exercise opportunities

Abstract

We introduce a new non-zero-sum game of optimal stopping with asymmetric exercise opportunities. Given a stochastic process modelling the value of an asset, one player observes and can act on the process continuously, while the other player can act on it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, while the other one gets nothing. We study how each player balances the maximisation of gains against the maximisation of the likelihood of stopping before the opponent. In such a setup, driven by a Lévy process with positive jumps, we not only prove the existence, but also explicitly construct a Nash equilibrium with values of the game written in terms of the scale function. Numerical illustrations with put-option payoffs are also provided to study the behaviour of the players' strategies as well as the quantification of the value of available exercise opportunities.

Figures (7)

• Figure 1: Illustration of player $C$ and player $P$'s stopping strategies. The solid black trajectory shows the path of $X$ and the piecewise horizontal blue lines show player $P$'s most recent exercise opportunity, whose observation times are shown by dotted vertical lines. Given some $l^* > a ^*$, player $P$ stops at the first observation time of $X$ below $l^*$ (indicated by red circles) and player $C$ stops at the classical hitting time below $a^*$ (indicated by green squares). Case 1 shows the scenario when player $P$ has an exercise opportunity when $X$ is in $(a^*, l^*]$ (shown by the red strip) and thus exercises first. Case 2 shows the scenario when player $P$ does not get an exercise opportunity before $X$ enters $(-\infty, a^*]$ (shown by the blue strip) and thus player $C$ exercises first.
• Figure 2: (i) Plot of $a \mapsto I(a;l)$ on $[{\underline{x}_c-0.5}, l \wedge \overline{x}_c]$ for $l = \underline{x}_c, \ldots, \overline{x}_p$. The roots of $I(\cdot; l) = 0$ are indicated by circles and the vertical dotted lines correspond to $a = \underline{x}_c, \overline{x}_c$. (ii) Plot of $l \mapsto J(l;a)$ on $[a, \overline{x}_p]$ for $a = \underline{x}_c, \ldots, \overline{x}_c$. The roots of $J(\cdot; a) = 0$ are indicated by circles and the vertical dotted lines correspond to $l = \underline{x}_p, \overline{x}_p$.
• Figure 3: (i) Plot of $a \mapsto I(a;\widetilde{l}(a))$ on $[\underline{x}_c, \overline{x}_c]$. The root of $I(\cdot;\widetilde{l}(\cdot)) = 0$ is indicated by the circle. (ii) Plot of $l \mapsto J(l;\widetilde{a}(l))$ on $[\underline{x}_c, \overline{x}_p]$. The root of $J(\cdot;\widetilde{a}(\cdot)) = 0$ is indicated by the circle.
• Figure 4: (i) Plots of $e^x \mapsto v_c(x; a^*,l^*)$ in red, along with $e^x \mapsto v_c(x; a,l^*)$ in dotted blue for $e^a = e^{\underline{x}_c}, (e^{\underline{x}_c} + e^{a^*})/2, ( e^{a^*} + K_c)/2, K_c$. The points at $a$ and $a^*$ are indicated by circles and a star, respectively. The value at $l^*$ is indicated by the dotted vertical line. (ii) Plots of $e^x \mapsto v_p(x; a^*,l^*)$ in red, along with $v_p(x; a^*,l)$ in dotted blue for $e^l = e^{a^*}, (e^{a^*} + e^{l^*})/2, (K_p+ e^{l^*})/2, K_p$. The points at $l$ and $l^*$ are indicated by circles and a star, respectively. The value at $a^*$ is indicated by the dotted vertical line and the green line depicts the (reward) mapping $e^x\mapsto K_p-e^x$.
• Figure 5: Symmetric rewards case with $K_c=K_p = 60$. The panels in the first, second and third rows plot the same functions as those in Figure \ref{['figure_C_l_a']}, \ref{['figure_J_I_2']} and \ref{['figure_optimality']}, respectively, when $K_c = 50$.

Theorems & Definitions (41)

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