From optimal martingales to randomized dual optimal stopping

Figures & data

Figure 1. Left panel: objective functions ${\mathcal{O}}_0\left(\alpha \right)$ (no randomization), ${\mathcal{O}}_1\left(\alpha \right)$ (optimal randomization), and ${\overline{\mathcal{O}}}_1$ (‘naive’ randomization); right panel: relative deviations of ${\mathcal{Z}}_0\left(\alpha \right)$ (without randomization), ${\mathcal{Z}}_1\left(\alpha \right)$ (optimal randomization), ${\overline{\mathcal{Z}}}_1\left(\alpha \right)$ (‘naive’ randomization)

Figure 2. Cash-flow Z₁ versus continuation value C₁ as a function of W₁ for (Pa1) (left) and (Pa2) (right).

Figure 3. Objective function for BS-Call (Pa1) without randomization as function of $({\alpha }_1,{\alpha }_2)$.

Figure 4. Objective function for BS-Call (Pa1) with optimal randomization as function of $({\alpha }_1,{\alpha }_2)$.

Figure 5. Left panel: objective functions of ${\alpha }_1$ with ${\alpha }_2=1$ fixed, for BS-Call (Pa1) without, optimal, and ‘naive’ randomization; right panel: relative deviation of ${\mathcal{Z}}_0({\alpha }_1,1)$ (i.e. without randomization).

Table 1. LP minimization results due to $M^{\mathrm{s}\mathrm{t}\mathrm{y}}$ and $M^{\mathrm{H}}$ for (Pa1).

	$M^{\mathrm{s}\mathrm{t}\mathrm{y}}$			$M^{\mathrm{H}}$
(Pa1)	${\hat{\mathit{\alpha }}}_0$	${\hat{\mathit{\alpha }}}_{{\theta }^{\mathrm{n}\mathrm{a}\mathrm{i}\mathrm{v}\mathrm{e}}}$	${\hat{\mathit{\alpha }}}_1$	${\hat{\mathit{\alpha }}}_0$	${\hat{\mathit{\alpha }}}_{{\theta }^{\mathrm{n}\mathrm{a}\mathrm{i}\mathrm{v}\mathrm{e}}}$	${\hat{\mathit{\alpha }}}_1$
$\hat{m}$	0.16243	0.16399	0.16403	0.16268	0.16560	0.16696
$\hat{\sigma }/\sqrt{N}$	0.00573	0.00036	0.00029	0.00574	0.00113	0.00118
$\hat{\sigma }$	0.25639	0.01608	0.01278	0.25676	0.05063	0.05293
$m^{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}$	0.16490	0.16445	0.16442	0.16709	0.16664	0.16685
${\sigma }^{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}/\sqrt{N^{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}}$	0.00026	0.00001	0.00001	0.00026	0.00005	0.00005
${\sigma }^{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}$	0.26096	0.01460	0.01064	0.26439	0.05083	0.05153

Table 2. LP minimization results due to $M^{\mathrm{s}\mathrm{t}\mathrm{y}}$ and $M^{\mathrm{H}}$ for (Pa2).

	$M^{\mathrm{s}\mathrm{t}\mathrm{y}}$			$M^{\mathrm{H}}$
(Pa2)	${\hat{\mathit{\alpha }}}_0$	${\hat{\mathit{\alpha }}}_{{\theta }^{\mathrm{n}\mathrm{a}\mathrm{i}\mathrm{v}\mathrm{e}}}$	${\hat{\mathit{\alpha }}}_1$	${\hat{\mathit{\alpha }}}_0$	${\hat{\mathit{\alpha }}}_{{\theta }^{\mathrm{n}\mathrm{a}\mathrm{i}\mathrm{v}\mathrm{e}}}$	${\hat{\mathit{\alpha }}}_1$
$\hat{m}$	0.48748	0.49471	0.49490	0.49329	0.50082	0.50546
$\hat{\sigma }/\sqrt{N}$	0.02064	0.00201	0.00152	0.02076	0.00318	0.00308
$\hat{\sigma }$	0.92301	0.08981	0.06801	0.92852	0.14222	0.13762
$m^{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}$	0.49820	0.49639	0.49633	0.51079	0.50870	0.50912
${\sigma }^{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}/\sqrt{N^{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}}$	0.00095	0.00009	0.00007	0.00097	0.00016	0.00015
${\sigma }^{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}$	0.95415	0.09038	0.06674	0.97272	0.16047	0.15103