Perpetual American binary options. a) Compute the price [C_{b}^{mathrm{Am}}left(t, S_{t} ight)=operatorname{Sup}_{substack{tau geqslant t tau text {

Perpetual American binary options.

a) Compute the price

\[C_{b}^{\mathrm{Am}}\left(t, S_{t}\right)=\operatorname{Sup}_{\substack{\tau \geqslant t \\ \tau \text { Stopping time }}} \mathbb{E}^{*}\left[\mathrm{e}^{-(\tau-t) r} \mathbb{1}_{\left\{S_{\tau} \geqslant K\right\}} \mid S_{t}\right]\]

of the perpetual American binary call option.

b) Compute the price

\[P_{b}^{\mathrm{Am}}\left(t, S_{t}\right)=\operatorname{Sup}_{\substack{\tau \geqslant t \\ \tau \text { Stopping time }}} \mathbb{E}^{*}\left[\mathrm{e}^{-(\tau-t) r} \mathbb{1}_{\left\{S_{\tau} \leqslant K\right\}} \mid S_{t}\right]\]

of the perpetual American binary put option.